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. Spin-down resulting from accretion could be more effective in non-imereing disk ealaxies hosting simaller SATBUs. as secular processes stochastically drive small amounts of eas toward the SMBIL. | Spin-down resulting from accretion could be more effective in non-merging disk galaxies hosting smaller SMBHs, as secular processes stochastically drive small amounts of gas toward the SMBH. |
The resulting low spin may be consistent with observations of disk galaxics as hosts for radio-quiet sources, if radio-loudness is associated with the spin of the black hole (?).. | The resulting low spin may be consistent with observations of disk galaxies as hosts for radio-quiet sources, if radio-loudness is associated with the spin of the black hole \citep{BertiVolonteri08}. |
Since black hole merecrs may also contribute to the spin-down of SMDIIs (?).. it is portant to quality the relative importance of gas accretion for determining black hole spin. | Since black hole mergers may also contribute to the spin-down of SMBHs \citep{Sijackietal09}, it is important to qualify the relative importance of gas accretion for determining black hole spin. |
The sunulatious here do not model the gas down to scales iuside the accretion disk. nor allow the black hole particle to acerete imaterial frou larger scales (as discussed. in Section. 2)). | The simulations here do not model the gas down to scales inside the accretion disk, nor allow the black hole particle to accrete material from larger scales (as discussed in Section \ref{sec:simtr}) ). |
For these reasons. the sinulatious do not follow the spin ofthe black hole. | For these reasons, the simulations do not follow the spin of the black hole. |
The simulations do model eas on parsec scales. enabling us to resolve the direction of the augular niomenutun vector of eas and to provide boundary couditious for simulations of SMIBIT accretion disks. | The simulations do model gas on parsec scales, enabling us to resolve the direction of the angular momentum vector of gas and to provide boundary conditions for simulations of SMBH accretion disks. |
The aneular momentum vector is measured with respect to the black hole particles position aud velocity. | The angular momentum vector is measured with respect to the black hole particle's position and velocity. |
Some unucertaüutv iu the direction of the angular momentum vector could arise because the position of the black hole does not necessarily coimcide exactly with the position of the center of the galaxy aud because the black hole may have a significant rotational velocity about the center. | Some uncertainty in the direction of the angular momentum vector could arise because the position of the black hole does not necessarily coincide exactly with the position of the center of the galaxy and because the black hole may have a significant rotational velocity about the center. |
During most of the simulation. the black hole particle lies in the densest cell. which coiucides with the minima of the ealaxw’s gravitational potential. | During most of the simulation, the black hole particle lies in the densest cell, which coincides with the minimum of the galaxy's gravitational potential. |
Therefore. the black hole particle typically provides a reasonable estimate of the location of ceuter of the ealaxy. | Therefore, the black hole particle typically provides a reasonable estimate of the location of center of the galaxy. |
We have also iieasured the augular mioacutui relative to a center whose velocity is defined bv an average over the central cell aud six of its neighbors. but find that the effect on the augular momentum vector is neeligible (except for the brief periods of black hole particle displacement). | We have also measured the angular momentum relative to a center whose velocity is defined by an average over the central cell and six of its neighbors, but find that the effect on the angular momentum vector is negligible (except for the brief periods of black hole particle displacement). |
Figure & shows a map projection of the direction of the normalized augular momentum vector as it evolves in time at 100. 10. and Lpe from the black hole particle for the :=l[ and :=3 simulations (eft and vieht. respectively). | Figure \ref{fig:jmap} shows a map projection of the direction of the normalized angular momentum vector as it evolves in time at $100$, $10$ , and $1 \dim{pc}$ from the black hole particle for the $z=4$ and $z=3$ simulations (left and right, respectively). |
