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However. about 60% of all the neutral diffuse gas. a much higher fraction than that of reported earlier. has temperature below 500 K. The CNM temperature distribution shows a clear peak near T 50 K and the cloud size for the neutral ISM shows a bi-modal statistical distribution. | However, about $60\%$ of all the neutral diffuse gas, a much higher fraction than that of reported earlier, has temperature below $500$ K. The CNM temperature distribution shows a clear peak near T $ \sim 50$ K and the cloud size for the neutral ISM shows a bi-modal statistical distribution. |
Derived magnetic field from the non-thermal velocity dispersion matches. within a factor of 2. with the magnetic field value estimated from the Zeeman splitting measurements. | Derived magnetic field from the non-thermal velocity dispersion matches, within a factor of 2, with the magnetic field value estimated from the Zeeman splitting measurements. |
The Kolmogorov-like scaling is consistent with the existing theoretical prediction. numerical simulations and earlier observational results. | The Kolmogorov-like scaling is consistent with the existing theoretical prediction, numerical simulations and earlier observational results. |
This research has made use of the data from the millennium Arecibo 21 em absorption-line survey measurements and NASA's Astrophysics Data System. | This research has made use of the data from the millennium Arecibo 21 cm absorption-line survey measurements and NASA's Astrophysics Data System. |
We are grateful to Rajaram Nityananda and Nissim Kanekar for their comments on an earlier version of the Letter. | We are grateful to Rajaram Nityananda and Nissim Kanekar for their comments on an earlier version of the Letter. |
We thank K. Subramanian. D. J. Saikia and R. Srianand for useful discussions. | We thank K. Subramanian, D. J. Saikia and R. Srianand for useful discussions. |
One of the authors (LP) would like to acknowledge the hospitality of all the staff members of the Nationa Centre for Radio Astrophysics (NCRA) during her stay for the Visiting Student Research Programme (2007). | One of the authors (LP) would like to acknowledge the hospitality of all the staff members of the National Centre for Radio Astrophysics (NCRA) during her stay for the Visiting Student Research Programme (2007). |
We are gratefu to the anonymous referee for prompting us into substantially improving this paper. | We are grateful to the anonymous referee for prompting us into substantially improving this paper. |
This research was supported by the Nationa Centre for Radio Astrophysics of the Tata Institute of Fundamenta Research (ΤΕΕ). | This research was supported by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research (TIFR). |
Equations for (he maxinun proper motions corresponding to this peculiar acceleration expressed in|. are obtained by dividing equations (19)) (21)) by 145-10.. which is a conversion [actor of units used in the calculation. | Equations for the maximum proper motions corresponding to this peculiar acceleration expressed in, are obtained by dividing equations \ref{a_pec.eq}) \ref{a_pec.eq3}) ) by $1.45\cdot 10^{-6}$, which is a conversion factor of units used in the calculation. |
We find that the residual proper motion fieldl caused by the peculiar acceleration of the solar svstem will respect to the LSR. is smaller in amplitude than 1.F.. that is about 10 times smaller than the secular aberration (??)) caused by the galactocentrie acceleration of the ΓΗ shown in equation (??)). | We find that the residual proper motion field caused by the peculiar acceleration of the solar system with respect to the LSR is smaller in amplitude than 1, that is about 10 times smaller than the secular aberration \ref{hb}) ) caused by the galactocentric acceleration of the LSR shown in equation \ref{ha}) ). |
A global discrete pattern of proper motions is a vector field j((/.b) on the celestial sphere which can be expanded in orthogonal vector Auctions of spherical coordinates sU.)]. where T, and Ss; are orthogonal vector harmonics. called magnetic and electric harmonics respectivelv (Thorne1980).. and /; and 5; are the coellicients of the expansion. | A global discrete pattern of proper motions is a vector field $\vec\mu(l,b)$ on the celestial sphere which can be expanded in orthogonal vector functions of spherical coordinates ], where $\vec{T_j}$ and $\vec{S_j}$ are orthogonal vector harmonics, called magnetic and electric harmonics respectively \citep{thorne}, and $t_j$ and $s_j$ are the coefficients of the expansion. |
