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These make such subtraction scheme impossible. | These make such subtraction scheme impossible. |
Therefore it is more important (to prove the predictions of IC model. i.e. the diffuse emission in various other energy bands. ie. radio. N-ravs and VIIE. | Therefore it is more important to prove the predictions of IC model, i.e. the diffuse emission in various other energy bands, i.e. radio, X-rays and VHE. |
In future we can use (hese data to constraint the relative contributions between these two different mocels. | In future we can use these data to constraint the relative contributions between these two different models. |
Finally. we want to remark that by using the 3 GCs reported by Abdo et al. ( | Finally, we want to remark that by using the 8 GCs reported by Abdo et al. ( |
2010b) and Y newly confirmed. gamma-ray GCs by Tam et al. ( | 2010b) and 7 newly confirmed gamma-ray GCs by Tam et al. ( |
2010). Hui et al. ( | 2010), Hui et al. ( |
2010b) have carried out a correlation analvsis between the observed 5-rav. luninosities £L. and various cluster properties to probe the origin of the high energv photons [rom these GCs. | 2010b) have carried out a correlation analysis between the observed $\gamma$ -ray luminosities $L_{\gamma}$ and various cluster properties to probe the origin of the high energy photons from these GCs. |
Thev find that L, is positively correlated with the encounter rate D. and the metallicity |Fe/1]. which is an alterative independent estimator lor number MSPs in the elobular clusters (ef. | They find that $L_{\gamma}$ is positively correlated with the encounter rate $\Gamma_{c}$ and the metallicity $\left[{\rm Fe/H}\right]$, which is an alterative independent estimator for number MSPs in the globular clusters (cf. |
Ini. Cheng and Taam 2010). | Hui, Cheng and Taam 2010). |
Thev also find a tendency that £. increases with the enerev densities of the soft photon at the cluster location which favors the scenario that the observed eamma-rays from these GC's are significantly contributed by the inverse Compton scattering. | They also find a tendency that $L_{\gamma}$ increases with the energy densities of the soft photon at the cluster location which favors the scenario that the observed gamma-rays from these GCs are significantly contributed by the inverse Compton scattering. |
It should be noticed that Iii et al. ( | It should be noticed that Hui et al. ( |
2010b) have used dillerent way to calculate ihe encounter rate in comparing with Abdo et al. ( | 2010b) have used different way to calculate the encounter rate in comparing with Abdo et al. ( |
2010b). | 2010b). |
Hii et al. ( | Hui et al. ( |
20105) have included the observed dispersion velocity in evaluating D. whereas Abdo et al. ( | 2010b) have included the observed dispersion velocity in evaluating $\Gamma_{c}$ whereas Abdo et al. ( |
2010b) have used the Iree fall velocity to approximate the dispersion velocity. | 2010b) have used the free fall velocity to approximate the dispersion velocity. |
For illustration purpose in Figure 7 we follow the definition of encounter rate given in Abdo et al. ( | For illustration purpose in Figure 7 we follow the definition of encounter rate given in Abdo et al. ( |
2010b). | 2010b). |
Figure τα | Figure 7a |
is paper is organisec as follows. | This paper is organised as follows. |
In Sa2 we presen Ix nmocel for the jet and for the magnetic field. | In 2 we present the model for the jet and for the magnetic field. |
83 we show the results for a homogeneous jet. in £4 those for a structured jet ancl in §5 those for a Gaussian Finallyjet. | In 3 we show the results for a homogeneous jet, in 4 those for a structured jet and in 5 those for a Gaussian jet. |
Phe comparison and discussion can be found in 86. | The comparison and discussion can be found in 6. |
in 87 we derive and discuss our conclusions. adding possible complications to the nmoclels. | Finally in 7 we derive and discuss our conclusions, adding possible complications to the models. |
‘Phroughout this paper the adopted cosmological parameters are Lf)= 65. O4=0.7 and Q,,,=0.3. | Throughout this paper the adopted cosmological parameters are $H_0=65$ , $\Omega_{\lambda}=0.7$ and $\Omega_{m}=0.3$. |
