text
stringlengths 448
13k
| label
int64 0
1
| doc_idx
stringlengths 8
14
|
|---|---|---|
---
abstract: 'We report the selective stabilization of chiral rotational direction of bacterial vortices, from turbulent bacterial suspension, in achiral circular microwells sealed by an oil-water interface. This broken-symmetry, originating from the intrinsic chirality of bacterial swimming near hydrodynamically different top and bottom surfaces, generates a chiral edge current of bacteria at lateral boundary and grows stronger as bacterial density increases. We demonstrate that chiral edge current favors co-rotational configurations of interacting vortices, enhancing their ordering. The interplay between the intrinsic chirality of bacteria and the geometric properties of the boundary is a key-feature for the pairing order transition of active turbulence.'
author:
- Kazusa Beppu
- Ziane Izri
- Tasuku Sato
- Yoko Yamanishi
- Yutaka Sumino
- 'Yusuke T. Maeda'
- Kazusa Beppu
- Ziane Izri
- Tasuku Sato
- Yoko Yamanishi
- Yutaka Sumino
- 'Yusuke T. Maeda'
title: Edge Current and Pairing Order Transition in Chiral Bacterial Vortex
---
Turbulent flows offer a rich variety of structures at large length scales and are usually obtained by driving flows out of equilibrium[@zhang] while overcoming viscous dampening. A peculiar class of out-of-equilibrium fluids, stimulated from the lower scales, also present turbulence-like structures called active turbulence [@ramaswamy; @marchetti]. For | 1 | member_24 |
example, a dense bacterial suspension is driven out of equilibrium by the autonomous motion of self-propelled bacteria suspended therein[@yeomans; @frey; @kokot; @Li]. The alignment between bacteria shapes the active turbulence patterns of collective swimming into vortices of similar size[@wioland1; @wioland2; @beppu; @hamby; @clement; @nishiguchi2]. However, this vortical order decays over distance, making it a long-standing issue for the development of ordered dynamics at larger scales. Hence, a growing attention is paid to novel strategies to control active turbulence with simple geometric design.
![**Chiral bacterial vortex.** (a) Experimental setup: a dense bacterial suspension confined in hydrophilic-treated PDMS microwells and sealed under oil/water interface stabilized with a surfactant. (b) Ensemble of chiral bacterial vortices, in microwells with radius (top row) and (bottom row). Color map codes for the direction of the velocity field. Scale bar, . Schematic illustration of a bacterial vortex in a single microwell. “+” defines the positive sign of CCW rotation. CCW and CW occurrences are displayed in blue and red arrows, respectively.[]{data-label="fig1"}](figure1_9.jpg)
Chirality, i.e. the non-equivalence of opposite handedness, is ubiquitous across scales[@bahr], and is commonly involved in active systems[@glotzer; @lowen; @levis; @lenz], either biological, such as bacteria[@whitesides; @haoran; @howard; @petroff], cytoskeletons and molecular motors[@tee; @frey2; @kim], or | 1 | member_24 |
non-biological, consisting of self-propelled colloids[@jiang; @bechinger1; @irvine]. One of the effects of chirality is the non-equivalence of clockwise (CW) and counter-clockwise (CCW) rotations. As for bacteria, broken mirror-symmetry in flagellar rotation (CCW rotation around the tail-to-head direction during swimming) results in the opposite rotation of the cell body, which generates a net torque onto the solid surface the bacterium swims over, and in turn bends its trajectory circularly[@whitesides]. Despite such intrinsic chirality in individual motion, active turbulence reported in the past showed CW and CCW global rotational directions have equal probability, indicating that mirror symmetry was recovered at the collective level[@wioland2; @beppu; @nishiguchi2; @hamby]. Can microscopic chirality of bacterial motion be transferred into the macroscopic order of collective swimming? Such question is a great challenge that would provide both fundamental understanding of active turbulence and technical applications for controlled material transport[@fabrizio; @aronson].
In this Letter, we report the chiral collective swimming of a dense bacterial suspension confined in an asymmetric (different top and bottom interfaces) but achiral (perfectly circular lateral interface) hydrodynamic boundary. Non-equivalence between CW and CCW collective swimming is enhanced as the bacterial density increases, and the selected CCW rotation with respect to the bottom-top direction, later referred | 1 | member_24 |
to as “top view”, induces persistent edge current mostly observed on bacteria swimming near the lateral boundaries. Such edge current can alter the geometric constraints ruling the self-organization of bacterial vortices by suppressing the anti-rotational mode. The extended geometric rule, which is in excellent agreement with experiment, brings new understanding of chiral active matter in order to organize larger scale flow.
![**Chiral edge current.** (a) Color map of the orientation of collective motion in a chiral bacterial vortex ($R = \SI{50}{\micro\meter}$). The edge, defined as the area within from the lateral boundary, is separated from the rest of the microwell (the bulk) by a dashed white line. Scale bar, . (b) Normalized azimuthal velocity $v_{\theta}$ in microwells of various sizes. The blue circles indicate the edge current, and the green squares indicate the motion in bulk. $v_{\theta}$ is averaged over 10 s and plotted with error bars representing standard deviation.[]{data-label="fig2"}](figure2_8.jpg)
A bacterial suspension of *Escherichia coli* was confined in microwells with the depth of , made of poly-dimethyl siloxane (PDMS) rendered hydrophilic with a polyethylene glycol coating, and sealed with an oil/water interface stabilized with a surfactant (Fig. \[fig1\](a))[@izri]. Top/bottom hydrodynamic asymmetry (later referred to as “asymmetric conditions”) consists in | 1 | member_24 |
the solid bottom interface of the microwell and its top fluidic interface (Fig. S1 in [@supplement]). Bacteria in the suspension collectively move following the circular boundary of the microwell, but show a surprisingly selective CCW rotational direction (top view). The orientation map $\theta_i$ of the velocity field $\bm{v}(\bm{r}_i)$ obtained from particle image velocimetry (PIV) shows vortical structure maintained across the microwell of $R = \SI{20}{\micro\meter}$ ( Fig. \[fig1\](b)) . CCW-biased vortex (called “chiral bacterial vortex” hereafter) occurs without built-in chirality of the confinement, e.g. ratchets[@fabrizio; @aronson; @leonardo]. CCW rotation is strongly favored at 95% ($N = 145$, Fig. S3) probability in chiral bacterial vortex, while bacterial vortex rotates at equal probability in CCW or CW direction in water-in-oil droplets between *symmetric solid (top)/solid (bottom)* interfaces (symmetric conditions, Fig. S2). Flow reversal in vortex was not observed during our observations (Fig. S3) which emphasizes that chiral bacterial vortex is much more stable than the bacterial vortices in droplets[@hamby](Fig. S3). Chiral vortical structure is also observed in larger microwells ($R = \SI{35}{\micro\meter}$) but only within a distance of away from the circular boundary. This chiral motion with a coherent orientation near the lateral boundary, called edge current, is known to be a | 1 | member_24 |
key feature in chiral many-body systems[@jiang; @bechinger1; @irvine]. We in turn examined the size-dependence of chiral vortices and the edge current in order to investigate the mechanism of such stability and selectivity in rotational direction. We define the tangential vector $\bm{t}(\theta_i)$ in CCW direction along the circular boundary and the azimuthal velocity $v_{\theta}(\bm{r}_i)=\bm{v}(\bm{r}_i)\cdot \bm{t}_i$. The orientation of the edge current along the boundary wall is analyzed by $\langle v_{\theta}(\bm{r}_i)/ |\bm{v}| \rangle$ where $\langle \cdot \rangle$ denotes the average over all possible site $i$ (Fig. \[fig2\](a)). Surprisingly, this edge current was maintained even in very large microwells ($R\geq$), the size of which is much larger than the critical size of a stable bacterial vortex in the bulk ($\approx\SI{35}{\micro\meter}$) (Fig. \[fig2\](b)).
This persistence of edge current motivates us to further investigate its physical origin, by analyzing the interplay between the boundary and the intrinsic chirality of bacteria, which is the only element having chirality in the present system. With respect to the tail-to-head direction, the flagella of the bacterium rotate in the CCW direction. Torque balance then imposes a CW rotation of the body. Those two opposite rotations result in opposite frictions against the bottom interface, which ultimately converts into a CCW | 1 | member_24 |
rotation (top view) of bacteria swimming near the top interface (CW rotation near bottom interface) (Fig. \[fig3\](a))[@whitesides]. Because of this CCW bias of their trajectories beneath the top, bacteria that collide with the lateral boundaries align with it and swim in a CCW rotation direction (CW rotation direction on the bottom). However, this effect alone does not determine net chirality because bacteria swim in opposite directions on the top and bottom interfaces of the microwell. In order to find the origin of net chirality in edge current, we recorded the trajectories of individual bacteria in dilute ($0.2\% v/v$) suspensions, such that interactions between bacteria do not affect their swimming. Fig. \[fig3\](b) presents the probability distribution function of the azimuthal velocities of individual bacteria, $P(v_{\theta})$, in a microwell ($R = \SI{35}{\micro\meter}$). Individual bacterial motion beneath the top fluidic interface shows visible nonequivalence between CW and CCW swimming, as more than half ($71\%$) of the tracked bacteria swim in CCW direction. By contrast, $58\%$ of individual bacteria onto the bottom solid interface swim in CW rotational direction, indicating weaker chiral bias. To compare those swimming chiralities, we define the chirality index $CI(v_{\theta})$ as antisymmetric part of $P(v_{\theta})$, i.e. $$CI(v_{\theta}) = P(v_{\theta}) - | 1 | member_24 |
P(-v_{\theta}).$$ Positive chirality index means CCW rotation is dominant, and the larger the absolute value of the index, the more biased the rotational direction. For bacteria swimming near the top interface, chirality index is positive, while it is negative near the bottom interface (Fig. \[fig3\](c)). This tells that bacterial motion is CCW-biased near the top interface, and CW near the bottom interface. In addition, near the top interface, $|CI(v_{\theta})|>0.05$ whereas near the bottom interface $|CI(v_{\theta})| < 0.02$. This indicates that bacterial swimming is more biased near the top (fluidic) interface than near the bottom (solid) interface. This difference in the amplitude of the bias at each interface is responsible for the chiral rotation of the whole bacterial vortex, and therefore is at the origin of chiral edge current (Fig. \[fig3\](d)).
![**Bacterial swimming in dilute ($0.2\% v/v$) suspension** (a) Schematic illustrations of chiral bacterial swimming near the top fluidic interface (left) and near the bottom solid interface (right). Near the top interface, individual bacterial swimming is CCW-biased (blue curved arrow) while near the bottom interface it is CW-biased (see Fig. S5). (b) Histograms of azimuthal velocity and their average (vertical dashed line) of single bacteria swimming near top interface (left, schematic | 1 | member_24 |
illustration in insert, $N=5004$, average ) and bottom interface (right, schematic illustration in insert, $N=3702$, average ). Proportions of CCW and CW occurrences are displayed in blue and red at the top of each plot. (c) Corresponding chirality indices plotted against azimuthal velocity, near the top (blue) and bottom (red) interfaces. (d) Chiral bias of swimming near the top and bottom interfaces have opposite signs but different amplitudes, under asymmetric conditions.[]{data-label="fig3"}](figure3_7.jpg)
![**Chiral bacterial vortex is collective effect.** A dense ($20\% v/v$) bacterial suspension is confined under asymmetric conditions. Represented chirality indices against azimuthal velocity of (a) individual fluorescent bacteria (schematic illustration in insert) observed under confocal microscopy near the top (left) and bottom (right) interface, (b) fluid flow with tracer particles also observed under confocal microscopy, and (c) collective bacterial motion observed under bright field. Two microwell sizes were considered: $R = \SI{35}{\micro\meter}$ (small blue circles) and $R = \SI{50}{\micro\meter}$. In the larger microwells were considered two regions: the boundary layer within from the lateral boundary (larger blue diamonds), and the bulk that is more than away from the lateral wall (green disks). Sample size and average azimuthal velocity of each case can be found in Fig. S6.[]{data-label="fig4"}](figure4_7.jpg)
As the | 1 | member_24 |
density of self-propelled particles increases, their mutual interactions affect more significantly their global motion, which ultimately gives rise to collective behavior. As implied already, the rotational biases of handedness of individual bacteria ($71\%$ in top and $58\%$ in bottom) are too weak to account for the 95%-selective chirality in the bacterial vortex (Fig. \[fig1\](c)). To resolve this gap, we therefore characterized the structure of chiral bacterial vortices in much denser ($20\% v/v$) suspensions. Fig. \[fig4\](a) presents chirality index of individual bacteria near the top and bottom interfaces in a microwell ($R = \SI{35}{\micro\meter}$). Interestingly, both interfaces present dominantly CCW rotational direction, although the top interface is more strongly biased. This indicates that the effect of interactions between bacteria forces the rotational direction to be the same across the microwell. Intriguingly, chirality index of individual motion becomes larger as the bacterial density increases. Trajectories of tracer particles were also predominantly CCW, indicating that fluid flow also has CCW handedness. Moreover, chirality index of the fluid velocity (Fig. \[fig4\](b)) and the collective velocity (Fig. \[fig4\](c)) are comparable to that of individual motion. This suggests that a slight chiral bias in individual bacteria is amplified by collective bacterial interactions, which turns into a | 1 | member_24 |
global vortex with a unidirectional rotation.
When we analyzed the vortex flow near the lateral boundary of larger microwells ($R = \SI{50}{\micro\meter}$), individual motion at high density has a large positive chirality index, similarly to smaller microwells. However, near the center of larger microwells, chirality is null for individual and collective bacterial motion as well as for the fluid flow (Fig. \[fig4\](a) to (c)). The necessity to be close to the lateral boundaries to observe coherent chiral swimming emphasizes the importance of spatial constraint to amplify the rotational bias and turn it into chiral edge current.
![**Edge current favors co-rotational vortex pairing** (a) Schematic illustration and definition of relevant geometric parameters. (b) Illustration of the co-rotational vortex pairing (FMV pattern, left) and the anti-rotational pairing (AFMV pattern, right) with edge current. The edge current deviates the orientation angle of bacteria $\theta$ around the *tip*. In FMV pattern, bi-particle alignment and edge current deviate bacteria in the same direction, but in AFMV pattern, those effects compete. (c) Vorticity map of bacterial vortex pairs at various values of $\Delta$ with $R = \SI{19}{\micro\meter}$. (d) Order parameter $\Phi_{FMV}$ of FMV pairs of interacting bacterial vortices, against $\Delta/R$. Under no edge current (full grey | 1 | member_24 |
circles), FMV to AFMV transition occurs at $\Delta/R\simeq\sqrt{2}$, while under CCW edge current (inverted black triangles) it occurs at a larger value of $\Delta/R\simeq1.9$.[]{data-label="fig5"}](figure5_8.jpg)
The edge current plays a crucial role in the ordering of interacting bacterial vortices. When two bacterial vortices interact with one another via near-field interaction, they have either the same rotational direction (defined as ferromagnetic vortices, FMV) or opposite rotational directions (anti-ferromagnetic vortices, AFMV) in a geometry-dependent manner[@wioland2; @beppu; @nishiguchi2]. To reveal how a chiral edge current affects the ordering of interacting bacterial vortices, we construct a theoretical model of interacting bacterial vortices in doublets of overlapping identical circular boundaries (Fig. \[fig5\](a))[@beppu].
Two identical overlapping circular microwells, with a radius $R$ and an inter-center $\Delta$, offer the means for a systematic investigation of the pairing order transition[@beppu]. The ratio $\Delta/R = 2\cos\Psi$ is an important geometric parameter characterizing the pairing order transition from FMV to AFMV: if $\Delta$/R is small enough, the two vortices align in co-rotational direction and pair into a FMV pattern. Transition from FMV to AFMV occurs at a predicted value $\Delta_c/R = 2 \cos \SI{45}{\degree} = \sqrt{2}$ because that is the only configuration at which the two pairing patterns are equiprobable[@beppu]. How | 1 | member_24 |
does edge current affect the previously established design principle?
To answer this question, the orientation dynamics of bacteria with the heading angle $\theta$ is considered at the vicinity of the sharp areas of the boundaries (“*tip*”). Given the CCW fluid flow near boundary reorients the bacteria at the *tip*, the effective torque for reorientation $\bm{\tau_e} = - \partial U_e/\partial \theta$ has geometry-dependent potential $U_e = -2 \gamma_e \sin\theta\cos\Psi$ with $\gamma_e$ representing the ratio of fluid flow to bacterial swimming ($\gamma_e > 0$). This reorientation maintains the CCW rotation of the edge current along boundary in both FMV pairing (Fig. \[fig5\](b), left) and AFMV pairing (Fig. \[fig5\](b), right). In addition, the result of the bi-particular collision near the *tip* between bacteria coming from different circular parts[@beppu] also affects the vortex pairing. Bacterial collision is ruled by a polar alignment as a source of geometry-dependence[@vicsek; @supplement]. By considering the most probable configurations (general case detailed in [@supplement]), the orientation near the *tip* is decided by the potential of FMV pairing $U_p^{FMV} = 2\gamma_p\sin\Psi$ at $\theta = 0$ (Fig. \[fig5\](b), left) or of AFMV pairing $U_p^{AFMV} = 2\gamma_p\cos\Psi$ at $\theta = \pi/2$ (Fig. \[fig5\](b), right), with the strength of the polar alignment $\gamma_p>0$. | 1 | member_24 |
The respective sums of the potentials coincide at the transition point, i.e. $U_p^{FMV} + U_e|_{\theta=0} = U_p^{AFMV} + U_e|_{\theta=\pi/2}$, which leads to $$\gamma_p \sin\Psi_c = (\gamma_p- \gamma_e) \cos\Psi_c,$$ where $\gamma_p- \gamma_e$ indicates the suppression of AFMV pairing by chiral edge current. Hence, $\Delta_c/R=2\cos\Psi_c$ at which FMV and AFMV pairings occur at equal probability is $$\label{chiral_geometric rule}
\frac{\Delta_c}{R} = \frac{2}{\sqrt{1+(1-\gamma_e/\gamma_p)^2}}\simeq \sqrt{2} \Bigl( 1 + \frac{\gamma_e}{2\gamma_p}\Bigr).$$ FMV pattern is stabilized in $\Delta/R \geq \sqrt{2}$ and the relative strength of the edge current $\gamma_e/\gamma_p$ determines the shift of the transition point.