The vector harmonics T, and S, can be expressed in terms of the partial derivatives of the scalar spherical harmonics V; (Arfken&Weber1995). with respect to galactic longitude and latitude. | The vector harmonics $\vec{T_j}$ and $\vec{S_j}$ can be expressed in terms of the partial derivatives of the scalar spherical harmonics $V_j$ \citep{arf} with respect to galactic longitude and latitude. |
specifically. one has (Vitvazev&Shuksto2004) where C';=ο ave constant normalization coefficients making the harmonics orthonormal. and the cumulative index j={rin} counts the spherical harmonics V;=Ἐν over orders n—0.1.....x: and degrees m—0.1.....n. | Specifically, one has \citep{vit}
where $C_j\equiv C_{nm}$ are constant normalization coefficients making the harmonics orthonormal, and the cumulative index $j=\{nm\}$ counts the spherical harmonics $V_j\equiv V_{nm}$ over orders $n=0,1,\ldots,\infty$ and degrees $m=0,1,\ldots,n$. |
We do not specily the normalization constants C'; because real observations will provide a vector field of proper motions sampled at a number of discrete points on the celestial sphere corresponding to the observed. quasars. | We do not specify the normalization constants $C_j$ because real observations will provide a vector field of proper motions sampled at a number of discrete points on the celestial sphere corresponding to the observed quasars. |
clusters to be detected over the entire range of cluster magnitudes. providing a sample that is elfectively complete. | clusters to be detected over the entire range of cluster magnitudes, providing a sample that is effectively complete. |
No trend between (he mass and metallicity is observed for the blue elobular cluster population. suggesting that the previously published results are influenced strongly by the much lower signal to noise in that data. | No trend between the mass and metallicity is observed for the blue globular cluster population, suggesting that the previously published results are influenced strongly by the much lower signal to noise in that data. |
Section 2. discusses the data and the photometric reduction used (to ensure accurate cluster measurements. | Section \ref{sec: obs} discusses the data and the photometric reduction used to ensure accurate cluster measurements. |
Section 3. reviews (he IXMM algorithm used to determine the peaks of the color distributions. aud (he sensitivity of (he results to changes in the binning methods. | Section \ref{sec: kmm} reviews the KMM algorithm used to determine the peaks of the color distributions, and the sensitivity of the results to changes in the binning methods. |
Finally. section 4 presents (he results for our sample of clusters. and section 5. discusses these results in the context of the formation history of the globular cluster svstenm. | Finally, section \ref{sec: results}
presents the results for our sample of clusters, and section \ref{sec:
discussion} discusses these results in the context of the formation history of the globular cluster system. |
The data for this project come from a 50 orbit observation program with the Advanced Camera lor Survevs (ACS) aboard HST (PI: Daltz. Proposal ID: 10543). | The data for this project come from a 50 orbit observation program with the Advanced Camera for Surveys (ACS) aboard HST (PI: Baltz, Proposal ID: 10543). |
The images are of the central region of the eiant. elliptical galaxy M87. extending out (o a projected radius of 8 kpc (taking i»—M=31.021 Macriοἱal. (1999))). | The images are of the central region of the giant elliptical galaxy M87, extending out to a projected radius of 8 kpc (taking $m - M = 31.021$ \citet{Macri}) ). |
The data were taken over the course of three months. as part of a search for microlensing events. which require repeat visits to lind changes in brightness. | The data were taken over the course of three months, as part of a search for microlensing events, which require repeat visits to find changes in brightness. |
These multiple images of the same field vield data that can be combined into very. deep exposures. | These multiple images of the same field yield data that can be combined into very deep exposures. |