The mean direction of the rotation axis of the ealaxy on kiloparsec scales. which remains colparatively constant over the ~1Myr duration of the zoom-in siuulation. is oriented toward the center 6© the map in each case. | The mean direction of the rotation axis of the galaxy on kiloparsec scales, which remains comparatively constant over the $\sim 1 \dim{Myr}$ duration of the zoom-in simulation, is oriented toward the center of the map in each case. |
The disk is slightlv warped on scales snaller than a kiloparsec. so that the axis of the cicunmnuclear disk is orieuted at an anele to the lurge-scale disk. | The disk is slightly warped on scales smaller than a kiloparsec, so that the axis of the circumnuclear disk is oriented at an angle to the large-scale disk. |
Figure 9. shows the iuclinatiou angle of the angular moment vectors shown in Figure 8 witlirespect to those measured at a kiloparsec. over time. | Figure \ref{fig:thet} shows the inclination angle of the angular momentum vectors shown in Figure \ref{fig:jmap}
with respect to those measured at a kiloparsec, over time. |
Together. Figures δ and 9 demonstrate the effect of the circtummuclear disk’s chaotic behavior ou the angular momentum of the accreting eas. | Together, Figures \ref{fig:jmap} and \ref{fig:thet} demonstrate the effect of the circumnuclear disk's chaotic behavior on the angular momentum of the accreting gas. |
At. = and a scale of LOOpc. the direction of the augular momentum vector changes slowly but shows little scatter over the course of the simulation. | At $z=4$ and a scale of $100 \dim{pc}$, the direction of the angular momentum vector changes slowly but shows little scatter over the course of the simulation. |
There is a sual amount of scatter during the displacement of the black hole particle. which is visible on cach scale slow. | There is a small amount of scatter during the displacement of the black hole particle, which is visible on each scale shown. |
Rather than interpolating the magnitude aud direction of the augular momentum vector of the gas over the duration of the black hole displacement. as was done witli he iuterior gas mass profile iu Figure 2.. we continue ο lneasure quantities relative to the black hole particle. | Rather than interpolating the magnitude and direction of the angular momentum vector of the gas over the duration of the black hole displacement, as was done with the interior gas mass profile in Figure \ref{fig:gmz}, we continue to measure quantities relative to the black hole particle. |
Ouce the black hole particle resettles iuto the bottom of he potential well. the excess scatter iu L/L vanishes. | Once the black hole particle resettles into the bottom of the potential well, the excess scatter in $\boldsymbol{L}/L$ vanishes. |
The slow change in direction at 100pc may correspond ο the increased warping of the disk as the simulation xogresses, | The slow change in direction at $100
\dim{pc}$ may correspond to the increased warping of the disk as the simulation progresses. |
Iu a similar analysis of simulations of the erowtli of a disk galaxy by ?.. the spin axis of the central eas disk also changed direction on 100pc scales (relative ο kiloparsec scales). | In a similar analysis of simulations of the growth of a disk galaxy by \citet{SaitohWada04}, , the spin axis of the central gas disk also changed direction on $100 \dim{pc}$ scales (relative to kiloparsec scales). |
The augular 1iomenutun vector at LO and 1pc starts out with the same approximate oricutation as the arecr scales but shows slightly inore scatter. | The angular momentum vector at $10$ and $1 \dim{pc}$ starts out with the same approximate orientation as the larger scales but shows slightly more scatter. |
There are two sudden changes in the direction of the angular uonmnentuni vector at the Lpc scale. | There are two sudden changes in the direction of the angular momentum vector at the $1 \dim{pc}$ scale. |
In the first inciceut. at ~0.55Myr. the angular ολοτα changes abruptly woimore than 1007. coincidiug with the time of the displacement of the black hole particle deseribed iu Section 3. (as a chunp of eas forms near and moves into the center). | In the first incident, at $\sim 0.55 \dim{Myr}$, the angular momentum changes abruptly by more than $100 \degree$, coinciding with the time of the displacement of the black hole particle described in Section \ref{sec:mass} (as a clump of gas forms near and moves into the center). |