To test those chirality effects, we examined the pairing of doublets of chiral vortices with $R = \SI{19}{\micro\meter}$ within the range of $0 \leq \Delta < 2R = \SI{38}{\micro\meter}$. FMV pattern of chiral vortices is dominant in $0 \leq \Delta/R \leq 1.9$ and exhibits CCW rotation (Fig. \[fig5\](c)). The pairing order transition is also analyzed by using the order parameter of FMV pairing $\Phi_{FMV}$ [@supplement]. $\Phi_{FMV}$, which reaches 1 for FMV while it goes down to 0 for AFMV, is defined as $\vert \langle \bm{p}_i \cdot \bm{u}_i \rangle \vert$ with the orientation of velocity $\bm{p}_i$ measured experimentally at site $i$ and the expected orientation of FMV pattern $\bm{u}_i$ calculated numerically at corresponding | 1 | member_24 |
site. Under asymmetric condition, $\Phi_{FMV}$ shows a transition from $1$ to $0$ at $\Delta/R\simeq1.9$, while it occurs at $\Delta/R\simeq1.4$ under symmetric condition (absence of chiral edge current) (Fig. \[fig5\](d)). FMV pairing pattern appears to be favored in the presence of edge current. Moreover, according to Eq. with the experimental values $\gamma_p=0.5$ (obtained from independent experiment, Fig. S7) and $\Delta_c/R\simeq1.9$, the coefficient of edge current is $\gamma_e=0.3$ that is reasonably comparable to the ratio of fluid flow to bacterial swimming velocity (Figs.\[fig4\](a) and 4(b)). The excellent agreement with experiment means that chiral edge current affects the pairing order transition.
In conclusion, we revealed that confining a dense bacterial suspension in microwells with asymmetric top/bottom interfaces but achiral circular lateral boundaries stabilizes a chiral vortex. Hydrodynamically different top/bottom interfaces give then rise to a subtle broken symmetry in bacterial swimming, that is amplified by collective bacterial motion and becomes an edge current persistent over larger length scale. We showed that it is possible to generate and control the break of mirror-symmetry without using built-in chirality, unlike most current experimental setup, e.g. with built-in chiral ratchet-shape[@fabrizio; @aronson; @leonardo]. Moreover, in present chiral bacterial vortex, rigid rod much longer than bacterial body can consistently | 1 | member_24 |
rotate in CCW direction over multiple rounds at (Fig. S8). Thus, asymmetric hydrodynamic boundary offers simple and fast material transport, even without built-in chirality. Finally, the edge current favors co-rotational FMV pairing pattern of doublets of identical vortices. Even in triplets of identical overlapping circular microwells, the pairing order transition from FMV to (frustrated) AFMV patterns is also shifted to higher values (from $1.7$ to $1.9$) (Fig. S9), suggesting that chiral bacterial vortex has fewer limitation from geometric frustration. Although beyond the scope of this study, understanding how the nature of an interface affects the amplitude of the chiral flow it generates could give the means of a finer tuning of the global rotation of confined bacterial vortices.
The finding of edge current in chiral bacterial vortex opens new directions for the tailoring of collective motion. Such asymmetric hydrodynamic boundaries are also involved in flocking and lane formation of active droplets[@stone]. The edge current observed in active droplets with similar collective effect may advance the generic understanding of chiral collective behavior[@glotzer; @tjhung]. Furthermore, stabilized co-rotational pairing order is a key to clarify how broad class of active matter amplifies microscopic chirality. As such constrained pairing order is also relevant to | 1 | member_24 |
chiral spinners[@tierno], controlling chirality-induced order with simple geometric rule would verify the validity of geometric approach to chiral many-body systems.
This work was supported by Grant-in-Aid for Scientific Research on Innovative Areas (JP18H05427 and JP19H05403) and Grant-in-Aid for Scientific Research (B) JP17KT0025 from MEXT.
[45]{} J. Zhang and A. Libchaber Phys. Rev. Lett. **84**, 4361 (2000). S. Ramaswamy, Annu. Rev. Condens. Matt. Phys. **1**, 323 (2010). M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. **85**, 1143 (2013). H.H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. Löwen, and J. M. Yeomans. Proc. Natl. Acad. Sci. U.S.A. **109**, 14308-14313 (2012). V. Bratanov, F. Jenko, and E. Frey, Proc. Natl. Acad. Sci. USA 112, 15048-15053 (2015). G. Kokot, S. Das, R.G. Winkler, G. Gompper, I.S. Aranson, and A. Snezhko, Proc. Natl. Acad. Sci. U.S.A. **114**, 12870-12875 (2017). H. Li, et al., Proc. Natl. Acad. Sci. U.S.A. **116**, 777-785 (2019). H. Wioland, F. G. Woodhouse, J. Dunkel, J. O. Kessler, and R. E. Goldstein, Phys. Rev. Lett. **110**, 268102 (2013). H. Wioland, F. G. Woodhouse, J. Dunkel, and R. E. Goldstein, Nat. Phys. **12**, 341-345 (2016). | 1 | member_24 |
K. Beppu, Z. Izri, J. Gohya, K. Eto, M. Ichikawa, and Y. T. Maeda, Soft Matt. **13**, 5038-5043 (2017). A.E. Hamby, D.K. Vig, S. Safonova, and C.W. Wolgemuth, Sci. Adv. **4**, eaau0125 (2018). B. Vincenti, C. Ramos, M.L. Cordero, C. Douarche, R. Soto, and E. Clement, Nat. Commun. **10**, 5082 (2019). D. Nishiguchi, I. Aranson, A. Snezhko and A. Sokolov, Nat. Commun. **9**, 4486 (2018). H-S Kitzerow and C. Bahr, Chirality in Liquid Crystal, Springer, New York (2001). N.H. Nguyen, D. Klotsa, M. Engel, and S.C. Glotzer, Phys. Rev. Lett. **112**, 075701 (2014). H. Löwen, Eur. Phys. J. **225**, 2319-2331 (2016). B. Liebchen, and D. Levis. Phys. Rev. Lett. **119**, 058002 (2017). A. Maitra and M. Lenz, Nat. Commun. **10**, 920 (2019). W.R. DiLuzio, et al. Nature **435**, 1271-1274 (2005). H. Xu, J. Dauparas, D. Das, E. Lauga and Y. Wu, Nat. Commun. **10**, 1792 (2019). I. H. Riedel, K. Kruse, and J. A. Howard, Science **309**, 300-303 (2005). A.P. Petroff, X-L. Wu, and A. Libchaber, Phys. Rev. Lett. **114**, 158102 (2015). Y.H. Tee, et al. Nat. Cell Biol. **17**, 445-457 (2015). J. Denk, L. Huber, E. Reithmann, and E. Frey, Phys. Rev. Lett. 116, 178301 (2016). K. Kim, et | 1 | member_24 |
al. Soft Matt. **14**, 3221-3231 (2018). H. Jiang, H. Ding, M. Pu and Z. Hou, Soft Matt. **13**, 836-841 (2017). N. Narinder, C. Bechinger, and J.R. Gomez-Solano, Phys. Rev. Lett. **121**, 078003 (2018). V. Soni, E.S. Bililign, S. Magkiriadou, S. Sacanna, D. Bartolo, M.J. Shelley and W.T.M. Irvine, Nat. Phys. **15**, 1188-1194 (2019). R. Di Leonardo, et al. Proc. Natl. Acad. Sci. U.S.A. **107**, 9541-9545 (2010). A. Sokolov, M.M. Apodaca, B.A. Grzybowski, and I.S. Aranson, Proc. Natl. Acad. Sci. U.S.A. **107**, 969-974 (2010). Z. Izri, D. Garenne, V. Noireaux, and Y.T. Maeda, ACS Synth. Biol. **8**, 1705-1712 (2019). See supplemental material for details of experimental methods, full theoretical details. N. Koumakis, A. Lepore, C. Maggi and R. Di Leonardo, Nat. Commun. **4**, 2588 (2013). T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, and O. Shochet, Phys. Rev. Lett. **75**, 1226 (1995). A. Ortiz-Ambriz, C. Nisoli, C. Reichhardt, C.J.O. Reichhardt, and P. Tierno, Rev. Mod. Phys. **91**, 041003 (2019). S. Thutupalli, D. Geyer, R. Singh, R. Adhikari, and H.A. Stone, Proc. Natl. Acad. Sci. U.S.A. **115**, 5403-5408 (2018). E. Tjhung, M.E. Cates, and D. Marenduzzo, Proc. Natl. Acad. Sci. U.S.A. **114**, 4631-4636 (2017).
$ $
Supplemental information for\
Edge Current and | 1 | member_24 |
Pairing Order Transition in Chiral Bacterial Vortex {#supplemental-information-for-edge-current-and-pairing-order-transition-in-chiral-bacterial-vortex .unnumbered}
====================================================================
Materials and Methods
=====================
Device microfabrication
-----------------------
The device used in this experiment is a flow cell that contains an array of microwells of various radii $R$ ($\SI{20}{\micro\meter} \leq R \leq \SI{150}{\micro\meter}$) and three types of shape: single circular microwells, pairs and triplets of overlapping identical circular microwells. Their depth is in all the experiments[@beppu]. The flow cell is made of a cover glass slide (Matsunami, S1127, ) and a cover slip (Matsunami, C218181, ) separated by a double-sided adhesive tape (NICHI-BAN, NW-10, ).
The microwells are fabricated using standard soft lithography techniques. Briefly, PDMS polymer and curing agent (Dow Corning, 98-0898, Sylgard184) at 90-to-10 mass ratio was spin-coated at 1000 rpm for 30 seconds to reach a thickness of , on a mold made of a photoresist (SU-8 3025, Microchem) pattern (thickness ) cured and developed through conventional photo-lithography on a silicon wafer (Matsuzaki, Ltd., $\phi$2-inch wafer). After a curing at for an hour, the PDMS film is cut around a single array of microwells and peeled off to be bonded on its unpatterned surface to the glass cover slide that has been exposed to air plasma beforehand (, | 1 | member_24 |
corona discharge gun, Shinko Denso).
This assemblage is then once again exposed to air plasma (, corona discharge gun), covered with a solution of polyethylene glycol-poly-L-lysine (PEG-PLL, Nanocs, PG2k-PLY), and left to rest for 30 minutes. Ungrafted PEG-PLL is then washed away with deionized water. PDMS microwells and glass cover slide are now treated hydrophilic, and PEG-PLL coating prevents non-specific adhesion of bacteria. Finally, the flow cell is completed with the glass cover slip (untreated) being attached to the glass cover slide with the adhesive spacer placed over the unpatterned areas of the PDMS sheet. It is used immediately after fabrication (Fig. \[fig.s1\] Step 1).
![**Schematic illustration of device fabrication.** Step 1: device construction. A thin PDMS layer patterned with microwells is transferred between the two glass slides of a flow cell and coated with PEG-PLL to avoid unspecific adhesion of bacteria. Step 2: bacterial suspension is injected from one side of the flow cell. Step 3: excess bacterial suspension is flushed out by a mixture of oil and surfactant. This mixture seals the microwells on their top side. Step 4: device is sealed at both ends with epoxy glue.[]{data-label="fig.s1"}](figureS1_3.jpg)
Filling protocol
----------------
Once a flow cell is ready, it | 1 | member_24 |
is filled with of a bacterial suspension (two volume fractions were available: dense at $20\% v/v$, dilute at $0.2\% v/v$). The PEG-PLL coating of the PDMS allows the bacterial suspension to reach the inside of the microwell and fill them properly. Once the flow cell is completely filled, oil (light mineral oil, Sigma Aldrich) with surfactant (SPAN80, Nacalai) at $2 wt\%$ is injected from the same side, while excess bacterial suspension over the microwells is flushed out and absorbed with filter paper from the other side of the flow cell. This seals the microwells under an oil/water interface. To suppress unwanted flow in the flow cell, both of its ends are sealed with epoxy glue (Huntsmann, Ltd.). The array of microwells is then ready to be observed under microscope (Fig. \[fig.s1\] Steps 2-4)[@izri].
Persistently straight-swimming bacterial strain
-----------------------------------------------
We used bacterial strain *Escherichia coli* RP4979 that lacks tumbling ability. These bacteria swim smoothly without tumbling and show persistent straight motion in the bulk of a fluid. Highly motile bacteria were obtained by inoculating a single bacterial colony into of LB medium (NaCl , yeast extract , tryptone , pH7.2) and incubating it overnight at . The next day, of this | 1 | member_24 |
overnight culture solution is transferred to of T-Broth (NaCl , trypton , pH7.2), and the inoculated cultures are incubated at and agitated at 150 rpm for about 6 hours. After reaching an optical density of $0.4$, the culture medium is centrifuged at 3000 rpm at room temperature for 10 minutes to concentrate the bacterial suspension density to $20-25\% v/v$.
The straight swimming of bacteria was measured in a bulk fluid, and the fluctuation of the heading angle $\theta(t)$ was analyzed. Single bacteria are considered as self-propelled points particle, with a position $\bm{r}(t) = (\bm{x}(t), \bm{y}(t))$ and an orientation $\bm{d}(t)$(Fig.\[figs\_diffusion\](a)). For two-dimensional coordinates, the orientation of single bacteria is expressed by the unit vector along the long-axis of bacteria $\bm{d} = \bm{d}(\theta(t)) = (\cos\theta(t) , \sin\theta(t))$. Because the mean-square angle displacement (MSD) is given by $\langle \bigl[\bm{d}(\theta(t)) - \bm{d}(\theta(0))\bigr]^2 \rangle_t = 2(1-\exp[-D t])$ with time interval $\delta t$ and the angular diffusion coefficient $D$ that reflects the fluctuation of bacterial orientation at single cell level[@jain](Fig.\[figs\_diffusion\](b)). The obtained coefficient $D$ is 0.12 rad$^2$/sec, indicating that RP4979 bacteria persistently swim in one direction at a low density in bulk.
![**Straight swimming of single bacteria in bulk.** (a) Typical trajectory of swimming single bacteria | 1 | member_24 |
in bulk. Scale bar is . (b) Mean square displacement of heading angle of single bacteria is plotted with time. The slope of this MSD curve is fitted with $2(1-\exp[-Dt])$, where $D$ reflects orientation fluctuation of smoothly swimming bacteria[@jain].[]{data-label="figs_diffusion"}](figureS_fluctuation.jpg)
Bacterial density measurement and image velocimetry
---------------------------------------------------
To measure bacterial density, we used a mixture - 99-to-1 ratio - of two genetically modified bacteria (strain RP4979) that constitutively express fluorescent protein (either YFP or dTomato). That fluorescent labeling allows the quantitative recording of the trajectories of individual bacterial swimming in either dilute (Fig. 3 in main text) or dense suspensions (Fig. 4(a) in main text). By tracking dTomato-expressing bacteria, we can record the individual trajectories of bacteria inside collective vortical motion.
In the analysis of single bacteria and tracer particles in a suspension, they were tracked by means of a plugin of Particle Tracker 2D/3D in the Fiji(ImageJ) software. Bright-field optical imaging and video-microscopy were performed by using an inverted microscope (IX73, Olympus) with a CCD camera (DMK23G445, Imaging Source) that enables us to record bacterial collective motion at 30 frames per second. The velocity field of bacterial collective motion $\bm{v}(r,t)$ was analyzed by Particle Image Velocimetry(PIV) with Wiener filter method | 1 | member_24 |
using PIVlab based on MATLAB software, and its grid size was $\times$. Acquired velocity fields were further smoothed by averaging over 1 sec.
In addition, polystyrene tracer particles with red fluorescence (Molecular probes) were used to track the flow field (Fig. 4(b) in main text). The tracer particles were dispersed at a low density of $0.026\% v/v$, where individual particles could be tracked inside the bacterial suspension. Recording of the trajectories of red-labeled bacteria and red tracer particles was done with a confocal microscope (IX73 inverted microscope from Olympus, and confocal scanning unit CSU-X1 from Yokogawa Electric Cor. Ltd., iXon-Ultra EM-CCD camera from Andor Technologies) under red fluorescence channel. All the recordings were done at 30 frames per second.
Experimental details
====================
Preponderance and stability of CCW bacterial vortex
---------------------------------------------------
![**Preponderance and stability of chiral bacterial vortex.** (a) A droplet of a dense bacterial suspension confined in a microwell ($R = \SI{20}{\micro\meter}$) between *asymmetric fluidic (top) and solid (bottom)* interfaces. (b) Control experiment for the effect of top/bottom interfaces on the directionality of the bacterial vortex. A droplets ($\SI{15}{\micro\meter}\leq R\leq\SI{35}{\micro\meter}$) of an emulsion of dense bacterial suspension in oil between *symmetric solid (top)/solid (bottom)* interfaces. (c) Distributions of vorticity averaged | 1 | member_24 |
---
abstract: |
We consider a Riemannian cylinder $\Omega$ endowed with a closed potential $1$-form $A$ and study the magnetic Laplacian $\Delta_A$ with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.