On each observing cay. four exposures in FRl4W were taken with dithered pointing along with a single matching exposure in F606W. The four F814W images provide a fully dithered image for each day. with the F606W vielding full coverage over (he entire set of observations. | On each observing day, four exposures in F814W were taken with dithered pointing along with a single matching exposure in F606W. The four F814W images provide a fully dithered image for each day, with the F606W yielding full coverage over the entire set of observations. |
The images were combined using Multidrizzle (Fruchter&Hook2002) to a resolution of 0705 !. the nominal resolution of ACS, | The images were combined using Multidrizzle \citep{FH} to a resolution of $0\farcs05$ $^{-1}$, the nominal resolution of ACS. |
Although the cither pattern for this dataset is more than sufficient to allow higher resolution final images to be constructed. this is not necessary for the identification and photometry of the globular clusters. | Although the dither pattern for this dataset is more than sufficient to allow higher resolution final images to be constructed, this is not necessary for the identification and photometry of the globular clusters. |
At the resolution used. the globular clusters are significantly broader than the ACS PSF. so retaining the nominal ACS resolution provides the highest possible signal to noise. | At the resolution used, the globular clusters are significantly broader than the ACS PSF, so retaining the nominal ACS resolution provides the highest possible signal to noise. |
In all. 49 F606W and 205 FSI4W images were combined to vield final images with exposure (mes of /y=24500 8 and /,;=T3800 s. making these some of the deepest images ever (aken wilh HST. | In all, 49 F606W and 205 F814W images were combined to yield final images with exposure times of $t_V = 24500$ s and $t_I = 73800$ s, making these some of the deepest images ever taken with HST. |
In addition to (he exposures used. 8 FGOGW and 13 ES14W were taken but exeluded [rom analysis clue (ο a loss in the guide star (racking. | In addition to the exposures used, 8 F606W and 13 F814W were taken but excluded from analysis due to a loss in the guide star tracking. |
The final drizzlecl image contains the strongly varving galaxy light. | The final drizzled image contains the strongly varying galaxy light. |
As the main source of noise in the final image is due to the variations in this galaxy lieht from pixel to pixel. constructing an accurate model of (he galaxy is essential to estimating the detection efficiency across (he image. | As the main source of noise in the final image is due to the variations in this galaxy light from pixel to pixel, constructing an accurate model of the galaxy is essential to estimating the detection efficiency across the image. |
We use a model of the galaxy. determined [rom isophote fitting. but to optimize (his fit. it is advantageous to remove sources other than the galaxy light. | We use a model of the galaxy determined from isophote fitting, but to optimize this fit, it is advantageous to remove sources other than the galaxy light. |
In addition | In addition |
as being listed in Table 1. | as being listed in Table 1. |
To numerically simulate this system we use a svmplectie integrator (Wisdom&Llolman1991:Mikkola&Palmer 2000).. which allows us to follow the orbital evolution of each body. and. simultaneously the stability of this orbit indicated by the Lyapunov Character Inclicator (LCL) (Proceschlé1984). | To numerically simulate this system, we use a symplectic integrator \cite{wis91,mik00}, which allows us to follow the orbital evolution of each body and simultaneously the stability of this orbit indicated by the Lyapunov Character Indicator (LCI) \cite{fro84}. |
. Lach planet orbit. has an LOL ancl we choose the largest one among the three to indicate the stability of this planetary svstem in this paper. | Each planet orbit has an LCI and we choose the largest one among the three to indicate the stability of this planetary system in this paper. |
The time step is set to be 0.3 dd. which is about 2% “the orbital period of the innermost planet. | The time step is set to be $0.3$ d, which is about $2\%$ of the orbital period of the innermost planet. |