The fip iu direction is also visible at LOpc. | The flip in direction is also visible at $10 \dim{pc}$. |
In the second incident. at —1.2Mw. the angular momentum shifts by ~1007 while the black hole remains stationary at the center of the galaxy. | In the second incident, at $\sim 1.2 \dim{Myr}$, the angular momentum shifts by $\sim 100 \degree$ while the black hole remains stationary at the center of the galaxy. |
The second shift is the result of eravitational imteraction with a ~ few 105ML clump of eas. which develops at 2LOOpe aud continues to move toward the ceuter of the disk at late times. | The second shift is the result of gravitational interaction with a $\sim$ few $\times 10^8
\dim{M}_\odot$ clump of gas, which develops at $> 100 \dim{pc}$ and continues to move toward the center of the disk at late times. |
The mass of the clump is comparable to the mass of the disk iuterior to LOpc. | The mass of the clump is comparable to the mass of the disk interior to $\sim 10 \dim{pc}$. |
The second shift is also visible at the LOpe scale but occurs less suddenly. | The second shift is also visible at the $10
\dim{pc}$ scale but occurs less suddenly. |
The two panels in Figure 10. show hree-dimensional volunuc-reuderiugs of the eas density at 30pc before aud after the flip (top and bottom. respectively). | The two panels in Figure \ref{fig:flip} show three-dimensional volume-renderings of the gas density at $30 \dim{pc}$ before and after the flip (top and bottom, respectively). |
The inset shows a zooed-out view of the disk. where the large chump of gas is visible z200pc roni the center. | The inset shows a zoomed-out view of the disk, where the large clump of gas is visible $\gtrsim 200 \dim{pc}$ from the center. |
Tf the clamp remains iutact as it reaches he center of the circumauuelear region. a corresponding cluporary displacement of the black hole particle is expected. similar to the earlier displaceiieut. | If the clump remains intact as it reaches the center of the circumnuclear region, a corresponding temporary displacement of the black hole particle is expected, similar to the earlier displacement. |
The black tole particle is likely to be displaced when the iu-alling sas nass is comparable to or ereater than the dack hole mass aud the mass of the eas disk iuterior othe in-falhug chuup. | The black hole particle is likely to be displaced when the in-falling gas mass is comparable to or greater than the black hole mass and the mass of the gas disk interior tothe in-falling clump. |
However. a displacement has rotvet occurred by the eud of the simulation. | However, a displacement has notyet occurred by the end of the simulation. |
Similar angular momentum flips occur over the course of runs ZIL20.BU3. ZIL20.DITIO. and ZIL20.0T (not shown). often correspoudiue to coincideut displacements of the | Similar angular momentum flips occur over the course of runs Z4L20.BH3, Z4L20.BH10, and Z4L20.OT (not shown), often corresponding to coincident displacements of the |
This heating rate was justified [rom equation (13)) for the time evolution of the photon intensity J. | This heating rate was justified from equation \ref{eq:FP}) ) for the time evolution of the photon intensity $J$. |
MMB assumed that the rate of heat transfer to the atoms is determined by the change in the spectrum due to the recoil effect. which is the second term in (he of equation (13)): Thev then computed the heating rate by integrating (he variation in photon enereyv density per unit physical time over [requency: where theintegral 5.is given in equation (20)). and tp is the Gunn-Peterson optical depth Gall svmbols are defined in 82 aud 3: (he last equality involves an integration by parts). | MMR assumed that the rate of heat transfer to the atoms is determined by the change in the spectrum due to the recoil effect, which is the second term in the right-hand-side of equation \ref{eq:FP}) ): They then computed the heating rate by integrating the variation in photon energy density per unit physical time over frequency: where theintegral $S_c$is given in equation \ref{eq:scati}) ), and $\tau_{GP}$ is the Gunn-Peterson optical depth (all symbols are defined in 2 and 3; the last equality involves an integration by parts). |
llowever. (he first terii in equation (13)) must also be included to compute the heating rate of the gas. | However, the first term in equation \ref{eq:FP}) ) must also be included to compute the heating rate of the gas. |