In this paper we show results obtained with two clillerent codes. | In this paper we show results obtained with two different codes. |
Vhe first one is. Lully described. in S03 while the ποσο: one is discussed in this section. | The first one is fully described in S03 while the second one is discussed in this section. |
The main dilference oetween the two codes is in the treatment of the sicleway expansion and dyvnanmics in the non-relativistic phase. | The main difference between the two codes is in the treatment of the sideway expansion and dynamics in the non-relativistic phase. |
In he folowing we only remind the reader of the dilferent assuy(ions adopted by 503 and refer the reader to the xiper for more details. | In the following we only remind the reader of the different assumptions adopted by S03 and refer the reader to the paper for more details. |
We assume that the energv. released from. the engine is in the form of two oppositeJets. | We assume that the energy released from the engine is in the form of two opposite jets. |
They are described. by he following distributions ofinitial Lorentz factor Du and enerev per unit solid angle ο with respect to the jet axis (6 =): where θα is the jet opening angle. 6. is the core angular size. c.=c(0) and P=F(0). | They are described by the following distributions ofinitial Lorentz factor $\Gamma_0$ and energy per unit solid angle $\epsilon$ with respect to the jet axis $\theta=0$ ): where $\theta_{jet}$ is the jet opening angle, $\theta_{c}$ is the core angular size, $\epsilon_{c}=\epsilon(0)$ and $\Gamma_{c}=\Gamma(0)$. |
In the following. in order to make the comparison with the homogeneous jet easier. we will use preferentially the local isotropic equivalent. energy. defined as £7;,,(0)=45x «(0). | In the following, in order to make the comparison with the homogeneous jet easier, we will use preferentially the local isotropic equivalent energy, defined as $E_{iso}(\theta)=4\,\pi\,\epsilon(\theta)$ . |
In Eqs. | In Eqs. |
1. and 2 a.ap Controls the shape of the energy and Vy distributions in the wings. while δντην controls the smoothness of thejoint. between thejjet core ancl its wings. | \ref{eq:E}
and \ref{eq:G} $\alpha_{\epsilon},\alpha_{\Gamma}$ controls the shape of the energy and $\Gamma_0$ distributions in the wings, while $\beta_{\epsilon},\beta_{\Gamma}$ controls the smoothness of the joint between the jet core and its wings. |
Ifa,ApΞ0. equaions 1 and 2 describe the standard top hat model with sharp edges{ (homogeneousnmjet). | If $\alpha_{\epsilon},\alpha_{\Gamma}=0$, equations \ref{eq:E}
and \ref{eq:G} describe the standard top hat model with sharp edges (homogeneous jet). |
i£ observerthe observer line of sight is located within thejet. detects the GRB prompt phase and its CilB afterglow (CLA): if the viewing angle 6, is larger then θε he observes what it is called an orphan afterglow. an afterglow not preceded by the prompt 5-ray emission. | If the observer line of sight is located within the jet, the observer detects the GRB prompt phase and its GRB afterglow (GA); if the viewing angle $\theta_o$ is larger then $\theta_{jet}$, he observes what it is called an orphan afterglow, an afterglow not preceded by the prompt $\gamma$ -ray emission. |
When a, the code describes a structured jet: in this case 6), is assumed be always much larger then the observer angle. | When $\alpha_{\epsilon}>0$, the code describes a structured jet; in this case $\theta_{jet}$ is assumed to be always much larger then the observer angle. |
In fact we consider here a boundless jet (the end of the wings are so dim that are undetectable) in contrast to the sharp edged homogeneous jet. | In fact we consider here a boundless jet (the end of the wings are so dim that are undetectable), in contrast to the sharp edged homogeneous jet. |
Loa,=2 and S05x. the structured is that described in RLRO2 while S03 adopts a,—2 and jet4—1. | If $\alpha_{\epsilon}=2$ and $\beta_{\epsilon}\to\infty$, the structured jet is that described in RLR02, while S03 adopts $\alpha_{\epsilon}=2$ and $\beta_{\epsilon}=1$. |
For the Gaussian jet we use instead For simplicity we assume axial symmetry. | For the Gaussian jet we use instead _0 For simplicity we assume axial symmetry. |