*2000 Mathematics Subject Classification. 58J50, 35P15.*
*Key words and phrases. Magnetic Laplacian, Eigenvalues, Upper and lower bounds, Zero magnetic field*
author:
- Bruno Colbois and Alessandro Savo
bibliography:
- 'CS.bib'
title: 'Lower bounds for the first eigenvalue of the magnetic Laplacian\'
---
Introduction {#intro}
============
Let $(\Omega,g)$ be a compact Riemannian manifold with boundary. Consider the trivial complex line bundle $\Omega\times\bf C$ over $\Omega$; its space of sections can be identified with $C^{\infty}(\Omega,\bf C)$, the space of smooth complex valued functions on $\Omega$. Given a smooth real 1-form $A$ on $\Omega$ we define a connection $\nabla^A$ on $C^{\infty}(\Omega,\bf C)$ as follows: $$\label{connection}
\nabla^A_Xu=\nabla_Xu-iA(X)u$$ for all vector fields $X$ | 1 | member_25 |
on $\Omega$ and for all $u\in C^{\infty}(\Omega,\bf C)$; here $\nabla$ is the Levi-Civita connection assocated to the metric $g$ of $\Omega$. The operator $$\label{magnetic laplacian}
\Delta_A=(\nabla^A)^{\star}\nabla ^A$$ is called the [*magnetic Laplacian*]{} associated to the magnetic potential $A$, and the smooth two form $$B=dA$$ is the associated [*magnetic field*]{}. We will consider Neumann magnetic conditions, that is: $$\label{mneumann}
\nabla^A_Nu=0\quad\text{on}\quad{\partial}\Omega,$$ where $N$ denotes the inner unit normal. Then, it is well-known that $\Delta_A$ is self-adjoint, and admits a discrete spectrum $$0\le {\lambda}_1(\Delta_A)\le {\lambda}_2(\Delta_A) \le ... \to \infty.$$
The above is a particular case of a more general situation, where $E\to M$ is a complex line bundle with a hermitian connection $\nabla^E$, and where the magnetic Laplacian is defined as $\Delta_E=(\nabla^E)^{\star}\nabla ^E$.
The spectrum of the magnetic Laplacian is very much studied in analysis (see for example [@BDP] and the references therein) and in relation with physics. For *Dirichlet boundary conditions*, lower estimates of its fundamental tone have been worked out, in particular, when $\Omega$ is a planar domain and $B$ is the constant magnetic field; that is, when the function $\star B$ is constant on $\Omega$ (see for example a Faber-Krahn type inequality in [@Er1] and the recent[@LS] and the references | 1 | member_25 |
therein, also for Neumann boundary condition). The case when the potential $A$ is a closed $1$-form is particularly interesting from the physical point of view (Aharonov-Bohm effect), and also from the geometric point of view. For Dirichlet boundary conditions, there is a serie of papers for domains with a pole, when the pole approaches the boundary (see [@AFNN; @NNT] and the references therein). Last but not least, there is a Aharonov-Bohm approach to the question of nodal and minimal partitions, see chapter 8 of [@BH].
For *Neumann boundary conditions*, we refer in particular to the paper [@HHHO], where the authors study the multiplicity and the nodal sets corresponding to the ground state ${\lambda}_1$ for non-simply connected planar domains with harmonic potential (see the discussion below).
Let us also mention the recent article [@LLPP] (chapter 7) where the authors establish a *Cheeger type inequality* for ${\lambda}_1$; that is, they find a lower bound for ${\lambda}_1(\Delta_A)$ in terms of the geometry of $\Omega$ and the potential $A$. In the preprint [@ELMP], the authors approach the problem via the Bochner method.
Finally, in a more general context (see [@BBC]) the authors establish a lower bound for ${\lambda}_1(\Delta_A)$ in terms of the *holonomy* of | 1 | member_25 |
the vector bundle on which $\Delta_A$ acts. In both cases, implicitly, the flux of the potential $A$ plays a crucial role.
[$\bullet\quad$]{}From now on we will denote by $\lambda_1(\Omega,A)$ the first eigenvalue of $\Delta_A$ on $(\Omega,g)$.
Main lower bound
----------------
Our lower bound is partly inspired by the results in [@HHHO] for plane domains. First, recall that if $c$ is a closed parametrized curve (a loop), the quantity: $$\Phi^A_c=\dfrac{1}{2\pi}\oint_{c}A$$ is called the [*flux*]{} of $A$ across $c$. (We assume that $c$ is travelled once, and we will not specify the orientation of the loop, so that the flux will only be defined up to sign: this will not affect any of the statements, definitions or results which we will prove in this paper). Let then $\Omega$ be a fixed plane domain with one hole, and let $\Phi^A$ be the flux of the harmonic potential $A$ across the inner boundary curve. In Theorem 1.1 of [@HHHO] it is first remarked that $\lambda_1(\Omega, A)$ is positive if and only if $\Phi^A$ is not an integer (but see the precise statement in Section 2.1 below). Then, it is shown that $\lambda_1(\Omega,A)$ is maximal precisely when $\Phi^A$ is congruent to $\frac 12$ modulo integers. | 1 | member_25 |
The proof relies on a delicate argument involving the nodal line of a first eigenfunction; in particular, the conclusion does not follow from a specific comparison argument, or from an explicit lower bound.
In this paper we give a geometric lower bound of $\lambda_1(\Omega,A)$ when $\Omega$ is, more generally, a [*Riemannian cylinder*]{}, that is, a domain $(\Omega,g)$ diffeomorphic to $[0,1]\times{{\bf S}^{1}}$ endowed with a Riemannian metric $g$, and when $A$ is a closed potential $1$-form : hence, the magnetic field $B$ associated to $A$ is equal to $0$. The lower bound will depend on the geometry of $\Omega$ and, in an explicit way, on the flux of the potential $A$.
Let us write $\partial\Omega=\Sigma_1\cup\Sigma_2$ where $$\Sigma_1=\{0\}\times{{\bf S}^{1}}, \quad \Sigma_2=\{1\}\times{{\bf S}^{1}}.$$ We will need to foliate the cylinder by the (regular) level curves of a smooth function $\psi$ and then we introduce the following family of functions.
$$\begin{aligned}{\cal F}_{\Omega}=\{\psi:\Omega\to{{\bf R}}: \quad &\text{\it $\psi$ is constant on each boundary component}\\
&\text{\it and has no critical points inside $\Omega$}\}
\end{aligned}$$
As $\Omega$ is a cylinder, we see that ${\cal F}_{\Omega}$ is not empty. If $\psi\in{\cal F}_{\Omega}$, we set: $$K=K_{\Omega,\psi}=\dfrac{\sup_{\Omega}{\lvert{\nabla\psi}\rvert}}{\inf_{\Omega}{\lvert{\nabla\psi}\rvert}}.$$ It is clear that, in the definition of the constant $K$, we can assume | 1 | member_25 |
that the range of $\psi$ is the interval $[0,1]$, and that $\psi=0$ on $\Sigma_1$ and $\psi=1$ on $\Sigma_2$. Note that the level curves of the function $\psi$ are all smooth, closed and connected; moreover they are all homotopic to each other so that the flux of a closed $1$-form $A$ across any of them is the same, and will be denoted by $\Phi^A$.
We say, briefly, that $\Omega$ is [*$K$-foliated by the level curves of $\psi$.*]{} We also denote by $d(\Phi^A,{\bf Z})$ the minimal distance between $\Phi^A$ and the set of integer $\bf Z$: $$d(\Phi^A,{\bf Z})^2=\min\Big\{(\Phi^A-k)^2: k\in\bf Z\Big\}.$$
Finally, we say that [*$\Omega$ is a Riemannian product*]{} if it is isometric to $[0,a]\times{{\bf S}^{1}}(R)$ for suitable positive constants $a,R$.
\[main3\]
a\) Let $(\Omega,g)$ be a Riemannian cylinder, and let $A$ be a closed $1$-form on $\Omega$. Assume that $\Omega$ is $K$-foliated by the level curves of the smooth function $\psi\in{\cal F}_{\Omega}$. Then: $$\label{cylinder}
\lambda_1(\Omega,A)\geq\dfrac{4\pi^2}{KL^2}\cdot d(\Phi^A,{\bf Z})^2,$$ where $L$ is the maximum length of a level curve of $\psi$ and $\Phi^A$ is the flux of $A$ across any of the boundary components of $\Omega$.
b\) Equality holds if and only if the cylinder $\Omega$ is a Riemannian product.
[$\bullet\quad$]{}It is clear | 1 | member_25 |
that we can also state the lower bound as follows: $$\lambda_1(\Omega,A)\geq\dfrac{4\pi^2}{\tilde K_{\Omega}}\cdot d(\Phi^A,{\bf Z})^2,$$ where $\tilde K_{\Omega}$ is an invariant depending only $\Omega$: $$\tilde K_{\Omega}=\inf_{\psi\in{\cal F}_{\Omega}}K_{\Omega,\psi}L_{\psi}^2\quad\text{and}\quad L_{\psi}=\sup_{r\in {\rm range}(\psi)}{\lvert{\psi^{-1}(r)}\rvert}.$$ It is is not always easy to estimate $K$. In Section \[estimate K\] we will show how to estimate $K$ in terms of the metric tensor. Note that $K\geq 1$; we will see that in many interesting situations (for example, for revolution cylinders, or for smooth embedded tubes around a closed curve) one has in fact $K=1$.
Doubly connected planar domains
-------------------------------
We now estimate the constant $K$ above when $\Omega$ is an annular region in the plane, bounded by the inner curve $\Sigma_1$ and the outer curve $\Sigma_2$.
[$\bullet\quad$]{}We assume that the inner curve $\Sigma_1$ is convex.
From each point $x\in\Sigma_1$, consider the ray $\gamma_x(t)=x+tN_x$, where $N_x$ is the exterior normal to $\Sigma_1$ at $x$ and $t\geq 0$. Let $Q(x)$ be the first intersection of $\gamma_x(t)$ with $\Sigma_2$, and let $$r(x)=d(x,Q(x)).$$ We say that $\Omega$ is [*starlike with respect to $\Sigma_1$*]{} if the map $x\to Q(x)$ is a bijection between $\Sigma_1$ and $\Sigma_2$; equivalently, if given any point $y\in\Sigma_2$, the geodesic segment which minimizes distance from $y$ to $\Sigma_1$ is | 1 | member_25 |
entirely contained in $\Omega$.
For $x\in\Sigma_1$, we denote by $\theta_x$ the angle between $\gamma'_x$ and the outer normal to $\Sigma_2$ at the point $Q(x)$, and we let $$m\doteq\min_{x\in\Sigma_1}{\cos\theta_x}.$$ Note that as $\Omega$ is starlike w.r.t. $\Sigma_1$, one has $\theta_x\in [0,\frac{\pi}2]$ and then $m\geq 0$.
[$\bullet\quad$]{}To have a positive lower bound, we will assume that $m>0$ (that is, $\Omega$ is [*strictly*]{} starlike w.r.t. $\Sigma_1$).
We also define $$\label{annulus}
\twosystem
{\beta=\min\{r(x): x\in\Sigma_1\}}
{B=\max\{r(x): x\in\Sigma_1\}}$$
We then have the following result.
\[main2\] Let $\Omega$ be an annulus in ${{\bf R}^{2}}$, which is strictly-starlike with respect to its inner (convex) boundary component $\Sigma_1$. Assume that $A$ is a closed potential having flux $\Phi^A$ around $\Sigma_1$. Then: $$\lambda_1(\Omega,A)\geq \dfrac{4\pi^2}{L^2} \dfrac{\beta m}{B} d(\Phi^A,{\bf Z})^2$$ where $\beta$ and $B$ are as in , and $L$ is the length of the outer boundary component. If $\Sigma_2$ is also convex, then $m\geq \beta/B$ and the lower bound takes the form: $$\lambda_1(\Omega,A)\geq \dfrac{4\pi^2}{L^2} \dfrac{\beta^2}{B^2} d(\Phi^A,{\bf Z})^2.$$
In section \[sharpness\], we will explain why we need to control $\dfrac{\beta}{B}$, $L$, and why we need to impose the starlike condition. If $\beta=B$ and $\Sigma_2$ is the circle of length $L$ we get the estimate $$\lambda_1(\Omega,A)\geq \dfrac{4\pi^2}{L^2} d(\Phi^A,{\bf Z})^2$$ which is the first | 1 | member_25 |
eigenvalue of the magnetic Laplacian on the circle with potential $A$ (see section \[riemannian circle\]). If $\Sigma_2$ and $\Sigma_1$ are two concentric circles of respective lengths $L$ and $L_{\epsilon} \to L$, the domain is a thin annulus with $\lambda_1 \to \dfrac{4\pi^2}{L^2} d(\Phi^A,{\bf Z})^2$ which shows that our estimate is sharp.
Our aim is to use these estimates on cylinders as a basis stone in order to study the same type of questions on compact surfaces of higher genus.
Proof of the main theorem
=========================
Preliminary facts and notation {#preliminary}
------------------------------
First, we recall the variational definition of the spectrum. Let $\Omega$ be a compact manifold with boundary and $\Delta_A$ the magnetic Laplacian with Neumann boundary conditions. One verifies that $$\int_{\Omega}(\Delta_Au)\bar u=\int_{\Omega}{\lvert{\nabla^Au}\rvert}^2,$$ and the associated quadratic form is then $$Q_A(u)=\int_{\Omega}{\lvert{\nabla^Au}\rvert}^2.$$ The usual variational characterization gives:
$$\lambda_1(\Omega,A)= \min\Big\{ \frac{Q_A(u)}{\Vert u\Vert^2}:\ u\in C^{1}(\Omega,\mathbb C) / \{0\}\Big\}$$
The following proposition (which is well-known) expresses the [*gauge invariance*]{} of the spectrum of the magnetic Laplacian.
\[basic facts\] a The spectrum of $\Delta_A$ is equal to the spectrum of $\Delta_{A+d\phi}$ for all smooth real valued functions $\phi$; in particular, when $A$ is exact, the spectrum of $\Delta_A$ reduces to that of the classical Laplace-Beltrami operator acting | 1 | member_25 |
on functions (with Neumann boundary conditions if ${\partial}\Omega$ is not empty).
b If $A$ is a closed $1$-form, then $A$ is gauge equivalent to a unique (harmonic) $1$-form $\tilde A$ satisfying $$\twosystem
{d\tilde A=\delta\tilde A=0\quad\text{on}\quad \Omega}
{\tilde A(N)=0\quad\text{on}\quad {\partial}\Omega}$$ The form $\tilde A$ is often called the [Coulomb gauge]{} of $A$. Note that $\tilde A$ is the harmonic representative of $A$ for the absolute boundary conditions.
a This comes from the fact that $
\Delta_A e^{-i\phi}=e^{-i\phi} \Delta_{A+d\phi}
$ hence $\Delta_A$ and $\Delta_{A+d\phi}$ are unitarily equivalent.
b Consider a solution $\phi$ of the problem: $$\twosystem{\Delta\phi=\delta A \quad\text{on}\quad \Omega,}
{{\dfrac{{\partial}\phi}{\bdN}}=A(N) \quad\text{on}\quad {\partial}\Omega.}$$ Then one checks that $\tilde A=A-d\phi$ is a Coulomb gauge of $A$. As $\phi$ is unique up to an additive constant, $d\phi$, hence $\tilde A$, is unique.
We now focus on the first eigenvalue. Clearly, if $A=0$, then $\lambda_1(\Omega,A)=0$ simply because $\Delta_A$ reduces to the usual Laplacian, which has first eigenvalue equal to zero and first eigenspace spanned by the constant functions. If $A$ is exact, then $\Delta_{A}$ is unitarily equivalent to $\Delta$, hence, again, $\lambda_1(\Omega,A)=0$. In fact one checks easily from the definition of the connection that, if $A=d\phi$ for some real-valued function $\phi$ then $
\nabla^{A}e^{i\phi}=0,
$ which | 1 | member_25 |
means that $u=e^{i\phi}$ is $\nabla^A$-parallel hence $\Delta_A$-harmonic. On the other hand, if the magnetic field $B=dA$ is non-zero then $\lambda_1(\Omega,A)>0$.
It then remains to examine the case when $A$ is closed but not exact. The situation was clarified in [@Sh] for closed manifolds and in [@HHHO] for Neumann boundary conditions.
\[shikegawa\]The following statements are equivalent:
a\) $\lambda_1(\Omega,A)=0$;
b\) $dA=0$ and $\Phi^A_c\in\bf Z$ for any closed curve $c$ in $\Omega$.
Thus, the first eigenvalue vanishes if and only if $A$ is a closed form whose flux around every closed curve is an integer; equivalently, if $A$ has non-integral flux around at least one closed loop, then $\lambda_1(\Omega,A)>0$.
Proof of the lower bound
------------------------
From now on we assume that $\Omega$ is a Riemannian cylinder. Fix a first eigenfunction $u$ associated to $\lambda_1(\Omega, A)$ and fix a level curve $$\Sigma_r=\{\psi=r\}, \quad\text{where $r\in [0,1]$.}$$ As $\psi$ has no critical points, $\Sigma_r$ is isometric to ${{\bf S}^{1}}(\frac{L_r}{2\pi})$, where $L_r$ is the length of $\Sigma_r$. The restriction of $A$ to $\Sigma_r$ is a closed $1$-form denoted by $\tilde A$; we use the restriction of $u$ to $\Sigma_r$ as a test-function for the first eigenvalue $\lambda_1(\Sigma_r,\tilde A)$ and obtain: $$\label{level}
\lambda_1(\Sigma_r,\tilde A)\int_{\Sigma_r}{\lvert{u}\rvert}^2\leq\int_{\Sigma_r}{\lvert{\nabla^{\tilde A}u}\rvert}^2.$$ By the estimate | 1 | member_25 |
on the eigenvalues of a circle done in Section \[sectioncircle\] below we see : $$\lambda_1(\Sigma_r,\tilde A)=\dfrac{4\pi^2}{L_r^2}d(\Phi^{\tilde A},{\bf Z})^2,$$ where $\Phi^{\tilde A}$ is the flux of $\tilde A$ across $\Sigma_r$. Now note that $\Phi^{\tilde A}=\Phi^{A}$, because $\tilde A$ is the restriction of $A$ to $\Sigma_r$; moreover $L_r\leq L$ by the definition of $L$. Therefore: $$\label{llower}
\lambda_1(\Sigma_r,\tilde A)\geq \dfrac{4\pi^2}{L^2}d(\Phi^{ A},{\bf Z})^2$$ for all $r$. Let $X$ be a unit vector tangent to $\Sigma_r$. Then: $$\begin{aligned}
\nabla^{\tilde A}_{X}u&=\nabla_{X}u-i\tilde A(X)u\\
&=\nabla_{X}u-iA(X)u\\
&=\nabla^A_{X}u.