In fact. the heating rate of the gas is not just determined by the recoil. but also by the fact that the background intensitv is greater on (he red wing of the line than on the blue wing. and scatterings have an average tendeney (o bring photons back into the line center. | In fact, the heating rate of the gas is not just determined by the recoil, but also by the fact that the background intensity is greater on the red wing of the line than on the blue wing, and scatterings have an average tendency to bring photons back into the line center. |
The energy change of the photons due to the first terim in equation (13)) can only go into the gas (whereas that of the third term is due to redshift and goes into the expansion of (he universe. and that of the fourth term is provided by a source). | The energy change of the photons due to the first term in equation \ref{eq:FP}) ) can only go into the gas (whereas that of the third term is due to redshift and goes into the expansion of the universe, and that of the fourth term is provided by a source). |
Thus. the correct heating rate is Under steady-state conditions (and no source term: here. we consider continuum photons only. to simplily the discussion). we can replace the quantity inside parenthesis by —7,podOr. and inleeraline by parts. we obtain which is the expression in equation (17)). | Thus, the correct heating rate is Under steady-state conditions (and no source term; here, we consider continuum photons only, to simplify the discussion), we can replace the quantity inside parenthesis by $- \tau_{GP}^{-1} \partial J/\partial x$, and integrating by parts, we obtain which is the expression in equation \ref{eq:heatr2c}) ). |
Note Chal. because jy=(nyc)(απ). the heating rate per unit volume is independent of the atomic density. except [or the weak dependence of J. on τρ. whereas MMB. assumed that the heating rate is independent of 2. | Note that, because $\tilde J_0 = (n_H c)/(4\pi \nu_{\alpha})$, the heating rate per unit volume is independent of the atomic density, except for the weak dependence of $I_c$ on $\tau_{GP}$ , whereas MMR assumed that the heating rate is independent of $n_H$ . |
integrated Le intensities. | integrated line intensities. |
Peak anteuna temperature ratios are calculated for Που 15)/TIC3N(5 1). hereafter denoted R46;5. | Peak antenna temperature ratios are calculated for $_{3}$ $_{3}$ N(5–4), hereafter denoted $_{16/5}$. |
B46;5 ranges frou less than 0.06 up to ~0.5. with CMC B having the highest value (Table 2)). | $_{16/5}$ ranges from less than 0.06 up to $\sim$ 0.5, with GMC B having the highest value (Table \ref{Trat}) ). |
The 5| and 1615 transitious eenerallv bracket the peak of the level populations. so we achieve eood coustraints on gas excitation. | The 5–4 and 16–15 transitions generally bracket the peak of the level populations, so we achieve good constraints on gas excitation. |
LTE excitation temperatures. T... nuplied by ος range from «10 k to >is Kk (chere we have neglected. Ἐν in this determination). | LTE excitation temperatures, $_{ex}$, implied by $_{16/5}$ range from $<$ 10 K to $>$ 18 K (where we have neglected $_{cmb}$ in this determination). |
Excitation is lowest towards GMCS D. D aud E. These excitation temperatures T, are similar to those found from the prestmably πιο less dense gas traced in C150 (Table2:Meier&Turner2001). | Excitation is lowest towards GMCs D, D' and E. These excitation temperatures $_{ex}$ are similar to those found from the presumably much less dense gas traced in $^{18}$ O \citep[Table
2;][]{MT01}. |
. Ouly the GAIC C clouds have significantly higher T. in Πουtwo towards CL - C3 κοT, (ICN) are about a factor of ercater than T, | Only the GMC C clouds have significantly higher $_{ex}$ in $_3$ N — towards C1 - C3 $_{ex}$ $_{3}$ N) are about a factor of two greater than $_{ex}$ $^{18}$ O). |
The resolution of the ΠΟ(109) data is significantly lower than it is for the 5.[ and 1615 lines. | The resolution of the $_{3}$ N(10–9) data is significantly lower than it is for the 5–4 and 16–15 lines. |
. conrare J=5L| aud J=10-9 line Ποιος, the 5Ll data were convolved to the resolution of the (10.9) data (5.79«5,71Meier&Turner 2001).. then iutegrated intensity ratios. hereafter B49;5. were sampled at the locations of Που». | To compare J=5–4 and J=10-9 line intensities, the 5–4 data were convolved to the resolution of the (10–9) data \citep[$5.\arcsec9\times5.\arcsec1$][]{MT01}, then integrated intensity ratios, hereafter $_{10/5}$ , were sampled at the locations of $_{16/5}$. |