We adopt the masses. semimajor axes and eccentricities of planets the values listed in Table 1. | We adopt the masses, semimajor axes and eccentricities of planets the values listed in Table 1. |
Phe initial orbital inclination of companion D is set to be zero. while the companion D and € have initial inclinations of 10. degrees. | The initial orbital inclination of companion D is set to be zero, while the companion B and C have initial inclinations of $10^{-5}$ degrees. |
Other angles (the ascending node. the periastron and. the mean anomaly) are randomly generated from 0.23). | Other angles (the ascending node, the periastron and the mean anomaly) are randomly generated from $[0,2\pi)$. |
Four hundred: simulations. with cifferent initial conditions. are integrated up to LO” vears. | Four hundred simulations, with different initial conditions, are integrated up to $10^6$ years. |
lenoring the companion D. we first integrate a svsten composed. of the central star ancl the companion D and C. Then. retaining the initial conditions of these three »odies and adding the companion D on an initial orbit with 10 given α.ο.{ and randomly. selected angle elements. we integrate the four-body svstem. | Ignoring the companion D, we first integrate a system composed of the central star and the companion B and C. Then, retaining the initial conditions of these three bodies and adding the companion D on an initial orbit with the given $a,e,i$ and randomly selected angle elements, we integrate the four-body system. |
As the simulations show. 10 companion D has nearly no influence on the motion of 1f inner planets. although it is much heavier. | As the simulations show, the companion D has nearly no influence on the motion of the inner planets, although it is much heavier. |
This is due ο its large semimagjor axis (5.461 aau). | This is due to its large semimajor axis $5.461$ au). |
However. in order to get reliable simulations for the real system. we report here 1e numerical results from the four-body mocel. | However, in order to get reliable simulations for the real system, we report here the numerical results from the four-body model. |
During the integrations. if the distance between any wo of the planets became smaller than half of the criterion for the “Lill stability” (Clacman1993).. the svstem was considered. become collapsed and the simulation was erminated. | During the integrations, if the distance between any two of the planets became smaller than half of the criterion for the “Hill stability” \cite{gla93}, the system was considered become collapsed and the simulation was terminated. |
Generally. the motion of a planet. is dominated. by he central star and its orbit is a conic section with small deviations due to the gravitational perturbation from other planet(s). | Generally, the motion of a planet is dominated by the central star and its orbit is a conic section with small deviations due to the gravitational perturbation from other planet(s). |
This perturbation can be described. by hefunction. which can be expanded. in terms of the orbital elements. | This perturbation can be described by the, which can be expanded in terms of the orbital elements. |
We use a.c.zc.Q0.A to denote he semi-major axis. eccentricity. inclination. Longitucle of periastron. longitude of ascending node. and. mean ongitude. respectively. | We use $a,e,i,\varpi,\Omega,\lambda$ to denote the semi-major axis, eccentricity, inclination, longitude of periastron, longitude of ascending node, and mean longitude, respectively. |
The disturbing function for a planet with mass m, and orbital elements. τν204.4. (61.6041.AL). »erturbed: by another planet. (indicated by subscripts 72). can be written as Llere fro=Gre and 4 isa linear combination of angles where Jp.οντττνjo are integers satisfving Jj|0. | The disturbing function for a planet with mass $m_1$ and orbital elements $(a_1,e_1,i_1,\varpi_1,\Omega_1,\lambda_1)$ , perturbed by another planet (indicated by subscripts `2'), can be written as Here $\mu_2={\cal G}m_2$ and $\varphi$ is a linear combination of angles where $j_1,j_2,\cdots,j_6$ are integers satisfying $j_1+j_2+\cdots+j_6=0$ . |
Particularly. when the orbital periods satisfy. {νιzm 3. that is. AyBAe&O (dots denote the time derivation). we have νεαον790 since generally cy.5A,Ae. | Particularly, when the orbital periods satisfy $P_2/P_1\approx 3$ , that is, $\dot\lambda_1-3\dot\lambda_2\approx 0$ (dots denote the time derivation), we have $\dot\varphi_{(j_1=1,j_2=-3)}\approx 0$ since generally $\dot\varpi_1, \dot\varpi_2, \dot\Omega_1,