\end{aligned}$$ The consequence is that: $$\label{energy}
{\lvert{\nabla^{\tilde A}u}\rvert}^2={\lvert{\nabla^{\tilde A}_{X}u}\rvert}^2={\lvert{\nabla^{A}_{X}u}\rvert}^2\leq {\lvert{\nabla^{A}u}\rvert}^2.$$ [$\bullet\quad$]{}[*Note that equality holds in iff $\nabla^A_{N}u=0$ where $N$ is a unit vector normal to the level curve $\Sigma_r$ (we could take $N=\nabla\psi/{\lvert{\nabla\psi}\rvert}$).*]{}
For any fixed level curve $\Sigma_r=\{\psi=r\}$ we then have, taking into account , and : $$\dfrac{4\pi^2}{L^2}d(\Phi^{ A},{\bf Z})^2\int_{\psi=r}{\lvert{u}\rvert}^2\leq \int_{\psi=r}{\lvert{\nabla^Au}\rvert}^2.$$ Assume that $B_1\leq{\lvert{\nabla\psi}\rvert}\leq B_2$ for positive constants $B_1,B_2$. Then the above inequality implies: $$\dfrac{4\pi^2}{L^2}d(\Phi^{ A},{\bf Z})^2\cdot B_1\int_{\psi=r}\dfrac{{\lvert{u}\rvert}^2}{{\lvert{\nabla\psi}\rvert}}\leq B_2\int_{\psi=r}\dfrac{{\lvert{\nabla^Au}\rvert}^2}{{\lvert{\nabla\psi}\rvert}}.$$ We now integrate both sides from $r=0$ to $r=1$ and use the coarea formula. Conclude that $$\dfrac{4\pi^2}{L^2}d(\Phi^{ A},{\bf Z})^2\cdot B_1\int_{\Omega}{{\lvert{u}\rvert}^2}\leq B_2\int_{\Omega}{\lvert{\nabla^Au}\rvert}^2.$$ As $u$ is a first eigenfunction, one has: $$\int_{\Omega}{\lvert{\nabla^Au}\rvert}^2=\lambda_1(\Omega,A)\int_{\Omega}{\lvert{u}\rvert}^2.$$ Recalling that $K=\frac{B_2}{B_1}$ we finally obtain the estimate .
Proof of the equality case {#equalitycase}
--------------------------
If the cylinder $\Omega$ is a | 1 | member_25 |
Riemannian product then it is obvious that we can take $K=1$ and then we have equality by Proposition \[cyl\] below. Now assume that we do have equality: we have to show that $\Omega$ is a Riemannian product. Going back to the proof, we must have the following facts.
[**F1.**]{} [*All level curves of $\psi$ have the same length $L$*]{}.
[**F2.**]{} [*${\lvert{\nabla\psi}\rvert}$ must be constant and, by renormalization, we can assume that it is everywhere equal to $1$.* ]{}Then, $\psi:\Omega\to [0,a]$ for some $a>0$ and we set $$N\doteq\nabla\psi.$$
[**F3.**]{} [*The eigenfunction $u$ on $\Omega$ restricts to an eigenfunction of the magnetic Laplacian of each level set $\Sigma_r=\{\psi=r\}$, with potential given by the restriction of $A$ to $\Sigma_r$.*]{}
[**F4.**]{} [*One has $\nabla^A_Nu=0$ identically on $\Omega$.* ]{}
### First step: description of the metric
$\Omega$ is isometric to the product $[0,a]\times {{\bf S}^{1}}(\frac{L}{2\pi})$ with metric $$\label{metric}
g={{\begin{pmatrix}}{1&0\\}{0&\theta^2(r,t)\\}{\end{pmatrix}}}, \quad (r,t)\in [0,a]\times [0,L]$$ where $\theta(r,t)$ is positive and periodic of period $L$ in the variable $t$. Moreover $\theta(0,t)=1$ for all $t$.
We first show that the integral curves of $N$ are geodesics; for this it is enough to show that $
\nabla_NN=0
$ on $\Omega$. Let $e_1(x)$ be a vector tangent to the level curve | 1 | member_25 |
of $\psi$ passing through $x$. Then, we obtain a smooth vector field $e_1$ which, together with $N$, forms a global orthonormal frame. Now $${\langle{\nabla_NN},{N}\rangle}=\dfrac 12 N\cdot{\langle{N},{N}\rangle}=0.$$ On the other hand, as the Hessian is a symmetric tensor: $${\langle{\nabla_NN},{e_1}\rangle}=\nabla^2\psi(N,e_1)=\nabla^2\psi(e_1,N)={\langle{\nabla_{e_1}N},{N}\rangle}=\dfrac 12e_1\cdot{\langle{N},{N}\rangle}=0.$$ Hence $\nabla_NN=0$ as asserted. As each integral curve of $N=\nabla\psi$ is a geodesic meeting $\Sigma_1$ orthogonally, we see that $\psi$ is actually the distance function to $\Sigma_1$. We introduce coordinates on $\Omega$ as follows. For a fixed point $p\in\Omega$ consider the unique integral curve $\gamma$ of $N$ passing through $p$ and let $x\in\Sigma_1$ be the intersection of $\gamma$ with $\Sigma_1$ (note that $x$ is the foot of the unique geodesic which minimizes the distance from $p$ to $\Sigma_1$). Let $r$ be the distance of $p$ to $\Sigma_1$. We then have a map $
\Omega\to [0,a]\times\Sigma_1
$ which sends $p$ to $(r,x)$. Its inverse is the map $F: [0,a]\times\Sigma_1\to\Omega$ defined by $$F(r,x)=\exp_x(rN).$$ Note that $F$ is a diffeomeorphism; we call the pair $(r,x)$ the [*normal coordinates*]{} based on $\Sigma_1$. We introduce the arc-length $t$ on $\Sigma_1$ (with origin in any assigned point of $\Sigma_1$) and recall that $L$ is length of $\Sigma_1$ (which is also the length of $\Sigma_2)$). Let us compute | 1 | member_25 |
the metric $g$ in normal coordinates. Since $N={\dfrac{{\partial}}{{\partial}r}}$ one sees that $g_{11}=1$ everywhere; for any fixed $r=r_0$ we have that $F(r_0,\cdot)$ maps $\Sigma_1$ diffeomorphically onto the level set $\{\psi=r_0\}$ so that ${\dfrac{{\partial}}{{\partial}r}}$ and ${\dfrac{{\partial}}{{\partial}t}}$ will be mapped onto orthogonal vectors, and indeed $g_{12}=0$. Setting $\theta(r,t)^2={\langle{{\dfrac{{\partial}}{{\partial}t}}},{{\dfrac{{\partial}}{{\partial}t}}}\rangle}$ one sees that the metric takes the form . Finally note that $
\theta(0,t)=1
$ for all $t$, because $F(0,\cdot)$ is the identity.
### Second step : Gauge invariance
Let $\Omega$ be any Riemannian cylinder and $A=f(r,t)\,dr+h(r,t)\,dt$ a closed $1$-form on $\Omega$. Then, there exists a smooth function $\phi$ on $\Omega$ such that $$A+d\phi=H(t)\,dt$$ for a smooth function $H(t)$ depending only on t. Hence, by gauge invariance, we can assume from the start that $A=H(t)\,dt$.
Consider the function $
\phi(r,t)=-\int_0^rf(x,t)\,dx.
$ Then: $$A+d\phi=\tilde h(r,t)\,dt$$ for some smooth function $\tilde h(r,t)$. As $A$ is closed, also $A+d\phi$ is closed, which implies that ${\dfrac{{\partial}\tilde h}{\bdr}}=0$, that is, $
\tilde h(t,r)
$ does not depend on $r$; if we set $H(t)\doteq\tilde h(t,0)$ we get the assertion.
[$\bullet\quad$]{}We point out the following consequence. If $u=u(r,t)$ is an eigenfunction, we know from [**F4**]{} above that $\nabla^A_Nu=0$, where $N={\dfrac{{\partial}}{\bdr}}$. As $
\nabla^A_Nu={\dfrac{{\partial}u}{\bdr}}-iA({\dfrac{{\partial}}{\bdr}})u
$ and $A=H(t)\,dt$ we obtain $A({\dfrac{{\partial}}{\bdr}})=0$ hence $
{\dfrac{{\partial}u}{\bdr}}=0
| 1 | member_25 |
$ at all points of $\Omega$. This implies that $$\label{uoft}
u=u(t)$$ depends only on $t$.
### Third step : spectrum of circles and Riemannian products {#sectioncircle}
In this section, we give an expression for the eigenfunctions of the magnetic Laplacian on a circle with a Riemannian metric $g$ and a closed potential $A$. Of course, we know that any metric $g$ on a circle is always isometric to the canonical metric $g_{\rm can}=\,dt^2$, where $t$ is arc-length. But our problem in this proof is to reconstruct the global metric of the cylinder and to show that it is a product, and we cannot suppose a priori that the restricted metric of each level set of $\psi$ is the canonical metric. The same is true for the restricted potential: we know that it is Gauge equivalent to a potential of the type $a\,dt$ for a scalar $a$, but we cannot suppose a priori that it is of that form.
We refer to Appendix \[riemannian circle\] for the complete proof of the following fact.
\[circle\] Let $(M,g)$ be the circle of length $L$ endowed with the metric $
g=\theta(t)^2\,dt^2
$ where $t\in [0,L]$ and $\theta(t)$ is a positive function, periodic of period | 1 | member_25 |
$L$. Let $A=H(t)\,dt$. Then, the eigenvalues of the magnetic Laplacian with potential $A$ are: $$\lambda_k(M,A)=\dfrac{4\pi^2}{L^2}(k-\Phi^A)^2, \quad k\in\bf Z$$ with associated eigenfunctions $$u_k(t)=e^{i\phi(t)}e^{\frac{2\pi i (k-\Phi^A)}{L}s(t)}, \quad k\in\bf Z.$$ where $\phi(t)=\int_0^tH(\tau)\,d\tau$ and $s(t)=\int_0^t\theta(\tau)\,d\tau$.
In particular, if the metric is the canonical one, that is, $g=dt^2$, and the potential $1$-form is harmonic, so that $A=\frac{2\pi \Phi^A}{L}dt$, then the eigenfunctions are simply : $$u_k(t)=e^{\frac{2\pi i k}{L}t}, \quad k\in\bf Z.$$
We remark that if the flux $\Phi^A$ is not congruent to $1/2$ modulo integers, then the eigenvalues are all simple. If the flux is congruent to $1/2$ modulo integers, then there are two consecutive integers $k,k+1$ such that $
\lambda_{k}=\lambda_{k+1}.
$ Consequently, the lowest eigenvalue has multiplicity two, and the first eigenspace is spanned by $$e^{i\phi(t)}e^{\frac{\pi i}{L}s(t)}, \, e^{i\phi(t)}e^{-\frac{\pi i}{L}s(t)}.$$ The following proposition is an easy consequence (for a proof, see also Appendix \[riemannian circle\]).
\[cyl\] Consider the Riemannian product $\Omega=[0,a]\times{{\bf S}^{1}}(\frac{L}{2\pi})$, and let $A$ be a closed $1-$form on $\Omega$. Then, the spectrum of $\Delta_A$ is given by $$\dfrac{\pi^2 h^2}{a^2}+\dfrac{4\pi^2}{L^2}(k-\Phi^A)^2, \quad h, k\in{\bf Z}, h\geq 0.$$ In particular, $$\lambda_1(\Omega,A)=\dfrac{4\pi^2}{L^2}d(\Phi^A,{\bf Z})^2.$$
### Fourth step : a calculus lemma
In this section, we state a technical lemma which will allow us to conclude. The proof is | 1 | member_25 |
conceptually simple, but perhaps tricky at some points; then, we decided to put it in Appendix \[technical lemma\].
\[calculus\] Let $s:[0,a]\times [0,L]\to {{\bf R}}$ be a smooth, non-negative function such that $$s(0,t)=t,\quad s(r,0)=0, \quad s(r,L)=L \quad\text{and}\quad {\dfrac{{\partial}s}{\bdt}}(r,t)\doteq\theta(r,t)>0.$$ Assume that there exist smooth functions $p(r),q(r)$ with $p(r)^2+q(r)^2>0$ such that $$p(r)\cos(\frac{\pi}{L}s(r,t))+q(r)\sin(\frac{\pi}{L}s(r,t))=F(t)$$ where $F(t)$ depends only on $t$. Then $p$ and $q$ are constant and $
{\dfrac{{\partial}s}{\bdr}}=0
$ so that $$s(r,t)=t$$ for all $(r,t)$.
### End of proof of the equality case
Assume that equality holds. Then, if $u$ is an eigenfunction, we know that $u=u(t)$ by the discussion in and $u$ restricts to an eigenfunction on each level circle $\Sigma_r$ for the potential $A=H(t)\,dt$ above (see Fact 3 at the beginning of Section \[equalitycase\] and the second step above).
We assume that $\Phi^A$ is congruent to $\frac 12$ modulo integers. This is the most difficult case; in the other cases the proof is a particular case of this, it is simpler and we omit it.
Recall that each level set $\Sigma_r$ is a circle of length $L$ for all $r$, with metric $g=\theta(r,t)^2\,dt$. As the flux of $A$ is congruent to $\frac 12$ modulo integers, we see that there exist complex-valued functions | 1 | member_25 |
$w_1(r),w_2(r)$ such that $$u(t)=e^{i\phi(t)}\Big(w_1(r)e^{\frac{\pi i}{L} s(r,t)}+w_2(r)e^{-\frac{\pi i}{L} s(r,t)}\Big),$$ which, setting $f(t)=e^{-i\phi(t)}u(t)$, we can re-write $$\label{rewrite}
f(t)=w_1(r)e^{\frac{\pi i}{L} s(r,t)}+w_2(r)e^{-\frac{\pi i}{L} s(r,t)}.$$ Recall that here $\phi(t)=\int_0^tH(\tau)\,d\tau$ and $$s(r,t)=\int_0^t\theta(r,\tau)\,d\tau.$$ We take the real part on both sides of and obtain smooth real-valued functions $F(t), p(r),q(r)$ such that $$F(t)=p(r)\cos({\frac{\pi}{L}}s(r,t))+q(r)\sin(\frac{\pi}{L} s(r,t)).$$ Since $\theta(0,t)=1$ for all $t$, we see $$s(0,t)=t.$$ Clearly $s(r,0)=0$; finally, $s(r,L)=\int_0^L\theta(r,\tau)\,d\tau=L$, being the length of the level circle $\Sigma_r$. Thus, we can apply Lemma \[calculus\] and conclude that $s(r,t)=t$ for all $t$, that is, $$\theta(r,t)=1$$ for all $(r,t)$ and the metric is a Riemannian product.
It might happen that $p(r)=q(r)\equiv 0$. But then the real part of $f(t)$ is zero and we can work in an analogous way with the imaginary part of $f(t)$, which cannot vanish unless $u\equiv 0$.
General estimate of $K_{\Omega,\psi}$ {#estimate K}
-------------------------------------
We can estimate $K_{\Omega,\psi}$ for a Riemannian cylinder $\Omega=[0,a]\times{{\bf S}^{1}}$ if we know the explicit expression of the metric in the normal coordinates $(r,t)$, where $t\in [0,2\pi]$ is arc-length : $$g=
\left(
\begin{array}{cc}
g_{11} & g_{12} \\
g_{21} & g_{22}
\end{array}
\right).$$ If $g^{ij}$ is the inverse matrix of $g_{ij}$, and if $\psi=\psi(r,t)$ one has: $${\lvert{\nabla\psi}\rvert}^2=g^{11}\Big({\dfrac{{\partial}\psi}{\bdr}}\Big)^2+2g^{12}{\dfrac{{\partial}\psi}{\bdr}}{\dfrac{{\partial}\psi}{\bdt}}+
g^{22}\Big({\dfrac{{\partial}\psi}{\bdt}}\Big)^2.$$ The function $\psi(r,t)=r$ belongs to ${\cal F}_{\Omega}$ and one | 1 | member_25 |
has: $
{\lvert{\nabla\psi}\rvert}^2=g^{11},
$ which immediately implies that we can take $$K_{\Omega,\psi}\leq \dfrac{\sup_{\Omega}g^{11}}{\inf_{\Omega}g^{11}}.$$ Note in particular that if $\Omega$ is rotationally invariant, so that the metric can be put in the form: $$g=\left(
\begin{array}{cc}
1 & 0 \\
0 & \alpha(r)^2
\end{array}
\right),$$ for some function $\alpha(r)$, then $K_{\Omega,\psi}=1$. The estimate becomes $$\label{simple}
\lambda_1(\Omega,A)\geq\dfrac{4\pi^2}{L^2}\cdot d(\Phi^A,{\bf Z})^2,$$ where $L$ is the maximum length of a level curve $r={\rm const}$.
Yet more generally, one can fix a smooth closed curve $\gamma$ on a Riemannian surface $M$ and consider the tube of radius $R$ around $\gamma$: $$\Omega=\{x\in M: d(x,\gamma)\leq R\}.$$ It is well-known that if $R$ is sufficiently small (less than the injectivity radius of the normal exponential map) then $\Omega$ is a cylinder with smooth boundary which can be foliated by the level sets of $\psi$, the distance function to $\gamma$. Clearly ${\lvert{\nabla\psi}\rvert}=1$ and holds as well.
A concrete example where we could estimate the width $R$ is the case of a compact surface $M$ of genus $\ge 2$ and curvature $-a^2\le K\le -b^2$, $a \ge b >0$. Let $\gamma$ be a simple closed geodesic. Then, using the Gauss-Bonnet theorem, one can show that $R$ is bounded below by an explicit positive | 1 | member_25 |
constant $R=R(\gamma,a)$, hence the $R$-neighborhood of $\gamma$ is diffeomorphic to the product $S^1 \times (-1,1)$ (see for example [@CF]). If we take $\Omega$ as the Riemannian cylinder of width $R(\gamma,a)$ having one boundary component equal to $\gamma$ then we can foliate $\Omega$ with the level sets of the distance function to $\gamma$ and so $K=1$ and holds, with $L$ given by the length of the other boundary component.