Though at lower resolution than B46;5. we make the approximation that τρις does not change on these sub-GAIC scales. | Though at lower resolution than $_{16/5}$, we make the approximation that $_{10/5}$ does not change on these sub-GMC scales. |
While leading to larger uncertainties. this xovides a wav to mclude all three transitions in modcling deuse gas excitation at very high resolution. | While leading to larger uncertainties, this provides a way to include all three transitions in modeling dense gas excitation at very high resolution. |
Ayy5 range roni 0.25 to L.l (Table D)). | $R_{10/5}$ range from 0.25 to 1.4 (Table \ref{Tinti}) ). |
GAIC D has the highest value of 1.1. | GMC B has the highest value of 1.4. |
The remainder of the CAICs have ratios of 0.3«το<0.6. | The remainder of the GMCs have ratios of $0.3 < R_{10/5}< 0.6$. |
Excitation temperatures mplied w these ratios range from T,= 616 Ik. consistent with those derived from τρις) separately. | Excitation temperatures implied by these ratios range from $_{ex} = $ 6–16 K, consistent with those derived from $R_{16/5}$ separately. |
Ty. derived vom Που are similar to those derived. from ClO. The only exception here is GAIC A. the cloud where PDRs (Photon-Dominated- Regions) dominate (Aleicr&Turner 2005). | $_{ex}$ derived from $_{10/5}$ are similar to those derived from $^{18}$ O. The only exception here is GMC A, the cloud where PDRs (Photon-Dominated Regions) dominate \citep[][]{MT05}. |
. Towards GMC A (0Ο) and ΜΕ ιο) are 12-15 Ik. twice that of T. (BR49;5). | Towards GMC A $_{ex}$ $^{18}$ O) and $_{ex}$ $_{16/5}$ ) are 12 - 13 K, twice that of $_{ex}$ $_{10/5}$ ). |
Before modeling the deusities implied by the line ratios. we test whether IR pumping can be respousible for the observed excitation. | Before modeling the densities implied by the line ratios, we test whether IR pumping can be responsible for the observed excitation. |
IR pumping is iuportaut if. where "D, is the Einstein B, of the corresponding vy=] vibrational trausitious at ~15 jan. £415 pan) is the 15 jan intensity as seen by the IICSN molecules and αν is the Einstein η of the rotational state (ee,estimateCostagliola&Aalto2010). | IR pumping is important if, where $^{vib}B_{ul}$ is the Einstein $B_{ul}$ of the corresponding $\nu_{7}=1$ vibrational transitions at $\sim 45 ~\mu$ m, $I_{\nu}$ (45 $\mu$ m) is the 45 $\mu$ m intensity as seen by the $_{3}$ N molecules and $^{rot}A_{ul}$ is the Einstein $A_{ul}$ of the rotational state \citep[e.g.][]{CA10}. |
. It is extremely difficult to the applicable 15 jan IR intensity. but it is expected to be most intense towards the starburst GAC (B). | It is extremely difficult to estimate the applicable 45 $\mu$ m IR intensity, but it is expected to be most intense towards the starburst GMC (B). |
A detailed. assessincut of IR pumping iust await high resolution MIR maps. but we constrain £(15 jin) iu several wavs. | A detailed assessment of IR pumping must await high resolution MIR maps, but we constrain $I_{\nu}$ (45 $\mu m$ ) in several ways. |
First. we take the 15 pau fux from Braucll and scale it bv the fraction of total 20 jaa flux that comes from within 24 of the starburst as ound bv Becklinetal.(1980). and then average over hat aperture. | First, we take the 45 $\mu$ m flux from \citet[][]{Brandl06} and scale it by the fraction of total 20 $\mu$ m flux that comes from within $^{''}$ of the starburst as found by \citet[][]{BGMNSWW80} and then average over that aperture. |
For the average νο gan) caleulated this wav. Du L415 jun) is 10| (10 7°) times too low o pump the 1615 (51) transition. | For the average $I_{\nu}$ (45 $\mu$ m) calculated this way, $^{vib}B_{ul}~I_{\nu}$ (45 $\mu$ m) is $10^{-4}$ $10^{-2.5}$ ) times too low to pump the 16–15 (5–4) transition. |
Alternatively if we (very conservatively) take the total MIR huuiuosity from he central and assume it comes from a blackbody of the observed. color temperature (e.ge.ΩςIk:Deck-iuetal.1980). then [ο jun) is still at least an order of magnitude too low to mect the inequality iu eq. | Alternatively if we (very conservatively) take the total MIR luminosity from the central $^{''}$ and assume it comes from a blackbody of the observed color temperature \citep[e.g. $\sim$50 K;][]{BGMNSWW80} then $I_{\nu}$ (45 $\mu$ m) is still at least an order of magnitude too low to meet the inequality in eq. |
1 for both transition. | \ref{pump} for both transition. |