\dot\Omega_2 \ll \dot\lambda_1, \dot\lambda_2$ . |
Then in the disturbing function. 2. those terms containing A,3À» can not be eliminated by the averaging technique and become the leading terms. | Then in the disturbing function $R$, those terms containing $\lambda_1-3\lambda_2$ can not be eliminated by the averaging technique and become the leading terms. |
Ane if the inclinations 7).72 are not too large. the leading terms in 2? are of OCF).O(te3).Olepes) and the corresponding angle ος. now calleLacsonantarguments, are 0;=AyBA»|Ὅτοι. fs=ALΌλο|2:z5.604 —Àj.BA|mydbwe. | And if the inclinations $i_1,i_2$ are not too large, the leading terms in $R$ are of $O(e_1^2), O(e_2^2), O(e_1e_2)$ and the corresponding angle $\varphi$, now called, are $\theta_1=\lambda_1-3\lambda_2+2\varpi_1$, $\theta_2=\lambda_1-3\lambda_2+2\varpi_2$, $\theta_3=\lambda_1-3\lambda_2+\varpi_1+\varpi_2$. |
Therefore the ibration of any one of 6,55 indicates the two planets are in an (eccentricily-fype) 3:1 MMB. | Therefore the libration of any one of $\theta_{1,2,3}$ indicates the two planets are in an ) 3:1 MMR. |
We note that no more than wo of them are linear independent and 85=(8,|63)/2 see or example Murray Dermott (1999). for more details]. | We note that no more than two of them are linear independent and $\theta_3=(\theta_1+\theta_2)/2$ [see for example Murray Dermott \shortcite{mur99} for more details]. |
Beside these resonant arguments. the relative apsidal ongitude Aw=σου is also a critical argument often o be discussed. | Beside these resonant arguments, the relative apsidal longitude $\Delta\varpi=\varpi_1-\varpi_2$ is also a critical argument often to be discussed. |
We call the libration of Ac in an MMIR hecorolalion. | We call the libration of $\Delta\varpi$ in an MMR the. |
ln a 3:1 AIALR. if both 06, ancl 06» ibrate with small amplitudes. Aw=4(4)65) will also necessarily librate. | In a 3:1 MMR, if both $\theta_1$ and $\theta_2$ librate with small amplitudes, $\Delta\varpi=
\frac{1}{2}(\theta_1 - \theta_2)$ will also necessarily librate. |
In this sense.the apsidal corotation is not independent ancl should be dilferent from the term ofresonance. which is in the context of a secular perturbation. | In this sense,the apsidal corotation is not independent and should be different from the term of, which is in the context of a secular perturbation. |
Lereafter we use subscripts 1.2 to label the orbital elements of companion D and € respectively. | Hereafter we use subscripts $1,2$ to label the orbital elements of companion B and C respectively. |
About one third (133 out of 400) of the simulations collapse during the integrations. and all the remainders survive for the LO? vvrintegration. | About one third $133$ out of $400$ ) of the simulations collapse during the integrations, and all the remainders survive for the $10^6$ yrintegration. |
Most of the survivors have very short (<107 vvr) e-folding time 2; (=L/LCL the reciprocal of LCD. while 38 of them have 7;Z105 vvr. and they are in this sense regarded. as stable. | Most of the survivors have very short $<10^2$ yr) e-folding time $T_e$ $=1/{\rm LCI}$, the reciprocal of LCI), while 38 of them have $T_e \gid 10^3$ yr, and they are in this sense regarded as stable. |
In. these stable systems. all the three planets have final inclinations =07.1. but those unstable (Zi«LO? vr) svstems may have inclinations of companion D and € as high as ~207. that is. the stable system prefers to retain to be coplanar. | In these stable systems, all the three planets have final inclinations $\la 0\degr.1$, but those unstable $T_e< 10^3$ yr) systems may have inclinations of companion B and C as high as $\sim 20\degr$, that is, the stable system prefers to retain to be coplanar. |
AE the stable systems are found. to be associated with the 3:1 ALAR between the two inner planets. | All the stable systems are found to be associated with the 3:1 MMR between the two inner planets. |
According to dillerent configurations of 8,ου and A the stable svstems can be divided into three groups.with representative cases illustrated in 11. | According to different configurations of $\theta_{1,2,3}$ and $\Delta\varpi$, the stable systems can be divided into three groups,with representative cases illustrated in 1. |