Our initial Lorentz factor distribution. satisfies αιX8. therefore regions on the shock front with clillerent D, and energy are causally disconnected. ancl they evolve. independently until L/h=8. | Our initial Lorentz factor distribution satisfies $1/\Gamma\le \theta$, therefore regions on the shock front with different $\Gamma_0$ and energy are causally disconnected and they evolve independently until $1/\Gamma=\theta$. |
In the numerical simulations we assume [ου all models a costant initial. Lorentz factor across the jet. with Py= | In the numerical simulations we assume for all models a costant initial Lorentz factor across the jet, with $\Gamma_0=10^{4}$. |
With this choice the lighteurves shown in this paper (for observed. times 15 min) are insensitive to the initial E distribution. | With this choice the lightcurves shown in this paper (for observed times $\gsim 15$ min) are insensitive to the initial $\Gamma$ distribution. |
As a matter of fact. for any Lu(8)250. the fireball ceecleration starts earlier. (£o245s(F3/no/V53)77) than the smallest time of the figures and. afterwards the evolution follows the BAL self similar solution and consequently the shown afterglow properties are independent from the initial Lorentz factor. | As a matter of fact, for any $\Gamma_0(\theta)\gsim 50$, the fireball deceleration starts earlier $t_d \simeq 245 s
(E_{53}/n_0/\Gamma_{0,2}^{8})^{1/3}$ ) than the smallest time of the figures and afterwards the evolution follows the BM self similar solution and consequently the shown afterglow properties are independent from the initial Lorentz factor. |
HE relativistic kinematic ellects. freeze out. the lateral expansion or the pressure gradient. prevents mixing. the dillerent parts of the How are virtually independent along their entire evolution. | If relativistic kinematic effects freeze out the lateral expansion or the pressure gradient prevents mixing, the different parts of the flow are virtually independent along their entire evolution. |
We allow each. point of spherical coordinates. (r.8.o) to evolve acliahatically aud as iit were part of a uniform jet with «"mi=«(»pendently.. i)—Eof) and semi-aperture angle 8. | We allow each point of spherical coordinates $r,\theta,\phi$ ) to evolve adiabatically and independently, as if it were part of a uniform jet with $\epsilon=\epsilon(\theta)$, $\Gamma_0=\Gamma_0(\theta)$ and semi-aperture angle $\theta$. |
VPherclore. if mixing of matter is unimportant this treatment is correct at any time. otherwise it gives an approximate solution for L/PAd. | Therefore, if the mixing of matter is unimportant this treatment is correct at any time, otherwise it gives an approximate solution for $1/\Gamma\gg\Delta \theta$. |
Actually numerical hvedrodynamical simulations seem to suggest that c(8) cloes not vary appreciably with time until the non-relativistic phase sets in (kumar Cranot 2003) thus supporting our numerical approach. | Actually numerical hydrodynamical simulations seem to suggest that $\epsilon(\theta)$ does not vary appreciably with time until the non-relativistic phase sets in (Kumar Granot 2003) thus supporting our numerical approach. |
The full set. of πο. letermine the dvnanmies of (rh pajeh ol: yejet is:í18RE&).e (og. BECPanaitescu Kumar 2000. thereafter PISO0) where the parameter f (the ratio of the swept-up mass to the initial fireball rest mass) is given by: Aly is the rest mass of the two (symmetric) jets. O(r)nmun2ἐπί cos0(r)) is their solid angle and p(r) is the medium matter density. | The full set of equations that determine the dynamics of each patch of the jet is:, (e.g. Panaitescu Kumar 2000, thereafter PK00) where the parameter $f$ (the ratio of the swept-up mass to the initial fireball rest mass) is given by:, where $M_0$ is the rest mass of the two (symmetric) jets, $\Omega(r)=4\pi\,(1-\cos\theta(r))$ is their solid angle and $\rho(r)$ is the ambient medium matter density. |
The evolution of the solid angle is describedby: | The evolution of the solid angle is describedby: . |
For the comoving lateral velocity ὃς we tested. threedifferent recipes. | For the comoving lateral velocity $c_s$ we tested threedifferent recipes. |
First. we analyze a non-sidewavs expanding (NSIZ) jet with | First, we analyze a non-sideways expanding (NSE) jet with c_s=0. |
then a . expandingD (SE)a. one. either.. with. a constant comoving sidewayssound speed (Rhoads 1999) or with a more accurate treatment. which takes into | then a sideways expanding (SE) one, either with a constant comoving sound speed (Rhoads 1999) or with a more accurate treatment, which takes into |