IR pumping rates only become comparable to 0774, for the 5.£ transition if the total IR huuinositv originates from a ~2.5 pc source with a source temperature of 2100 Ik. We couclude that IR piping is not important for the 15 transition of Που in anv reasonable geometry of the IR feld. | IR pumping rates only become comparable to $^{rot}A_{ul}$ for the 5–4 transition if the total IR luminosity originates from a $\sim$ 2.5 pc source with a source temperature of $\gsim$ 100 K. We conclude that IR pumping is not important for the 16–15 transition of $_{3}$ N in any reasonable geometry of the IR field. |
For the 5IL transition to be sensitive to IR pumping. the IR source lust be wari. opaque and extremely compact. | For the 5–4 transition to be sensitive to IR pumping, the IR source must be warm, opaque and extremely compact. |
Therefore IR pumping is neglected for all clouds iu the following discussion. | Therefore IR pumping is neglected for all clouds in the following discussion. |
The values of the excitation temperature coustrain the deusity aud kinetic temperatures. ayy, aud Tj. aud the plivsical conditions of the clouds driving the excitation. | The values of the excitation temperature constrain the density and kinetic temperatures, $n_{H_{2}}$ and $T_k$ , and the physical conditions of the clouds driving the excitation. |
A series of Large Velocity Caacdient (ΤΑ) radiative transfer gnodels were run to predict the observed iutcusitics aud line ratios for a given πμ. T; /aud filling factor. f,=Quei. of the dense component (og.VandenBoutetal. 1983).. | A series of Large Velocity Gradient (LVG) radiative transfer models were run to predict the observed intensities and line ratios for a given $n_{H_{2}}$, $_{k}$ and filling factor, $f_{a}=\Omega_{source}/\Omega_{beam}$ , of the dense component \citep[e.g.,][]{VLSW83}. . |
Single componcut LVG inodels are instructive. particularly when lines are optically thin. as is the case for Που. The LVG model used is that of Meieretal.(2000).. adapted to IIC43N. with levels up to J=20 included. | Single component LVG models are instructive, particularly when lines are optically thin, as is the case for $_{3}$ N. The LVG model used is that of \citet{MTH00}, adapted to $_{3}$ N, with levels up to J=20 included. |
Collision cocticicuts are from Creen&Chapman(1978). | Collision coefficients are from \citet[][]{GC78}. |
.. A range of densitics. Hg, = 10? 109ey7. and kinetic temperatures. T, = Q100 Is. was explored. | A range of densities, $n_{H_{2}}$ = $^{2}$ $^{6}~cm^{-3}$, and kinetic temperatures, $_{k}$ = 0–100 K, was explored. |
Που colin censitics based ou LTE excitation (Table 2)). are caleulated at 2" vesolutiou frou: using molecular data of Laffertv&Lovas(1978).. IIC4N(5.1) inteusities. and T,(IC4N) from Table 2.. | $_{3}$ N column densities based on LTE excitation (Table \ref{Trat}) ), are calculated at $^{''}$ resolution from: using molecular data of \citet[][]{LL78}, $_{3}$ N(5–4) intensities, and $_{ex}$ $_{3}$ N) from Table \ref{Trat}. |
ΠουΝ abundances are found to be. NUICRN)τ-?410 5. with the highest values towards Cl and D. While uncertain these abundances agree with hose found iu Meier&απο(2005) aud are typical of Calactic center Που abundances (e.g.Morrisctal.1976:deVicenteet2000) ancl good cnoug1 or constraining ον, | $_{3}$ N abundances are found to be, $X(HC_{3}N) \simeq 10^{-9.1}~-~10^{-8.5}$ , with the highest values towards C1 and D. While uncertain these abundances agree with those found in \citet[][]{MT05} and are typical of Galactic center $_{3}$ N abundances \citep[e.g.][]{MTPZ76,DMNC00} and good enough for constraining $X/dv/dr$. |
The ratio9 of cloud linewidt1 o core size is ~13duostpe+ for the CAICs. | The ratio of cloud linewidth to core size is $\sim~1-3 \rm ~km~s^{-1}pc^{-1}$ for the GMCs. |
Therefore. we adopt a standard model abundance per velocity: eradicut⋡⋅ of DxN/defdr = 79 kins 1. but run nodels for values ofN/defdrLot - 10/7. | Therefore, we adopt a standard model abundance per velocity gradient of $X/dv/dr$ = $^{-9}$ km $^{-1}$, but run models for values of $X/dv/dr ~10^{-11}$ - $10^{-9}$. |
Autennua temperatures are sensitive to the unknown filline factor. | Antenna temperatures are sensitive to the unknown filling factor. |
Iu these extragalactic observations. the (|a corresponds to scales large compared to cloud structure. and lence fllius factors are not directly shown. | In these extragalactic observations, the beam corresponds to scales large compared to cloud structure, and hence filling factors are not directly known. |