In all thecases. the companion D is in a steady motion (we don't illustrate this here)and. the semi-major axes of companion D and € vibrate with very small variations (an example is shown in 22). | In all thecases, the companion D is in a steady motion (we don't illustrate this here)and, the semi-major axes of companion B and C vibrate with very small variations (an example is shown in 2). |
Case 11) shows a strong resonance with 6). 6» and 6; librating around. 215. 75me and 325.2n respectively. | Case 1) shows a strong resonance with $\theta_1$ , $\theta_2$ and $\theta_3$ librating around $215\degr$ $75\degr$ and $325\degr$ , respectively. |
Simultaneously. the Aco also librates around 2507. | Simultaneously, the $\Delta\varpi$ also librates around $250\degr$. |
The well | The well |
sirrounding it. | surrounding it. |
After subtraction. the total counts over an area enclosing the central star (as a percentage of the pre-subtracted values) are aand v])) and (Ha). | After subtraction, the total counts over an area enclosing the central star (as a percentage of the pre-subtracted values) are and ) and ). |
It is likely therefore that there is some residual stellar fhux in the central regions of the PSF-subtracted images. | It is likely therefore that there is some residual stellar flux in the central regions of the PSF-subtracted images. |
Using the profile of ΠΟ 166215 as the PSF. and also that generated byTn. deconvolution using the Lucy-Richardson method was performed on the RS Oph data for each emission line. | Using the profile of HD 166215 as the PSF, and also that generated by, deconvolution using the Lucy-Richardson method was performed on the RS Oph data for each emission line. |
Tests using both the CLEAN and Maximum Entropy techniques produced similar results. | Tests using both the CLEAN and Maximum Entropy techniques produced similar results. |
However. for the lower signal-to-noise of the stellar PSF meant that only the PSF could be used effectively in the deconvolution. | However, for the lower signal-to-noise of the stellar PSF meant that only the PSF could be used effectively in the deconvolution. |
As part of this study. a re-analvsis was carried out of pre-outburst WEPC2 observations of the line through the 502N filler on 2000 June 12 (Prop. | As part of this study, a re-analysis was carried out of pre-outburst WFPC2 observations of the line through the 502N filter on 2000 June 12 (Prop. |
ID 8332). | ID 8332). |
No extended emission. even al very faint levels. was detectable. | No extended emission, even at very faint levels, was detectable. |
This confirms the results of the analvsis of these data undertaken by Brocksoppetal.(2003). | This confirms the results of the analysis of these data undertaken by \cite{bro03}. |
. As can be seen in Figure 3.. extended structure was clearly visible in the line in the PSF-subtractecl image (and was indeed visible in the raw image). | As can be seen in Figure \ref{images}, extended structure was clearly visible in the line in the PSF-subtracted image (and was indeed visible in the raw image). |
Deconvolution revealed more detailed structure in both aandA. | Deconvolution revealed more detailed structure in both and. |
.. There was also a hint of possible extended emission close to the central source in the Ila line. but this was not present αἱ a significant level. | There was also a hint of possible extended emission close to the central source in the $\alpha$ line, but this was not present at a significant level. |
In the deconvolved aand iimages. the most striking feature is a double ring structure with major axis lving E-W and total (peak-to-peak) extent 360€30 mas (580£50 AU at d=1.6 kpc). | In the deconvolved and images, the most striking feature is a double ring structure with major axis lying E-W and total (peak-to-peak) extent $360 \pm 30$ mas $580 \pm 50$ AU at $d = 1.6$ kpc). |
The most extended structures seen in the radio (theouterlobes:O'Brienetal.2006) lie along this axis. | The most extended structures seen in the radio \citep[the outer lobes;][]{obr06} lie along this axis. |
Assuming ejection at the time of the outburst. the expansion rate (from the center) of the optical emission along this axis is 1.2220.1 mas ! (equivalent to vsp=3200x300 kin Fin the plane of the skv). | Assuming ejection at the time of the outburst, the expansion rate (from the center) of the optical emission along this axis is $1.2 \pm 0.1$ mas $^{-1}$ (equivalent to $v_{exp} = 3200 \pm 300$ km $^{-1}$ in the plane of the sky). |