Proof of Theorem \[main2\]: plane annuli {#convex}
========================================
Let $\Omega$ be an annulus in ${{\bf R}^{2}}$, which is starlike with respect to its inner convex boundary component $\Sigma_1$. Assume that $A$ is a closed potential having flux $\Phi^A$ around $\Sigma_1$. Recall that we have to show: $$\label{annuliestimate}
\lambda_1(\Omega,A)\geq \dfrac{4\pi^2}{L^2} \dfrac{\beta m}{B} d(\Phi^A,{\bf Z})^2$$ where $\beta, B$ and $m$ will be recalled below and $L$ is the length of the outer boundary component. If we assume that $\Sigma_2$ is also convex, then we show that $m\geq \beta/B$ and the lower bound takes the form: $$\label{annuliestimatetwo}
\lambda_1(\Omega,A)\geq \dfrac{4\pi^2}{L^2} \dfrac{\beta^2}{B^2} d(\Phi^A,{\bf Z})^2.$$
Before giving the proof let us recall notation. For $x\in\Sigma_1$, the ray $\gamma_x$ is the geodesic segment $\gamma_x(t)=x+tN_x$, where $N_x$ is the exterior normal to $\Sigma_1$ at $x$ and $t\geq 0$. The ray | 1 | member_25 |
$\gamma_x$ meets $\Sigma_2$ at a first point $Q(x)$, and we let $
r(x)=d(x,Q(x)).
$ For $x\in\Sigma_1$, we denote by $\theta_x$ the angle between the ray $\gamma'_x$ and the outer normal to $\Sigma_2$ at the point $Q(x)$, and we let $$m\doteq\min_{x\in\Sigma_1}{\cos\theta_x}.$$ We assume that $\Omega$ is strictly starlike, that is, $m>0$; in particular $Q(x)$ is unique. Recall also that: $$\label{annulus}
\beta=\min_{x\in\Sigma_1}r(x), \quad B=\max_{x\in\Sigma_1}r(x).$$ We construct a suitable smooth function $\psi$ and estimate the constant $K=K_{\Omega,\psi}$ with respect to the geometry of $\Omega$. The starlike assumption implies that each point in $\Omega$ belongs to a unique ray $\gamma_x$. Then we can define a function $\psi:\Omega\to [0,1]$ as follows: $$\psi=\threesystem
{0\quad\text{on}\quad\Sigma_1}
{1\quad\text{on}\quad\Sigma_2}
{\text{linear on each ray from $\Sigma_1$ to $\Sigma_2$}.}$$ Estimates and now follow from Theorem \[main3\] together with the following Proposition.
\[estimate doubly convex\] a At all points of $\Omega$ one has: $
\frac{1}{B}\leq{\lvert{\nabla\psi}\rvert}\leq\frac{1}{\beta m}.
$ Therefore: $$K_{\Omega,\psi}=\dfrac{\sup_{\Omega}{\lvert{\nabla\psi}\rvert}}{\inf_{\Omega}{\lvert{\nabla\psi}\rvert}}\leq\dfrac{B}{\beta m}.$$ b One has $$\sup_{r\in [0,1]}{\lvert{\psi^{-1}(r)}\rvert}=L={\lvert{\Sigma_2}\rvert}.$$ c If $\Sigma_2$ is also convex, then $m\geq \beta/B$ hence we can take $K=\beta^2/B^2$.
The proof of the Proposition \[estimate doubly convex\] depends on the following steps.
[**Step 1.**]{} [*On the ray $\gamma_x$ joining $x$ to $Q(x)$, consider the point $Q_t(x)$ at distance $t$ from $x$, and let | 1 | member_25 |
$\theta_x(t)$ be the angle between $\gamma'_x$ and $\nabla\psi(Q_t(x))$. Then the function $$h(t)=\cos(\theta_x(t))$$ is non-increasing in $t$. As $\theta_x(r(x))=\theta_x$ we have in particular: $$\cos(\theta_x(t))\geq \cos(\theta_x)\geq m$$ for all $t\in [0,r(x)]$ and $x\in\Sigma_1$.*]{}
[**Step 2.**]{} [*The function $r\to{\lvert{\psi^{-1}(r)}\rvert}$ is non-decreasing in $r$.*]{}
[**Step 3.**]{} [*If $\Sigma_2$ is also convex we have $m\geq \beta/B$.*]{}
We will prove Steps 1-3 below.
[**Proof of Proposition \[estimate doubly convex\]**]{}. a) At any point of $\Omega$, let $\nabla^R\psi$ denote the radial part of $\nabla\psi$, which is the gradient of the restriction of $\psi$ to the ray passing through the given point. As such restriction is a linear function, one sees that $$\dfrac{1}{B}\leq{\lvert{\nabla^R\psi}\rvert}\leq \dfrac{1}{\beta}.$$ Since ${\lvert{\nabla\psi}\rvert}\geq{\lvert{\nabla^R\psi}\rvert}$ one gets immediately $${\lvert{\nabla\psi}\rvert}\geq\dfrac{1}{B}.$$ Note that $\theta_x(t)$, as defined above, is precisely the angle between $\nabla\psi$ and $\nabla^R\psi$, so that, using Step 1,
$${\lvert{\nabla^R\psi}\rvert}={\lvert{\nabla\psi}\rvert}\cos\theta_x(t)\geq m{\lvert{\nabla\psi}\rvert}$$ hence: $${\lvert{\nabla\psi}\rvert}\leq \dfrac{1}{m}{\lvert{\nabla^R\psi}\rvert}\leq\dfrac{1}{\beta m}.$$ as asserted. It is clear that b) and c) are immediate consequences of Steps 2-3.
**Proof of Step 1.** We use a suitable parametrization of $\Omega$. Let $l$ be the length of $\Sigma_1$ and consider a parametrization $\gamma:[0,l]\to \Sigma_1$ by arc-length $s$ with origin at a given point in $\Sigma_1$. Let $N(s)$ be the outer normal vector to $\Sigma_1$ at the point $\gamma(s)$. | 1 | member_25 |
Consider the set: $$\tilde\Omega=\{(t,s)\in [0,\infty)\times [0,l): t\leq \rho(s)\}$$ where we have set $\rho(s)=r(\gamma(s))$. The starlike property implies that the map $
\Phi:\tilde\Omega\to \Omega
$ defined by $$\Phi(t,s)=\gamma(s)+tN(s)$$ is a diffeomorphism. Let us compute the Euclidean metric tensor in the coordinates $(t,s)$. Write $\gamma'(s)=T(s)$ for the unit tangent vector to $\gamma$ and observe that $N'(s)=k(s)T(s)$, where $k(s)$ is the curvature of $\Sigma_1$ which is everywhere non-negative because $\Sigma_1$ is convex. Then: $$\twosystem
{d\Phi(\dfrac{{\partial}}{{\partial}t})=N(s)}
{d\Phi(\dfrac{{\partial}}{{\partial}s})=(1+tk(s))T(s)}$$ If we set $\Theta(t,s)=1+t k(s)$ the metric tensor is: $$g={{\begin{pmatrix}}{1&0\\}{0&\Theta^2\\}{\end{pmatrix}}}$$ and an orthonormal basis is then $(e_1,e_2)$, where $$e_1=\dfrac{{\partial}}{{\partial}t}, \quad e_2=\dfrac{1}{\Theta}\dfrac{{\partial}}{{\partial}s}.$$ In these coordinates, our function $\psi$ is written: $$\psi(t,s)=\dfrac{t}{\rho(s)}.$$ Now $$\twosystem
{{\langle{\nabla\psi},{e_1}\rangle}={\dfrac{{\partial}\psi}{\bdt}}=\dfrac{1}{\rho(s)}}
{{\langle{\nabla\psi},{e_2}\rangle}=\dfrac{1}{\Theta}{\dfrac{{\partial}\psi}{\bds}}=-\dfrac{t\rho'(s)}{\Theta(t,s)\rho(s)^2}}.$$ It follows that $${\lvert{\nabla\psi}\rvert}^2=\dfrac{1}{\rho^2}+\dfrac{t^2\rho'^2}{\Theta^2\rho^4}=
\dfrac{\Theta^2\rho^2+t^2\rho'^2}{\Theta^2\rho^4}.$$ Recall the radial gradient, which is the orthogonal projection of $\nabla\psi$ on the ray, whose direction is given by $e_1$. If we fix $x\in\Sigma_1$, we have $$\theta_x(t)=\text{angle between $\nabla\psi$ and $e_1$}$$ and we have to study the function $$h(t)=\cos\theta_x(t)=\dfrac{{\langle{\nabla\psi},{e_1}\rangle}}{{\lvert{\nabla\psi}\rvert}}=\dfrac{1}{\rho(s){\lvert{\nabla\psi}\rvert}}$$ for a fixed $s$. From the above expression of ${\lvert{\nabla\psi}\rvert}$ and a suitable manipulation we see $$h(t)^2=\dfrac{\Theta^2}{\Theta^2+t^2g^2}$$ where $g=\rho'(s)/\rho(s)$. Now $$\begin{aligned}
\dfrac{d}{dt}\dfrac{\Theta^2}{\Theta^2+t^2g^2}&=\dfrac{2t\Theta g^2}{(\Theta^2+t^2g^2)^2}
(t{\dfrac{{\partial}\Theta}{\bdt}}-\Theta)\\
\end{aligned}$$ As $\Theta(t,s)=1+tk(s)$ one sees that $t{\dfrac{{\partial}\Theta}{\bdt}}-\Theta=-1$ hence $$\dfrac{d}{dt}h(t)^2=-\dfrac{2t\Theta g^2}{(\Theta^2+t^2g^2)^2}\leq 0$$ Hence $h(t)^2$ is non-increasing and, as $h(t)$ is positive, it is itself non-increasing.
[**Proof | 1 | member_25 |
of Step 2.**]{} In the coordinates $(t,s)$ the curve $\psi^{-1}(r)$ is parametrized by $\alpha:[0,l]\to\tilde\Omega$ as follows: $$\alpha(u)=(r\rho(u),u)\quad u\in [0,l].$$ Then: $$\begin{aligned}
{\lvert{\psi^{-1}(r)}\rvert}&=\int_0^l\sqrt{g(\alpha'(u),\alpha'(u))}\,du\\
&=\int_0^l\sqrt{r^2\rho'(u)^2+(1+rk(u)\rho(u))^2}\,du
\end{aligned}$$ Convexity of $\Sigma_1$ implies that $k(u)\geq 0$ for all $u$; differentiating under the integral sign with respect to $r$ one sees that indeed $\frac{d}{dr}{\lvert{\psi^{-1}(r)}\rvert}\geq 0$ for all $r\in [0,1]$.
[**Proof of Step 3.**]{} Let $T_x$ be the tangent line to $\Sigma_2$ at $Q(x)$ and $H(x)$ the point of $T_x$ closest to $x$. As $\Sigma_2$ is convex, $H(x)$ is not an interior point of $\Omega$, hence $$d(x,H(x))\geq\beta.$$ The triangle formed by $x, Q(x)$ and $H(x)$ is rectangle in $H(x)$, then we have: $$r(x)\cos\theta_x=d(x,H(x)).$$ As $r(x)\leq B$ we conclude: $$B \cos\theta_x\geq \beta,$$ which gives the assertion.
Sharpness of the lower bound {#sharpness}
============================
An upper bound
--------------
In this short paragraph, we give a simple way to get an upper bound when the potential $A$ is *closed*. Then, we will use this in different kinds of examples, in order to show that the assumptions of Theorem \[main2\] are sharp. The geometric idea is the following: if we have a region $D \subset \Omega$ such that the first absolute cohomology group $H^1(D)$ is $0$, then we can estimate from | 1 | member_25 |
---
abstract: 'During the epoch of large galaxy formation, thermal instability leads to the formation of a population of cool fragments which are embedded within a background of tenuous hot gas. The hot gas attains a quasi-hydrostatic equilibrium. Although the cool clouds are pressure confined by the hot gas, they fall into the galactic potential, and their motion is subject to drag from the hot gas. The release of gravitational energy due to the infall of the cool clouds is first converted into their kinetic energy, and is subsequently dissipated as heat. The cool clouds therefore represent a potentially significant energy source for the background hot gas, depending upon the ratio of thermal energy deposited within the clouds versus the hot gas. In this paper, we show that most of dissipated energy is deposited in the tenuous hot halo gas, providing a source of internal energy to replenish losses in the hot gas through bremsstrahlung emission and conduction into the cool clouds. The heating from the motion of the cool clouds allows the multi-phase structure of the interstellar medium to be maintained.'
author:
- 'Stephen D. Murray'
- 'Douglas N. C. Lin'
title: 'Energy Dissipation in Multi-Phase Infalling Clouds in | 1 | member_26 |
Galaxy Halos'
---
Introduction
============
The stellar velocity dispersion in the halos of galaxies similar to the Milky Way exceeds 100 km s$^{-1}$. The gravitational potential that binds the stars to their host galaxies is dominated by collisionless dark matter. According to the widely adopted cold dark matter (CDM) scenario, these normal galaxies are formed through the mergers of much smaller entities, dwarf galaxies. After violent relaxation, the dark matter is well mixed in phase space and attains an extended 3-D spatial distribution. In spiral galaxies, the formation and concentration of stars in extended, flattened, rotating disks requires the detachment of ordinary matter from the dark-matter halos of the original host dwarf galaxies. The dominance of the dark-matter halo to the galactic potential at large radii, and the separation of the ordinary matter imply that, during the epoch of galactic buildup, the ordinary matter was primarily in the form of gas which dissipated a substantial fraction of its initial potential energy.
In a previous paper (Lin & Murray 2000), we considered the dynamical evolution of infalling gas in the halos of normal galaxies. We showed that for typical values ($\sim 10^6$ K) of the virial temperature, the cooling timescale increases | 1 | member_26 |
with temperature, and the protogalactic clouds (hereafter PGC’s) are thermally unstable (Field 1965). Thermal instability leads to the rapid growth of perturbations and fragmentation of a PGC (Murray & Lin 1990). The result is that a two-phase medium develops during the initial cooling of the PGC, in which a population of warm fragmentary clouds (WFC’s) are confined by the pressure of hot, residual halo gas (RHG) (Burkert & Lin 2001). The RHG is cooled by radiative emission and conductive transport into the WFC’s (which are efficient radiators). In our earlier work, we assumed that the RHG is heated primarily by the release of the gravitational energy as the WFC’s into the central region of the halo potential, due both to their collective gravity as well as that of the dark matter. The WFC’s are unable to cool below 10$^4$ K until their density reaches a sufficiently high value that the WFC’s become self-shielded from external photo-dissociating UV radiation (Couchman & Rees 1986; Haiman, Rees, & Loeb 1997; Dong, Lin, & Murray 2003).
In the above picture, the evolution of the WFC’s is similar to that of Lyman-$\alpha$ clouds and high velocity clouds (HVC’s). Both of those systems have been proposed | 1 | member_26 |
as representatives of late-time accretion of material in an ongoing process of galaxy buildup by mergers [@MI94b; @Blitzetal99; @Manning99]. Because they evolve at an earlier time and closer to the centers of the parent galaxies, however, the WFC’s would evolve in an environment of higher pressures and UV fluxes, compared to either Lyman-$\alpha$ clouds or HVC’s. Their environment may, instead, more closely resemble that of cooling flows (e.g. Sarazin 1986; Loewenstein & Mathews 1987; Sarazin & White 1987, 1988), and many of our results may have relevance to those systems. Additionally, the Ly-$\alpha$ clouds have been proposed as being contained within dark matter “minihaloes,” (e.g. Rees 1986; Ikeuchi 1986; Mo, Miralda-Escude, & Rees 1993) whereas the WFC’s are either pressure confined, or at most weakly self-gravitating
In this paper, we verify our basic conjecture that most of the gravitational energy released by the infalling WFC’s is dissipated within the RHG. That process is crucial to the assumption that the RHG is in quasi-thermal equilibrium. Without this heating source, the background gas would gradually be depleted due to loss of thermal energy and precipitation into WFC’s. A reduction in the pressure of the background gas would also enable the WFC’s to | 1 | member_26 |
expand and eventually eliminate the multi-phase structure of the gas. In order to simulate this process in detailed, we adopt a 2-D numerical hydrodynamic scheme with a multi-phase medium.
The motion of clouds relative to, and their interaction with an external medium has been studied by numerous authors. [@MI94a] examined ram-pressure stripping due to the supersonic motion of gas past clouds confined within minihalos, a very different situation from that described above for the evolution of the WFC’s. Tenorio-Tagle et al. (1986, 1987) examined the interactions of clouds hitting relatively high density galactic disks at high speeds. Again, that is a very different situation from the evolution of the WFC’s, which move slowly through a low density medium with a smooth density distribution. [@MWBL93] examined the loss of gas from a cloud due to the growth of Kelvin-Helmholtz instability for transsonic motions. As with the above studies, however, the energy transfer between the cloud and the background gas was not examined.
We proceed by briefly describing our method and the model parameters in §2. In §3, we analyze the results of our computations. Finally we discuss the implication of these results in §4.
Numerical Method and Model Parameters
=====================================
Equation | 1 | member_26 |
of Motion
------------------
Following its collapse into the potential of the galactic dark matter halo, the RHG is shock-heated to the virial temperature of the potential, and rapidly attains a quasi hydrostatic equilibrium. For computational simplicity, we adopt a Cartesian coordinate system in which the galactic potential $g$ is imposed in the $y$ direction. For a spherically symmetric potential, $y$ corresponds to the radial direction. The equation of motion of the RHG becomes $${d V_{h x} \over d x} = - {1 \over \rho_h} {d P_t \over d x}$$ $${d V_{h y} \over d y} = - {1 \over \rho_h} {d P_t \over d y} -g$$ where $\rho_h$, $P_h$, $V_{h, x}$, and $V_{h,y}$ are respectively the density, pressure, two-velocity components of the RHG, $P_t = P_h +
P_w$ is the total pressure, $\rho_w$, $P_w$, $V_{w, x}$, and $V_{w,y}$ are respectively the density, pressure and two velocity components of WFC’s. The equation of motion for the WFC’s is similar, $${d V_{w x} \over d x} = - {1 \over \rho_w} {d P_t \over d x}$$ $${d V_{w y} \over d y} = - {1 \over \rho_w} {d P_t \over d y} -g+F_D,$$ where $F_D$ is a drag force term, which is | 1 | member_26 |
a function of the speed and geometry of the WFC’s, and of their density contrast with the RHG.