To first order. the line ratios. Ryyss aud Riges are independent of filling factor if we assunie that Που(2 (109) and 15) originate iu the same gas. | To first order, the line ratios, $R_{10/5}$ and $_{16/5}$ are independent of filling factor if we assume that $_{3}$ N(5--4), (10–9) and (16–15) originate in the same gas. |
So uodel ratios are compared to the observed data to coustrain parameter space. | So model ratios are compared to the observed data to constrain parameter space. |
For the pariueter space nupled by the line ratiosmocel brightuess temperatures are deteriined. | For the parameter space implied by the line ratiosmodel brightness temperatures are determined. |
A comparison of the model brightucss eniperature to the observed brightuess teniperature sets he required areal filliug factors for that solution. | A comparison of the model brightness temperature to the observed brightness temperature sets the required areal filling factors for that solution. |
Figure 1. displavsthe results of theLVC modeling. | Figure \ref{lvg} displaysthe results of theLVG modeling. |
Acceptable cxlo (Tp. yy.) parameter spaces are shown | Acceptable $\pm1\sigma$ $_{k}$ $n_{H_{2}}$ ) parameter spaces are shown |
component of the displacement current is essentially unchanged [rom its value in vacuo. | component of the displacement current is essentially unchanged from its value in vacuo. |
It follows that the neglect of the perpendicular component of the inductive electric field in models for pulsar electrodynanmies is not justifiable. | It follows that the neglect of the perpendicular component of the inductive electric field in models for pulsar electrodynamics is not justifiable. |
huplications of including the inductive electric field are discussed in the remainder of (his paper. | Implications of including the inductive electric field are discussed in the remainder of this paper. |
The neglect of the inductive electric field in an obliquely rotating pulsar is not justified. and its inclusion implies (hat the magnetosphere cannot be in rigid rotation. | The neglect of the inductive electric field in an obliquely rotating pulsar is not justified, and its inclusion implies that the magnetosphere cannot be in rigid rotation. |
The perpendicular component of the inductive electric field implies an electric drift Av. determined by equation (10)). | The perpendicular component of the inductive electric field implies an electric drift $\Delta{\bi v}$, determined by equation \ref{EIperp}) ). |
It is convenient to introduce spherical polar coordinates. r.0.o delined by the rotation axis. | It is convenient to introduce spherical polar coordinates, $r, \theta,\phi$ defined by the rotation axis. |
The spherical polar components of the dipolar fielcl are απ where (he initial conditions are chosen such that the magnetic axis is in the plane ó=0 al /—0. | The spherical polar components of the dipolar field are ( ( ), where the initial conditions are chosen such that the magnetic axis is in the plane $\phi=0$ at $t=0$. |
The angle 6,,. defined by cos9,,=cosa8+sinasin@cos(ó—wl). corresponds io the magnetic colatitude. which varies periodically as the star rotates. | The angle $\theta_m$, defined by $\cos\theta_m=\cos\alpha\cos\theta+\sin\alpha\sin\theta\cos(\phi-\omega t)$, corresponds to the magnetic colatitude, which varies periodically as the star rotates. |
At the phases ὡ—wlng. n—O0.cl... the field line is in an azimuthal plane. but at other phases the field has a nonzero azimuthal component (23,4 0). | At the phases $\phi-\omega t=n\pi$, $n=0,\pm1,\cdots$, the field line is in an azimuthal plane, but at other phases the field has a nonzero azimuthal component $B_\phi\ne0$ ). |
At very soft energies. the equilibrium distribution of binary energies found. by Goodman&αι(1903). converges to a power law with dNg/dexe a consequence of a Alaxwellian clistribution of velocities and a uniform number densitv (Hleggie1975). | At very soft energies, the equilibrium distribution of binary energies found by \citet{goodman93a} converges to a power law with ${\rm d}N_B/{\rm d}\epsilon \propto \epsilon^{-5/2}$, a consequence of a Maxwellian distribution of velocities and a uniform number density \citep{heggie75}. |
. Here. we illustrate the οσοι of a non-uniform density distribution. where we assume that the typical star sees à number density that is à power law with padius. n(r)xrU. | Here we illustrate the effect of a non-uniform density distribution, where we assume that the typical star sees a number density that is a power law with radius, $n(r) \propto r^{-q}$. |
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