The optical emission is also detectable above background in the deconvolved images to a total extent of 520450 mas. corresponding to an expansion rate of 1.72:0.2 mas !.. | The optical emission is also detectable above background in the deconvolved images to a total extent of $520 \pm 50$ mas, corresponding to an expansion rate of $1.7 \pm 0.2$ mas $^{-1}$. |
We may compare this to 1.4 220.3 mas ! [or the E-W lobes seen in the radio. taken over 4 epochs from /=21.5 to 62.7 days in the 2006 outburst (ODrien et al.. | We may compare this to 1.4 $\pm0.3$ mas $^{-1}$ for the E-W lobes seen in the radio, taken over 4 epochs from $t = 21.5$ to 62.7 days in the 2006 outburst (O'Brien et al., |
in preparation) and 1.3 mas ! [or the equivalent features derived from VLBI observations in | in preparation) and 1.3 mas $^{-1}$ for the equivalent features derived from VLBI observations in |
(e.g.. axions) are well defined energy states of the system. | (e.g., axions) are well defined energy states of the system. |
However. once the interaction term is introduced. the only well defined energy states of the system are mixed photon-particle states. | However, once the interaction term is introduced, the only well defined energy states of the system are mixed photon-particle states. |
The equation of motion for the photon-particle system takes the form (e.g.. 009, Raffelt Stodolsky where @ is the photon energy and the magnetic field in the direction of the photon polarizationB (the photon's E field). | The equation of motion for the photon-particle system takes the form (e.g., C09, Raffelt Stodolsky where $\omega$ is the photon energy and $B_\|$ the magnetic field in the direction of the photon polarization (the photon's $E$ field). |
Clearly. neither pure photon nor pure ALP states are eigenstates of the system but rather some combination of them. | Clearly, neither pure photon nor pure ALP states are eigenstates of the system but rather some combination of them. |
Let us now focus on the limit where Big=B10!° GG. g=107condition-uGeV-, and the photon wavelength. A= mn, This iis met either near resonance where n,Anicm; or when both masses are individually smaller than V‘9B,O (which limitis actually met is irrelevant). | Let us now focus on the limit where $B_{16}=B_\|/10^{16}$ G, $g=10^{-14}g_{-14}\,{\rm GeV}^{-1}$, and the photon wavelength, $\lambda=\lambda_m$ m. This condition is met either near resonance where $m_\gamma^2 \simeq m_a^2$ or when both masses are individually smaller than $\sqrt{gB_\| \omega}$ (which limit is actually met is irrelevant). |
The eigenstates of oequation 3. are then given by where [aj is the axion state and |? is the photon state. | The eigenstates of equation \ref{mat} are then given by where $\left \vert a \right >$ is the axion state and $\left \vert \gamma \right >$ is the photon state. |
The eigenvalues are m]= o. By analogy with optics. these masses are related to xgB,effective refractive indices: 24=|Ón;=dEη/2@7 (for |8n,|« 1) meaning that different paths through a refractive medium would be taken by the rays. | The eigenvalues are $m_\pm^2= \pm gB_\| \omega$ By analogy with optics, these masses are related to effective refractive indices: $n_\pm=1+\delta n_\pm\simeq 1-{m_\pm^2}/{2\omega^2}$ (for $\vert \delta n_\pm\vert \ll1$ ) meaning that different paths through a refractive medium would be taken by the rays. |
We note that there is no dependence on the particle or photon mass so long as equation 4 1s satisfied. | We note that there is no dependence on the particle or photon mass so long as equation \ref{cond2} is satisfied. |
In terms of the refractive Index. the equation of motion for aray may be found by minimizing the action {dsr(s). | In terms of the refractive index, the equation of motion for a ray may be found by minimizing the action $\int d{\bf s} n({\bf s})$. |
This is completely analogous to mechanics where we substituteon. | This is completely analogous to mechanics where we substitute. |
In this case. a force is 22/ds=figuret(g/2)(0B/Os). | In this case, a force is $\partial \mathfrak{L}/\partial {\bf s} = \pm (g/2)(\partial B /\partial {\bf s})$. |
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