Parameters for the Residual Halo Gas
------------------------------------
We consider four models. The parameters are listed in Table \[tab:models\], which lists, for each model, value of $g$, the polytropic index for the cloud, $\gamma_w$, the density contrast between the cloud and the background, $D_\rho$, and the initial downward speed of the cloud, normalized to the sound speed of the background. In all cases, the RHG is initialized with the same temperature throughout, but thereafter evolves with a polytropic equation of state, in which $P_h = K_h \rho_h ^{\gamma_h}$ where $K_h$ is the adiabatic constant, and the polytropic index $\gamma_h=5/3$ for each model. We also assume that the RHG is initially in hydrostatic equilibrium, such that $$\rho_h=\rho_0{\rm e}^{{g\over{C_h^2}}\left(y-y_0\right)},
\label{eq:den}$$ where $\rho_0$ is the density at a reference height $y_0$, and $C_h$ is the isothermal sound speed of the RHG. Because the RHG initially has the same temperature throughout, the magnitude of $K_h$ is a function of $y$, such that $$K_h = C_h^2 \rho(y)^{1-\gamma_h}.$$ The density scale height of the RHG is $$r_h = { C_h^2 \over g}.$$
In all models the velocities are normalized | 1 | member_26 |
to $C_h=1$. The initial location of the WFC’s is set to be at $x_0=y_0=0$. The value of $g$ is uniform throughout the grid, justified by the fact that the computational domain represents a small fraction of a galaxy. For Models 1-3, we set $g=0.1$, so that $r_h = 10$, while in Model 4, $g=0.05$, giving $r_h = 20$. We set $\rho_0=1$. In these units, Equation (\[eq:den\]) reduces to $$\rho_h = {\rm e}^{-g\left(y - y_0\right)}.$$ The computational domain extends from -0.75 to 0.75 in $x$, and from -15 to 1 in $y$. At the base of the computational domain, $\rho_h
/\rho_0 \sim 4.5$ for Models 1-3, and 2.1 for Model 4.
Parameters for the Warm Fragmentary Clouds
------------------------------------------
The density ratio at the launch point, $D_{\rho}\equiv \rho_w/\rho_h =10^{2}$ in Models 1-3, while $D_\rho = 25 $ in Model 4. The magnitude of $D_\rho$ would be constant throughout the simulation time if 1) the WFC’s retain their integrity, 2) $\gamma_h = \gamma_w$, and 3) there is no shock dissipation to modify $K_h$ and $K_w$. In general, however, $D_\rho$ is a function of $y$, depending on the equation of state for both the WFC’s and RHG.
The value $D_{\rho}=100$ is selected to represent | 1 | member_26 |
ionized clouds at temperatures of 10$^4$ K in a Milky Way-sized galaxy, for which the RHG is heated to the virial temperature of $\approx$10$^6$ K. The smaller value of $D_{\rho}$ for Model 4 would be appropriate for either cooler backgrounds or warmer clouds.
The physical dimensions of the WFC’s are set by dynamical and thermal processes [@LM00]. Clouds below a minimum radius, $S_{min}$, are re-heated by conduction from the RHG. The maximum radius, $S_{max}$, is set by the point at which the clouds become self-gravitating. Such clouds have negative specific heats, and so are unstable to external heating. The lower limit upon the cloud size translates to [@LM00] $$S_{min,s}=1.6{\ T}^{3/2}_6{\ D}_{100}^{-1}{\ n}_{10}^{-1}
{\ }\Lambda_{25}^{-1}{\ \rm pc},$$ in the limit of saturated conduction, where $T_6$ in the temperature of the RHG in units of 10$^6$ K, $D_{100}$ is the density contrast between the cloud and RHG in units of 100, n$_{10}$ is number density of the cloud in units of 10 cm$^{-3}$, and $\Lambda_{25}$ is the cooling efficiency in units of 10$^{-25}$ ergs cm$^3$ s$^{-1}$, characteristic of low metallicity gas near 10$^4$ K [@DM72]. In the limit of unsaturated conduction, $$S_{min,u}=4{\rm T}^{7/4}_6{ \rm n}_{10}^{-1}
{\ }\Lambda_{25}^{-1/2}{\ \rm pc}.$$ The maximum cloud | 1 | member_26 |
size is set by the Bonner-Ebert criterion for self-gravity to become important, $$S_{max}=350{\ T}_4\left({{nT}\over{10^5}}\right)^{-1/2}{\ \rm pc},$$ where $T_4$ is the temperature of the WFC’s in units of 10$^4$ K, and $nT$ is the pressure of the clouds (assumed to be in pressure equilibrium with the RHG).
In the two-phase model discussed by [@LM00], the WFC’s are heated by UV emission from massive stars, in a self-regulated star formation process. For the parameters given above, the total column densities of the clouds range from $5\times10^{19}$ to $10^{22}$ cm$^{-2}$.
In Models 1-2 and 4, we adopt a polytropic equation of state for the WFC’s with a power index $\gamma_w = \gamma_h$. In Model 3 we attempt to maintain $\gamma_w\approx1$ throughout the evolution. This is done by allowing the cloud to cool, but turning cooling off below a set minimum temperature, which we select as $10^4$ K. The high cooling efficiency in this temperature regime ensures that the cloud temperature cannot significantly exceed the cutoff temperature. Cooling is not allowed to proceed in the background gas.
Because we are using a single-fluid code (see below), zones where cloud and background gas mix are a concern. In order to prevent significant cooling of the | 1 | member_26 |
background gas, cooling is turned off whenever the background gas exceeds a volume fraction of 0.5, as measured by the relative amounts of the two tracers initially placed within the cloud and background. Cooling is also not allowed whenever the temperature of a cell exceeds 0.2 times the initial temperature of the background gas.
The models with polytropic equations of state permit the use of dimensionless numbers, as used here, to scale the results to a wide range of systems. The presence of cooling in Model 3, however, introduces some dimensions into the problem. In that model, we take $T_h=10^6$ K, and $T_w=10^4$ K, appropriate, as discussed above, for an L$_\ast$ galaxy. The corresponding sound speeds are $C_h=130$ km s$^{-1}$, and $C_w=13$ km s$^{-1}$ (assuming ionized gas). The initial density of the cloud, $n_w=6$ cm$^{-3}$. A length dimension is not explicitly imposed upon the problem, because the heating and cooling are strictly local processes. Typical values can, however, be computed. For an isothermal potential, $g=V_c^2/R$, where $V_c$ is the circular speed and $R$ is the galactocentric radius. Taking $V_c=220$ km s$^{-1}$, the physical scale height of the gas $$R_h={{C_h^2}\over g}={130\over220}R=350~R_{kpc}{\ \rm pc},$$ where $R_{kpc}$ is the galactocentric radius in kpc. | 1 | member_26 |
In code units, $R_h=10$ (see previous section), and so one unit of distance in the code corresponds to $l_{unit}=35$ $R_{kpc}$ pc in physical units. The cloud radius is initially set to be 0.2 in code units, or 7 $R_{kpc}$ pc in physical dimensions. The unit of time for Model 3 is given by the ratio $l_{unit}/C_h=0.26~R_{kpc}$ Myr.
In Models 1, 3, and 4, we assume that the WFC is initially at rest. The evolution of the WFC’s is, however, dynamic, with clouds continually colliding and merging, being disrupted by dynamical instabilities, being reheated by conduction from the RHG, cooling to form stars, and condensing out of the RHG. When the clouds form from the RHG, they would be expected to have typical speeds up to the sound speed of the RHG. In Model 2, therefore, we take the WFC to be initially falling in the $-y$ direction, at a velocity equal to $C_h$.
Due to their negative buoyancy, the WFC’s fall through the RHG in all models. If the background RHG is not perturbed, it induces a drag force $F_D$ on WFC’s. For WFC’s with sizes $S$ which are larger than the mean free path of particles in the RHG, | 1 | member_26 |
$$F_D = {1\over2}C_D \pi S^2 \rho_h V_w ^2,$$ where $C_D$ is the drag coefficient, and $S$ is the cloud radius (Batchelor 2000). In flows with high Reynolds number, the turbulent wake behind the body provides an effective momentum transfer mechanism, dominating $C_D$. For example, the experimentally measured $C_D$ for a hard sphere in a nearly inviscid fluid is 0.165 (Whipple 1972). For compressible gas clouds, $C_D$ is probably closer to unity.
When $F_D\approx g$, the WFC’s attain a terminal speed $$V_t \approx \left( {8 D_{\rho} {S g} \over 3 C_D } \right)^{1/2}.
\label{eq:vterm}$$ At the launch point, the size of the WFC is set to be $S (y_0) = 0.2$ in models 1-3. If $C_D =1$ and the WFC preserves its integrity, $V_t =
2.3$, which would exceed sound speed of the RHG. Once the Mach number of the WFC exceeds unity, however, shock dissipation would greatly increase the drag relative to the above estimate. Prior to the WFC achieving $V\approx1$, however, Rayleigh Taylor instability causes it to break up into smaller pieces. For smaller fragments, the value of $V_t$ is reduced, as seen in Equation (\[eq:vterm\]). Due to shock dissipation, the sound speed in the RHG is also slightly | 1 | member_26 |
larger than that in Equation (\[eq:vterm\]). Both of the above factors may prevent the falling WFC’s from attaining $V_t>C_h$. Because the WFC’s are pressure confined, however, their internal sound speed $C_w = D_\rho ^{-1/2} C_h =0.1 \ll V_t$, and so internal shock dissipation is likely to occur within the WFC’s. In order to examine the role of the relative magnitudes of the speeds, we choose, in model 4, $S (y_0) =0.1$, $g=0.05$, and $D_\rho (y_0) = 25$ such that $V_t < C_h$ throughout the computational domain. Shock dissipation does not occur in the RHG, but it is present interior to the WFC.
In Model 3, we assume the same initial condition as model 1, but adopt an effectively isothermal equation of state for the WFC’s by allowing the gas to cool to 10$^4$ K. The resulting energy drainage would lead to a greater dissipation rate within the WFC’s but it should not significantly modify the energy deposition rate into the RHG.
Numerical Method
----------------
The models discussed below are calculated using Cosmos, a multi-dimensional, chemo-radiation-hydrodynamics code developed at Lawrence Livermore National Laboratory (Anninos, Fragile & Murray 2003). For the current models, radiative emission is not included. In order to maximize | 1 | member_26 |
the resolution, the models are run in two dimensions. Because Cosmos runs on a Cartesian grid, this means that the clouds simulated are actually slices through infinite cylinders, rather than spheres. This limitation should not, however, significantly alter our conclusions, and allows us to run the simulations at significantly higher resolution than would be possible in three dimensions. The resolutions of the models are 300x3200 zones. The clouds are therefore resolved by 80 zones across their diameters. This is somewhat poorer than the resolution found necessary by Klein, McKee, & Colella (1994) for their study of shock-cloud interactions. Because the clouds in our models are not subject to extreme shocks, however, lower resolutions should be adequate, and reductions in resolution by a factor of two have not been found to have any affect upon our results.
Because we are concerned with energy transfer and dissipation, the form of the artificial viscosity used in the models might be expected to play a significant role. In order to examine that possibility, we have computed versions of Model 1 using both scalar and tensor forms of the artificial viscosity, with the coefficient varied by a factor of two, and both with and without | 1 | member_26 |
linear artificial viscosity. The energy changes in the cloud and background were found to differ among the models by no more than 10%. We therefore conclude that the form of the artificial viscosity does not dominate our results. The lack of sensitivity is most likely to to the absence of strong shocks in the models.
The models are run with reflecting boundary conditions on all sides. This choice of boundaries serves to isolate the system, eliminating potential ambiguities in the interpretation of the energies of the two components.
Results of the Numerical Simulations
====================================
Model 1: Transsonic sedimentation of adiabatic clouds
-----------------------------------------------------
In Model 1, we adopt a polytropic equation of state for both the cloud and background. For the values of $D_\rho$, $S$, and $g$ of the model, $V_t \sim C_h$ during the descent.
In Figure \[fig:mod1rho\], we show the evolution of the density of Model 1. The model is shown from time 0 to 16, at intervals of 2 (the horizontal sound crossing time in the RHG $\Delta x/C_h=1.5$). The WFC rapidly accelerates to a speed $\vert V_y\vert \approx C_h$, at which point the increasing drag causes it to achieve a terminal speed. The deceleration of the cloud | 1 | member_26 |
as it approaches terminal speed leads to the growth of Rayleigh-Taylor instability, causing rapid breakup of the cloud. For an incompressible fluid, the Rayleigh-Taylor instability grows, in the linear regime, as ${\rm e}^{\omega t}$, where $$\omega^2={{2\pi g}\over \lambda}\left({{\rho_h-\rho_l}
\over{\rho_h+\rho_l}}\right),
\label{eq:rt}$$ $\lambda$ is the wavelength of the perturbation, and the subscripts $h$ and $l$ refer, respectively, to the heavy and light fluids (Chandrasekhar 1961, p. 428). For subsonic flows, the growth rate is similar for compressible fluids. Perturbations with the shortest wavelengths grow most rapidly, but saturate quickly when their amplitudes $A\approx\lambda$. As a result, wavelengths $\lambda\sim S$ lead most strongly to cloud breakup. For such perturbations, the above relation gives $\omega\approx17$, in fair agreement with the rate of breakup observed in the cloud, though the latter is complicated by the additional growth of Kelvin-Helmholtz instability due to the flow of gas around the cloud (cf. Murray et al. 1993).
Figure \[fig:mod1en\] shows the energy evolution of Model 1. Shown are the evolution of the total (internal, kinetic, and gravitational), the kinetic plus internal, kinetic, and internal energies. Values for the background gas are given by the solid curves, while those for the cloud are indicated by the dashed curves. The | 1 | member_26 |
energies are in code units, and are plotted as changes relative to their initial values. The energies of the cloud and background are calculated as sums across the entire computational grid, with the contribution from each zone weighted by the fractional amount of the appropriate tracer present in each zone. This should minimize any confusion due to mixing of the cloud and background gas. The high order of the advection scheme also minimizes numerical diffusion (Anninos, Fragile, & Murray 2003).
As can be seen in Figure \[fig:mod1en\], the total energy of the cloud decreases as it falls in the gravitational potential. The increase in $E_{Tot}$ at late times is due to the upward motion of cloud material entrained within the vortices that form behind the cloud. The kinetic energy of the cloud increases until it reaches a terminal infall speed at $t\approx10$, According to Equation (\[eq:vterm\]), the terminal speed of infalling clouds is an increasing function of their size. As a cloud breaks up into many smaller pieces, its kinetic energy decreases along with $V_t$. The internal energy of the cloud does not change significantly during its descent and breakup. The distance travelled before breakup is $\sim 30
S(y_0)$. The | 1 | member_26 |
effective cross section of the cloud is $\sim 2 S(y_0)$, implying that the mass of the RHG that is encountered by the falling cloud is smaller than, but comparable to the mass of the cloud (Murray et al. 1993). During break up, the terminal velocity of the fragments $V_t\propto S^{1/2}$, in accordance with Equation (\[eq:vterm\]). The fragments therefore trail behind the remaining clouds.
The kinetic, and especially the internal energy of the background gas are substantially increased by the end of the simulation. In this model, therefore, the majority of the energy released by the infall of the cloud is deposited into the internal energy of the background gas, primarily by the action of weak shock waves generated by the motion of the infalling cloud. This result supports the assumption of [@LM00] that the rate of energy deposition throughout the galaxy is directly proportional to the total infall rate of WFC’s throughout the system.
Model 2: Supersonic impact of WFC’s
-----------------------------------
In Model 1, the WFC attains a terminal speed which is a significant fraction of both $C_h$, and the value of $V_t$ predicted from Equation \[eq:vterm\]. It might also be expected that WFC’s which condense from the RHG would | 1 | member_26 |
have initial speeds comparable to the sound speed of the RHG. In order to examine the possible effects of nonzero initial speeds upon the evolution, we consider in Model 2 an initial condition in which the WFC is already falling at the sound speed at the start of the numerical calculation.
The density of Model 2 is shown in Figure \[fig:mod2rho\]. Due to the more rapid motion of the cloud, as compared to that of Model 1, the simulation is only carried out to $t=12$. The initial motion of the cloud can be seen to drive a weak shock ahead of it. Behind the shock, the leading edge of the cloud continues to move downwards at almost $C_h$, slowing down gradually until the very end of the simulation, when it rapidly decelerates as it breaks up, due to the combined action of Rayleigh-Taylor and Kelvin-Helmholtz instabilities. These results suggest that the infalling WFC’s quickly settle to $V_t$ irrespective of the initial conditions, as we have assumed previously (Lin & Murray 2000). The breakup of the cloud proceeds at nearly the same vertical height as in Model 1. The similarity arises because the models have the same gravitational accelerations and density | 1 | member_26 |
contrast. Prior to breakup, the downward motion of the cloud is more rapid than the value of $V_t$ found for Model 1. The differences are due to the modification in the drag caused by the leading shock in Model 2.
The energy evolution is shown in Figure \[fig:mod2en\]. The initial kinetic energy of the cloud, $E_{K,0}=6.3$, is almost entirely dissipated by $t=10$. Over the same time interval, the cloud is also able to penetrate to a greater depth than the cloud in Model 1, increasing the release of gravitational energy relative to that model. Together, these effects lead to a gain of internal energy for the background gas by a approximately a factor of two larger than seen in Model 1.
However, the depth at which the cloud breaks up is similar in the two models. As in Model 1, the break up occurs when the cloud encounters a column of RHG that is comparable in mass to that of the cloud. Thereafter, the fragments’ rate of sedimentation is significantly reduced in accordance with Equation (\[eq:vterm\]). The asymptotic rate of RHG’s internal energy increase in Model 2 is comparable to that in Model 1.
Model 3: Efficient energy loss within | 1 | member_26 |
the cool clouds
-----------------------------------------------------
In Model 3, we approximate an isothermal equation of state for the cloud, as described above, in order to represent the limit in which cooling is highly efficient. The evolution of the density is shown in Figure \[fig:mod3rho\], while the energies are shown in Figure \[fig:mod3en\]. The isothermal behavior of the cloud leads to nonconservation of the total energy of the cloud plus background, and so we do not plot that here, focusing instead upon the kinetic and internal energies.
As expected, cooling within mixed cells does lead to some cooling of the background gas, as well as some overcooling within the cloud, both of which can be seen in Figure \[fig:mod3en\]. The lack of heating within the cloud leads to additional compression relative to the previous models, reducing its breakup. Overall, however, the transfer of kinetic energy of the cloud to the internal energy of the background gas is very similar to the adiabatic models described above, indicating that efficient cooling within the clouds does not have a strong effect upon the energy deposition rate.
Fragmentation of the cloud also occurs in Model 3. The efficient cooling enhances the density contrast between the cloud and | 1 | member_26 |
the RHG, such that the cloud retains smaller volume and cross section. Consequently, the cloud encounters a smaller gas mass along the path of its descent, and fragmentation occurs at a greater depth. On small wavelengths, the infalling cloud appears to be better preserved than in the previous models. But on the scale of the cloud size, the cloud again fragments after encountering a column similar to its own mass, as above.
Model 4: Subsonic Sedimentation of WFC’s
----------------------------------------
For Model 4, $D_\rho=25$, and $g=0.05$, such that $V_t$ is predicted by Equation \[eq:vterm\] to be subsonic. The evolution of the density of Model 4 is shown in Figure \[fig:mod4rho\]. The cloud rapidly reaches a terminal speed, $V_t\approx0.3$, smaller than predicted if $C_D=1$. As in Model 1, however, expansion of the cloud enhances the drag coefficient to $C_D>1$. The cloud therefore never achieves the terminal speed predicted for a hard sphere, even in the absence of strong supersonic dissipation. From Equation \[eq:rt\], $\omega\approx6$, and the cloud breaks up even more rapidly than the more dense clouds considered in Models 1-3, due to its reduced density contrast relative to those models.
The downward displacement of the cloud in Model 4 is reduced | 1 | member_26 |
by a factor of a few relative to that of Model 1. As a result, the gravitational energy released by the settling of the cloud, and dissipated into the background gas, is reduced by an order of magnitude relative to Model 1, as can be seen in Figure \[fig:mod4en\].
In the absence of trans/supersonic motion of the cloud through the background, shock dissipation cannot be a strong mechanism for the dissipation of energy due to motion of the cloud. The primary mechanism involves the wake of the cloud. In the simulations, the vortical motions behind the cloud dissipate energy on small scales, due to artificial and numerical viscosity. In three dimensions, the high Reynolds numbers would lead to the formation of turbulent wakes, which would lead to the dissipation of energy by viscous stress on sufficiently small length scales, leading to the same outcome as observed in Model 4. The observed outcome of the energy deposition is not, therefore, sensitive to the exact physical process responsible for it.
Summary and Discussion
======================
In this paper, we examine the interactions of a two-phase medium in a passive gravitational potential. This situation represents the physical environment that occurs naturally in the context | 1 | member_26 |
of galaxy formation, cooling flows, and during the transition of gas clouds from quasi-hydrostatic contraction to dynamical collapse. It is a natural consequence of thermal instability, which generally leads to the emergence of a population of relatively cool, dense clouds (warm, fragmentary clouds, or WFC’s) that are pressure confined by an ambient hot, tenuous gas (residual hot gas, or RHG). In such a state, the hot gas establishes a quasistatic equilibrium with the background gravitational potential, and the cold clouds settle into it under the action of their negative buoyancy. In the present investigation, we neglect the self-gravity of the gas, and consider the potential to be due to a time-invariant background distribution of dark matter or stars.
Through a series of numerical simulations, we demonstrate the following evolutionary outcomes.
1\) During their descent, the WFC’s break up on the same timescale as is required for them to attain a terminal speed.
2\) Most of the energy released from the sedimentation of the WFC’s into the background gravitational potential is deposited into the RHG.
These results provide justifications for the assumptions we made in our earlier model for the evolution of multi-phase medium during the epoch of galaxy formation (Lin | 1 | member_26 |
---
abstract: 'One of the main scientific objectives of the ongoing [[*Fermi*]{}]{} mission is unveiling the nature of the unidentified $\gamma$-ray sources (UGSs). Despite the large improvements of [[*Fermi*]{}]{} in the localization of $\gamma$-ray sources with respect to the past $\gamma$-ray missions, about one third of the [[*Fermi*]{}]{}-detected objects are still not associated to low energy counterparts. Recently, using the Wide-field Infrared Survey Explorer ([[*WISE*]{}]{}) survey, we discovered that blazars, the rarest class of Active Galactic Nuclei and the largest population of $\gamma$-ray sources, can be recognized and separated from other extragalactic sources on the basis of their infrared (IR) colors. Based on this result, we designed an association method for the $\gamma$-ray sources to recognize if there is a blazar candidate within the positional uncertainty region of a generic $\gamma$-ray source. With this new IR diagnostic tool, we searched for $\gamma$-ray blazar candidates associated to the UGS sample of the second [[*Fermi*]{}]{} $\gamma$-ray catalog (2FGL). We found that our method associates at least one $\gamma$-ray blazar candidate as a counterpart each of 156 out of 313 UGSs analyzed. These new low-energy candidates have the same IR properties as the blazars associated to $\gamma$-ray sources in the 2FGL catalog.'
author:
| 1 | member_27 |
- 'F. Massaro, R. D’Abrusco, G. Tosti, M. Ajello, A. Paggi, D. Gasparrini.'
title: 'Unidentified gamma-ray sources: hunting $\gamma$-ray blazars'
---
Introduction {#sec:intro}
============
More than half of the $\gamma$-ray sources detected by [*Compton*]{} Gamma-Ray Observatory (CGRO), and present in the third EGRET (3EG) catalog were not associated with known counterparts seen at low energies [@hartman99]. Whatever the nature of the unidentified $\gamma$-ray sources (UGSs), these objects could provide a significant contribution to the isotropic gamma-ray background (IGRB) [e.g., @abdo10a]. Solving the puzzle of the origin of the UGSs, together with a better knowledge of other IGRB contributions estimated from known sources, is crucial also to constrain exotic high-energy physics phenomena, such as dark matter signatures, or new classes of sources.
With the advent of the [[*Fermi*]{}]{} mission the localization of $\gamma$-ray sources has significantly improved with respect to the past $\gamma$-ray missions, thus simplifying the task of finding statistically probably counterparts at lower energies. New association methods also have been developed and applied, so that the number of UGSs has significantly decreased with respect to the 3EG catalog [@hartman99]; however, according to the second [[*Fermi*]{}]{} $\gamma$-ray catalog (2FGL), about one third of detected gamma-ray sources in the energy range | 1 | member_27 |
above 100 MeV is still unassociated [@abdo11]. It is worth noting that the most commonly detected sources in the $\gamma$-ray sky, since the epoch of CGRO, are blazars, one of the most enigmatic classes of Active Galactic Nuclei (AGNs) [e.g., @hartman99]. Within the 2FGL, there are 576 UGSs out of a total number of 1873 sources detected, while among the 1297 associated sources, $\sim$ 1000 have been associated with AGNs [@abdo11; @ackermann11a].
Blazar emission extends over the whole electromagnetic spectrum and is generally interpreted as non-thermal radiation arising from particles accelerated in relativistic jets closely aligned to the line of sight [@blandford78]. They come in two flavors: the BL Lac objects, with featureless optical spectra or only with absorption lines of galactic origin and weak and narrower emission lines, and the Flat Spectrum Radio Quasars, with a optical spectra showing broad emission lines. In the following, we indicate the former as BZBs and the latter as BZQs, respectively, according to the ROMA-BZCAT[^1] nomenclature [@massaro09; @massaro10; @massaro11a].
The first step to improve our knowledge on the origin of the UGSs and of their associations with low-energy counterparts, is to recognize those that could have a blazar within their $\gamma$-ray positional uncertainty | 1 | member_27 |
regions.
Recently, we developed a procedure to identify blazars using their infrared (IR) colors within the preliminary data release of the Wide-field Infrared Survey Explorer ([[*WISE*]{}]{}) survey [@wright10] [^2]. In particular, we discovered that the IR color space distribution of the extragalactic sources dominated by non-thermal emission, as blazars, can be used to distinguish such sources from other classes of galaxies and/or AGNs and/or galactic sources [@massaro11b hereinafter Paper I]. We also found that $\gamma$-ray emitting blazar delineate a narrow, distinct region of the IR color-color plots, denominated as the [[*WISE*]{}]{} Gamma-ray blazar Strip ([[*WGS*]{}]{}) [@dabrusco12 hereinafter Paper II]. There is a peculiar correspondence between the IR and $\gamma$-ray spectral properties of the blazars detected in the 2FGL (Paper II). Then, on the basis of our previous investigation of these IR-$\gamma$-ray properties of blazars, we built a parametrization of the [[*WGS*]{}]{} to evaluate how many AGNs of Uncertain type (AGUs) have a counterpart associated with a $\gamma$-ray blazar candidate in the 2FGL [@massaro12a hereinafter Paper III].
In this paper, we present a new association method based on the IR colors of the $\gamma$-ray emitting blazars and the [[*WGS*]{}]{} parametrization. Then we apply this new association procedure to search for $\gamma$-ray | 1 | member_27 |
blazar candidates within the $\gamma$-ray positional error regions of the UGSs. One of the main advantages of our method is that it reduces the number of potential counterparts for the UGSs and provides their positions with arcsec resolution, thus restricting the search regions for future followup observations necessary to confirm their blazar nature. Unfortunately, only a restricted number of UGSs falls within the portion of the sky currently covered by the IR observations of the [[*WISE*]{}]{} Preliminary Data Release corresponding to $\sim$ 57% of the whole sky. Then, when the [[*WISE*]{}]{} survey will be completely released in March 2012[^3], it will be possible to apply the method to the whole sky, even in regions not covered at radio, optical and X-ray frequencies, where the other methods for establishing counterpart associations for the 2FGL cannot be used.
This paper is organized as follows: in Section \[sec:sample\] we describe the samples used in our investigation; in Section \[sec:method\] we illustrate the new association method; then, in Section \[sec:ugs\] we apply the new association technique to the UGSs and describe the subset of sources that has been associated with $\gamma$-ray blazar candidates. In Section \[sec:comparison\], we also compare our results with those found | 1 | member_27 |
adopting different statistical approaches for a subsample of UGSs. Finally, conclusions are presented in Section \[sec:summary\].
The sample selection {#sec:sample}
====================
To build our association procedure we considered a sample of blazars selected from the combination of the ROMA-BZCAT [@massaro09; @massaro10] and the 2FGL [@abdo11], as described and used in Paper II and used in Paper III to parametrize the [[*WGS*]{}]{}, denoted the 2FB sample. It contains 284 $\gamma$-ray blazars (135 BZBs and 149 BZQs) that have optical and radio counterparts as reported in the ROMA-BZCAT, and also having a [[*WISE*]{}]{} counterpart within 2.4 $^{\prime\prime}$ radius (see Paper I and III). The blazars in the 2FB sample are detected by [[*WISE*]{}]{} with a signal to noise ratio higher than 7 in at least one band and do not have any upper limits in all the [[*WISE*]{}]{} bands. We excluded from our analysis all the blazars with a [[*Fermi*]{}]{} analysis flag, according to the 2FGL and the 2LAC [@abdo11; @ackermann11a]. The blazars of uncertain type (BZUs) have been excluded from our analysis, while the BL Lac candidates have been considered as BZBs. More details on the 2FB sample and the source selections are given in Papers II and III.
Then, we | 1 | member_27 |
applied our association procedure to the sample of the UGS defined as follows. The number of UGSs in the 2FGL is 576, but only 410 of these $\gamma$-ray sources lie in the region of the sky available in the [[*WISE*]{}]{} Preliminary Data Release. These sources can be analyzed according to our method based on the IR [[*WISE*]{}]{} colors. We adopted a more conservative selection restricting our sample to 313 UGSs out of 410, excluding sources with a [[*Fermi*]{}]{} analysis flag, since these sources might not be real and/or could be affected by analysis artifacts [see e.g. @abdo11 for more details].
The [[*WGS*]{}]{} association method {#sec:method}
====================================
In Paper III, working on the AGUs, we built the [[*WGS*]{}]{} parametrization to verify if the low-energy counterparts of the AGUs, associated in 2FGL, is consistent with the [[*WGS*]{}]{}, so being a $\gamma$-ray blazar candidate. With respect to the previous analysis, the following proposed association procedure aims at providing new $\gamma$-ray blazar candidates, possible counterparts of the UGSs, that lie within their $\gamma$-ray positional uncertainty regions, on the basis of our previous results on the IR-$\gamma$-ray blazar properties. In this Section, we report the basic details of our [[*WGS*]{}]{} parametrization together with the definition | 1 | member_27 |
of different classes of $\gamma$-ray blazar candidates. Then we describe our new association procedure.
The [[*WGS*]{}]{} parametrization {#sec:parameter}
---------------------------------
In Paper II, we found that $\gamma$-ray emitting blazars (i.e., those in the 2FB sample) cover a narrow region in the 3D color space built with the [[*WISE*]{}]{} magnitudes delineating the so-called [[*WISE*]{}]{} Gamma-ray blazar Strip ([[*WGS*]{}]{}).
In Paper III, using the 2FB sample, we presented the parametrization of the [[*WGS*]{}]{} based on the [*strip parameter*]{} $s$. This parameter, ranging between 0 and 1, provides a measure of the distance between the [[*WGS*]{}]{} and the location of a [[*WISE*]{}]{} source in the three dimensional IR color parameter space. For example, sources with high values of $s$ (e.g., $\geq$ 0.50) are consistent with the [[*WGS*]{}]{}. We also distinguished between [[*WISE*]{}]{} sources that lie in the subregion of the [[*WGS*]{}]{}occupied by the BZBs and BZQs using the $s_b$ and $s_q$ parameters separately (Paper III).
The IR color space has been built using the archival data available in the 2011 [[*WISE*]{}]{} Preliminary Data Release, in four different bands centered at 3.4, 4.6, 12, and 22 $\mu$m with an angular resolution of 6.1, 6.4, 6.5 & 12.0$^{\prime\prime}$, respectively and achieving 5$\sigma$ point source sensitivities of | 1 | member_27 |
0.08, 0.11, 1 and 6 mJy. In addition, the absolute (radial) differences between [[*WISE*]{}]{} source-peaks and “true" astrometric positions anywhere on the sky are no larger than $\sim$ 0.50, 0.26, 0.26, and 1.4$^{\prime\prime}$ in the four [[*WISE*]{}]{} bands, respectively [@cutri11][^4].
$\gamma$-ray blazar candidate definition {#sec:association}
----------------------------------------
Based on the $s_b$ and $s_q$ distributions of all [[*WISE*]{}]{} sources in different random regions of the sky, at both high and low Galactic latitudes (Paper III), the critical threshold of the $s$ parameters, used to define the above classes, have been arbitrarily determined on the basis of the following considerations:
- [class A: [[*WISE*]{}]{} sources with 0.24 $<s_b<$ 1.00 and 0.38 $<s_q<$ 1.00;]{}
- [class B: [[*WISE*]{}]{} sources with 0.24 $<s_b<$ 1.00 or 0.38 $<s_q<$ 1.00;]{}
- [class C: [[*WISE*]{}]{} sources with 0.10 $<s_b<$ 0.24 and 0.14 $<s_q<$ 0.38.]{}
All the [[*WISE*]{}]{} sources with $s_b<$0.10 or $s_q<$0.14 are considered [*outliers*]{} of the [[*WGS*]{}]{} and, for this reason, discarded. All the above thresholds are then used to select the [[*WISE*]{}]{} sources that are associated to the UGSs and that can be considered potential $\gamma$-ray blazar candidates.
The above choice of threshold have been adopted for the analysis of the $\gamma$-ray blazar content within the | 1 | member_27 |
AGUs (Paper III). From the distributions of the $s_b$ and $s_q$ parameters for the generic IR [[*WISE*]{}]{} sources, we note that 99.9% of them have $s_b<$0.24 and $s_q<$0.38. Then, for the BZBs in the 2FB sample only 6 sources out of 135 have $s_b<$ 0.24, and in the case of the BZQs only 33 sources out of 149 show $s_q$ values lower than 0.38. We also note that 99.0% of the generic IR [[*WISE*]{}]{} sources have $s_b<$0.10 and only 2 BZBs are below this value, while 97.2% of the generic IR [[*WISE*]{}]{} sources together with only 5 BZQs out of 149 have $s_q<$0.14.
The [[*WISE*]{}]{} objects of class A are the most probable blazar counterpart of the unidentified $\gamma$-ray sources, because their WISE colors are more consistent with the [[*WGS*]{}]{} in both the BZBs and BZQs subregions than the colors of sources of class B or C. Based on the distributions of the $s_b$ and $s_q$ parameters for [[*WISE*]{}]{} sources in random region of the sky, the sources of class A are, as expected, rarer than the sources belonging to the other two classes (see Section \[sec:ugs\] for more details).
The association procedure {#sec:procedure}
-------------------------
![The position of the region | 1 | member_27 |
of comparison (ROC) for a generic [[*Fermi*]{}]{} source, with respect to the searching region (SR) centered on the position reported in the 2FGL catalog. The radius of both regions is $R=\theta_{999}$ and they are separated by 2.5$\sqrt{2}$ deg of distance.[]{data-label="fig:roi"}](./roi.pdf){height="6.8cm" width="9.5cm"}
For each unidentified $\gamma$-ray source we defined the [*searching region*]{} (SR) corresponding to a circular region of radius $R$=$\theta_{999}$, centered on the position given in the 2FGL, where $\theta_{999}$ is the major axis of the elliptical source location region corresponding to the 99.9% level of confidence. In addition, we also considered a [*region of comparison*]{} (ROC) defined as a circular region of the same radius $R$, but lying at 2.5$\sqrt{2}$ deg angular distance from the 2FGL position. A schematic view of the locations of the SR and the ROC is shown in Figure \[fig:roi\].
Successively, for every unassociated gamma-ray source in the 2FGL catalog, we ranked all the [[*WISE*]{}]{} sources within its SR on the basis of the classification described above and we selected as $\gamma$-ray blazar candidates the positionally closest sources with the highest class. In our analysis we considered only sources of the [[*WISE*]{}]{} preliminary catalog detected in all the four [[*WISE*]{}]{} bands, without any upper limit.
| 1 | member_27 |
The ROCs are used to assess the association confidence that a [[*WISE*]{}]{} source in a random region in the sky, where no $\gamma$-ray source is located, has IR colors compatible with the WGS. To provide an estimate of the association confidence, we considered the distribution of the strip parameters $s_b$ and $s_q$ for all the [[*WISE*]{}]{} sources within each ROC associated to an UGS. For these [[*WISE*]{}]{} sources we estimated the confidence $\pi$ that a generic [[*WISE*]{}]{} source belongs to the same class as the $\gamma$-ray blazar candidate selected within the SR. Thus the $\pi$ value will be expressed as the ratio between the number of [[*WISE*]{}]{} sources of a particular class and the total number of [[*WISE*]{}]{} sources that lie in the ROC.
Testing the association method with blazars {#sec:test}
-------------------------------------------
We performed a test to evaluate the completeness of our association method searching for the $\gamma$-ray blazar candidates that are potential counterparts of the 2FB sample, and verifying whether our procedure correctly finds the same associations as in the 2FB sample.
Assuming that the 284 blazars in the 2FB sample have been associated to the real low-energy counterparts, we run our association procedure considering the IR colors for | 1 | member_27 |
all the [[*WISE*]{}]{} sources within the SRs for all these sources. We found that for the population of BZBs, consisting of 135 BL Lacs, our association procedure is able to recognize 123 sources as the 2FGL, 62 of class A, and 61 of class B. Within the remaining 12 BZBs, 3 objects are associated to WISE sources of higher class than the original 2FGL associated sources, while for 9 sources we only found outliers of the [[*WGS*]{}]{} within their SRs.
For the BZQs, our method finds the same associations as in the 2FGL catalog for 124 of the sources, with 85 sources classified as class A, 32 classified as class B and 7 as class C. For the remaining 25 sources, we found 11 outliers and 14 $\gamma$-ray sources associated to a [[*WISE*]{}]{} source with higher classes.
Our procedure re-associates 247 out of 284 $\gamma$-ray blazars of the 2FB sample in agreement with the 2FGL analysis, with a completeness of 87.0% (91.0% for the BZBs and 83.0% for the BZQs). We found that 7.1% are outliers of the [[*WGS*]{}]{}, but this number can be expected because the [[*WGS*]{}]{} parametrization was built to require at least 90% of the 2FB sources | 1 | member_27 |
inside each 2-dimensional [[*WGS*]{}]{} projection (see Paper III for more details).
It is interesting to note that 17 out of 284 $\gamma$-ray sources in the 2FGL have a “better", on the basis of our method, $\gamma$-ray blazar candidate within the SR. These associations need to be verified with followup observations, as for example in the X-rays, and a deeper analysis to check their reliability relative to the 2FGL association method will be performed in a forthcoming paper[@massaro12b].
Results {#sec:ugs}
=======
The application of our association procedure to the 313 UGSs selected from the 2FGL (see Section \[sec:sample\] for more details), led to the associations of 156 UGSs with a low-energy candidate $\gamma$-ray blazar counterpart within their SRs. According to our criteria (see Section \[sec:association\]), these 156 new associations consist of 44 sources of class A, 74 of class B and 38 of class C. Thus our procedure finds associations with likely $\gamma$-ray blazar candidates for 49.8% of the UGSs analyzed. We also list of all the $\gamma$-ray blazar candidates with lower class for each UGSs, if more than one is present within the SRs. Among these 156 new associations, for 86 sources, 12 of class A, 43 of class B | 1 | member_27 |
and 31 of class C, have only a single $\gamma$-ray blazar candidate within the SR. In Figure \[fig:strip\_pln1\] we show the [[*WISE*]{}]{} colors of the 156 $\gamma$-ray blazar candidates in comparison with those of the blazars in the 2FB sample for the \[3.4\]-\[4.6\]-\[12\] $\mu$m 2D projection of the [[*WGS*]{}]{}.
![The \[3.4\]-\[4.6\]-\[12\] $\mu$m 2D projection of the [[*WGS*]{}]{} is shown. Red dashed lines show the boundaries of the [[*WGS*]{}]{} used in our analysis (see Paper III for more details). The orange background filled circles are the blazars associated with the 2FGL constituting the 2FB sample while the balck filled circles indicate the 156 $\gamma$-ray blazars that have been associated by our procedure.[]{data-label="fig:strip_pln1"}](./strip_pln1.pdf){height="6.0cm" width="9.7cm"}
By restricting our sample of UGSs only to those at high Galactic latitudes, i.e. $|b|>$15$^\circ$, we found a $\gamma$-ray blazar candidate for 72 UGSs, 16 of class A, 29 of class B and 27 of class C; where for 34 out of these 74, the low energy counterpart associated with our method is univocal.
![The distribution of the Galactic latitude for all the UGSs analyzed in comparison with that for the 156 associated by our procedure.[]{data-label="fig:glat_distrib"}](./glat_distrib.pdf){height="6.0cm" width="9.7cm"}
In Figure \[fig:glat\_distrib\], we shown the distribution of the Galactic latitude | 1 | member_27 |
(i.e., sin $b$ ) for all the UGSs analyzed in comparison with those 156 associated by our method. At high Galactic latitude, the method seems to be less efficient given the ratio between the number of UGSs analyzed and those associated. This could be due to the non uniform exposure of the archival [[*WISE*]{}]{} observations in the [[*WISE*]{}]{} Preliminary Data Release[^5], and will be re-analyzed once the whole [[*WISE*]{}]{} archive will be available. In addition, we note that our association method could be more efficient at low Galactic latitudes where the blazar catalogs, as the ROMA-BZCAT, are less complete [@massaro09].
We also remark that within the 313 regions of comparison chosen for the UGSs there are 55195 [[*WISE*]{}]{} sources, but only 49 of class A, 213 of class B and 129 of class C, all of them detected in all four [[*WISE*]{}]{} bands and with a signal to noise ratio higher than seven in at least one band, as the blazars in the 2FB sample. The distributions of the $s_b$ and $s_q$ parameters for all the 55195 [[*WISE*]{}]{} sources within the 313 ROCs are shown in Figure \[fig:histogram\]. A blind search of all the possible $\gamma$-ray blazar candidates in the | 1 | member_27 |
[[*WISE*]{}]{} archive on the basis of the [[*WGS*]{}]{} properties will be performed once it will be completely available [@massaro12b]. However, the $s_b$ and $s_q$ distributions reported in Figure \[fig:histogram\] strongly suggest that the density of [[*WISE*]{}]{} blazar candidates is low over the sky.
![The distribution of the $s_b$ (black) and $s_q$ (red) parameters for all the 55195 [[*WISE*]{}]{} source within all the ROCs defined for the 313 UGSs analyzed. The vertical lines corresponds to the thresholds for the $s_b$ and $s_q$ parameters to determine the blazar classes (see Section \[sec:association\]).[]{data-label="fig:histogram"}](./histogram.pdf){height="6.0cm" width="9.7cm"}
In Table \[tab:example\] we show three cases of [[*WISE*]{}]{} sources that have been associated with our procedure to three UGSs. We report both the $s_b$ and $s_q$ values, the [[*WGS*]{}]{} class and the association confidence $\pi$. In this example, the source 2FGL J0038.8+6259 is associated to one [[*WISE*]{}]{} source of a class A, J003818.70+630605.0, that has been selected as a single $\gamma$-ray blazar candidate out of 791 [[*WISE*]{}]{} sources within its SR. The corresponding association confidence $\pi$, expressed in terms of number of sources with an higher $s_b$ or $s_q$ values than J003818.70+630605.0 within the region of comparison and estimated considering 830 [[*WISE*]{}]{} sources, is 2/830.
Similarly, the source | 1 | member_27 |
2FGL J0616.6+2425 has been associated to the [[*WISE*]{}]{} source J061609.79+241911.0, that belongs to class B with a low association confidence estimated on 5465 [[*WISE*]{}]{} sources in the region of comparison. The 2FGL source 2FGL JJ0312.8+2013 has a [[*WISE*]{}]{} class C source associated following our procedure, with a lower confidence of finding a similar source in a region of comparison where there are 512 [[*WISE*]{}]{} sources.
Within the 313 UGS analyzed there are 14 sources that have a variability index [@abdo11] higher than the value of 41.6 corresponding to the 99% of confidence that the source is variable. It is worth noting that 13 out of these 14 variable UGSs have been successfully associated here with a $\gamma$-ray blazar candidate, strongly supporting the blazar nature.
[|lrlcccc|]{} 2FGL & Sources & [[*WISE*]{}]{} & $s_b$ & $s_q$ & class & $\pi$\
name & in SR & name & & &\
J0038.8+6259 & 791 & J003818.70+630605.2 & 0.89 & 0.99 & A & 2/830\
J0616.6+2425 & 6021 & J061623.95+241809.2 & — & 0.57 & B & 1/5465\
J0312.8+2013 & 453 & J031223.00+200749.5 & 0.19 & 0.15 & C & 1/512\
\[tab:example\]
The entire list of the UGSs analyzed can be found in Table 2. | 1 | member_27 |
For each UGS, we report all the $\gamma$-ray blazar candidates with their IR colors (i.e., $c_{12}$ = \[3.4\]-\[4.6\] $\mu$m, $c_{23}$ = \[4.6\]-\[12\] $\mu$m and $c_{34}$ = \[12\]-\[22\] $\mu$m, together with their errors, $\sigma_{12}$, $\sigma_{23}$, $\sigma_{34}$, respectively), the distances in arc seconds between the $\gamma$-ray position and the selected [[*WISE*]{}]{} source, the $s_b$ and $s_q$ values, the class and the association confidence $\pi$ that there is a [[*WISE*]{}]{} source of the same class within the ROC (see Section \[sec:association\]).
In addition, we found that there are 157 unidentified $\gamma$-ray sources that do not have clear $\gamma$-ray blazar counterpart within their SRs and are classified as outliers of the [[*WGS*]{}]{}. The lack of association for these sources could be due to a lower accuracy of the $\gamma$-ray position that might occur close to the Galactic plane or to the systematic uncertainties of the diffuse emission model used in the 2FGL analysis. The whole UGS sample will be reconsidered for associations with $\gamma$-ray blazar candidates when the all-sky [[*WISE*]{}]{} survey will be available.
Assuming that all the 2FB blazar associations are correct, on the basis of our test (see Section \[sec:test\]), we can argue that within our sample we would expect about 41 | 1 | member_27 |
($\sim$ 13.0%) not recognized low-energy counterparts, for a total of 197 $\gamma$-ray blazar candidates within the 313 UGSs analyzed.
Finally, it is worth stressing that our association procedure provides also interesting information on the sources that do not have a $\gamma$-ray blazar candidates in the SR. The absence of $\gamma$-ray blazar candidates selected according to our association procedure could direct to better use the follow-up resources for identifying other $\gamma$-ray source candidates. For example in the case of the unidentified $\gamma$-ray source: 2FGL J1446.8$-$4701 within the 1604 [[*WISE*]{}]{} sources that lie in its SR, we did not find any $\gamma$-ray blazar candidates. This source has been recently identified with the pulsar PSR 1446-4701 (see Public List of LAT-Detected Gamma-Ray Pulsars) [^6].
Comparison with other methods {#sec:comparison}
=============================
We note that among the 313 UGSs analyzed, there are 70 sources that were also unidentified according to the investigation performed in the first Fermi $\gamma$-ray catalog (1FGL), and 48 of them have been associated with a $\gamma$-ray blazar candidates in our analysis. In particular, a recent analysis of the 1FGL unidentified $\gamma$-ray sources has been carried out using two different statistical approaches: the Classification Tree and the Logistic regression analyses [see @ackermann11b | 1 | member_27 |
and references therein].
For 44 out of the 48 UGSs, that have been analyzed on the basis of the above statistical methods, it is also possible to perform a comparison with our results to verify if the 2FGL sources that we associated to a $\gamma$-ray blazar candidates have been also classified as AGNs following the Ackermann et al. (2011b) procedures. By comparing the results of our association method with those in Ackermann et al. (2011b), we found that 27 out of 44 UGSs that we associate to a $\gamma$-ray blazar candidate are also classified as AGNs, all of them with a probability higher than 71% and 18 of them higher than 80%. Among the remaining 17 out of 44 sources, 7 have been classified as pulsars, with a very low probability with respect to the whole sample; in particular, 3 of these pulsar candidates are classified with a probability lower than 41% and all of them lower than 71%, making these classifications less reliable than those of the AGNs. The last 10 UGSs did not have a classification in Ackermann et al. (2011b). Consequently, we emphasize that our results are in good agreement with the classification suggested previously by Ackermann | 1 | member_27 |
et al. (2011b) consistent with the $\gamma$-ray blazar nature of the [[*WISE*]{}]{} candidates proposed in our analysis.
Summary and Conclusions {#sec:summary}
=======================
Recently, we discovered that blazars have peculiar mid-IR colors with respect to other galactic sources or different classes of AGNs. In particular, we found that within the 3-dimensional IR parameter space they delineate a distinct, well-defined, region known as [[*WISE*]{}]{} Blazar Strip (Paper I). Moreover, this distinction, mostly due to the non-thermal emission that dominates the IR radiation of blazars, appears to be more evident when considering those blazars selected on the basis of their $\gamma$-ray properties (Paper II) so defining the [[*WISE*]{}]{} Gamma-ray blazar Strip ([[*WGS*]{}]{}). Then, in Paper III, we built the [[*WGS*]{}]{} parametrization to test the consistency of the low energy counterpart of the AGUs, associated in 2FGL with the [[*WGS*]{}]{}.
On the basis of these results, in the present work, we developed a new association method to search for blazar counterparts of $\gamma$-ray sources and we applied this method to the blazars of the 2FGL sample. We also provide new $\gamma$-ray blazar candidates, potential counterparts of the UGSs, that lie within their $\gamma$-ray positional error region, having the same mid-IR colors as the $\gamma$-ray | 1 | member_27 |
blazars already associated. We also tested our new procedure [*a posteriori*]{} trying to re-associate all the blazars in the 2FB sample and we found that our results are in good agreement with different association procedures.
The application of our association procedure to the UGSs has led to the selection of possible blazar counterparts for 156 of 313 UGSs analyzed.
As also noted in Section \[sec:ugs\], our association procedure provides also interesting information on the sources that do not have a $\gamma$-ray blazar candidates in the SRs as the case of the unidentified $\gamma$-ray source: 2FGL J1446.8$-$4701, recently identified with the pulsar PSR 1446-4701.
Several developments will be considered to improve our association procedure, such as taking into account not only the IR colors, correspondent to flux ratios, but also the IR fluxes as well as the IR-$\gamma$-ray spectral index correlation (Paper II) and the sky distribution of the $\gamma$-ray blazar candidates, once the whole [[*WISE*]{}]{} data archive will be released. Then, it will be also possible to calibrate our association procedure choosing the different thresholds for the $s$ parameters at different Galactic latitudes to take into account of the [[*WISE*]{}]{} background.
Moreover, our association method is complementary to those adopted | 1 | member_27 |
in the 2FGL catalog analysis, because it is based on different multifrequency information. For this reason, these methods could be in principle combined to increase the fraction of associated UGSs and the efficiency of the association. Further developments of this new association method will be investigated in a forthcoming paper [@massaro12b].
We thank the anonymous referee for the his/her comments. F. Massaro is grateful S. Digel for their fruitful discussions for all the comments helpful toward improving our presentation. We also thank to A. Cavaliere, D. Harris, J. Grindlay, J. Knodlseder, P. Giommi, N. Omodei, H.Smith and D. Thompson for their suggestions. The work at SAO and at Stanford University is supported in part by the NASA grant NNX10AD50G, NNH09ZDA001N and NNX10AD68G. R. D’Abrusco gratefully acknowledges the financial support of the US Virtual Astronomical Observatory, which is sponsored by the National Science Foundation and the National Aeronautics and Space Administration. F. Massaro acknowledges the Fondazione Angelo Della Riccia for the grant awarded him to support his research at SAO during 2011 and the Foundation BLANCEFLOR Boncompagni-Ludovisi, n’ee Bildt for the grant awarded him in 2010 to support his research. TOPCAT[^7] [@taylor2005] was used extensively in this work for the preparation | 1 | member_27 |
and manipulation of the tabular data. Part of this work is based on archival data, software or on-line services provided by the ASI Science Data Center. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
Abdo, A. A. et al. 2010a ApJ, 720, 435 Abdo, A. A. et al. 2010b ApJS 188 405 Abdo, A. A. et al. ApJS submitted http://arxiv.org/abs/1108.1435 Ackermann, M. et al. 2011a ApJ, 743, 171 Ackermann, M. et al. 2011b ApJ submitted http://arxiv.org/abs/1108.1202 Blandford, R. D. & Rees, M. J., 1978, Proc. “Pittsburgh Conference on BL Lac objects", 328 Cutri, R. M. 2011, wise.rept, 1 D’Abrusco, R., Massaro, F., Ajello, M., Grindlay, J. E., Smith, Howard A. & Tosti, G. 2012 ApJ accepted Hartman, R.C. et al., 1999 ApJS 123 Laurino, O. & D’Abrusco 2011 MNRAS in press Massaro, E., Giommi, P., Leto, C., Marchegiani, P., Maselli, A., Perri, M., Piranomonte, S., Sclavi, S. 2009 A&A, 495, 691 Massaro, E., Giommi, P., Leto, C., Marchegiani, P., Maselli, A., Perri, M., Piranomonte, S., Sclavi, S. | 1 | member_27 |