Knowledge of the physical factors that control the conversion of interstellar gas into stars is of fundamental importance for both developing a predictive physical theory. of star formation and understanding the evolution of galaxies from the earliest epochs of cosmic history to the present time. | Knowledge of the physical factors that control the conversion of interstellar gas into stars is of fundamental importance for both developing a predictive physical theory of star formation and understanding the evolution of galaxies from the earliest epochs of cosmic history to the present time. |
An essential first step to obtaining such knowledge is to establish empirically the underlving relation or relationships that most clirectly connect the rate of star formation in a galaxy (o some general physical property of the interstellar eas [rom which stars form. | An essential first step to obtaining such knowledge is to establish empirically the underlying relation or relationships that most directly connect the rate of star formation in a galaxy to some general physical property of the interstellar gas from which stars form. |
A little more than a hall-century. ago. Schmidt. (1959) conjectured that this might take the form of a scaling relation between the rate of star lormation and some power. n. of the surface densitv of atomic (111) gas. | A little more than a half-century ago, Schmidt (1959) conjectured that this might take the form of a scaling relation between the rate of star formation and some power, n, of the surface density of atomic (HI) gas. |
From evaluation of the distributions of local HI gas and stars orthogonal to the Galactic plane. he suggested Chat ne 2. | From evaluation of the distributions of local HI gas and stars orthogonal to the Galactic plane, he suggested that $\approx$ 2. |
Subsequent studies comparing the surface densities of OD stars and HII regions with Chose of atomic gas within nearby external galaxies produced scaling laws with similar. super-linear. power-law indices (e.g.. Sanduleak 1969: Hamajina Tosa 1915). | Subsequent studies comparing the surface densities of OB stars and HII regions with those of atomic gas within nearby external galaxies produced scaling laws with similar, super-linear, power-law indices (e.g., Sanduleak 1969; Hamajima Tosa 1975). |
By the 1980s it became clear that molecular. not atomic. clouds were the sites of star formation in galaxies. | By the 1980s it became clear that molecular, not atomic, clouds were the sites of star formation in galaxies. |
The ability to make sensitive CO molecular-line observations enabled. [or the first time. the measurement of total gas surface densities (My, yy) in external galaxies | The ability to make sensitive CO molecular-line observations enabled, for the first time, the measurement of total gas surface densities $\Sigma_{HI + H_2}$ ) in external galaxies |
Residual proper motion maps for the reference stars listed in Tables 3-6. | Residual proper motion maps for the reference stars listed in Tables 3-6. |
The dispersion around the mean is + 0.31. £0.79. £ 0.5L and + 0.11 mas tin BLA. and = 0.52. 0.71. + 0.58. £ 0.62 mas vr.tin Decl. | The dispersion around the mean is $\pm$ 0.34, $\pm$ 0.79, $\pm$ 0.54, and $\pm$ 0.41 mas $^{-1}$ in R.A., and $\pm$ 0.52, $\pm$ 0.71, $\pm$ 0.58, $\pm$ 0.62 mas $^{-1}$ in Decl., |
for QO159-6127. Q0557-6713. QU558-6707 aud Q0615-6615. Relative positions in Rieht Ascension (Aacosd) es. epoch of observation Lor the stucied fields. | for Q0459-6427, Q0557-6713, Q0558-6707 and Q0615-6615, Relative positions in Right Ascension $\Delta\alpha$ $\delta$ ) $vs.$ epoch of observation for the studied fields. |
The values of Aacosd represent the individual positious of the QSO on different CCD [rames relative to the barycenter of the SRE. | The values of $\Delta\alpha$ $\delta$ represent the individual positions of the QSO on different CCD frames relative to the barycenter of the SRF. |
Syiubol sizes are proportional to the uumber of times the measurements vielled the same coordinate value for a particular epoch (extra small. sinall. mecitun. large. aud extra large sizes indicate 1 through 5 measurements per epoch. respectively). | Symbol sizes are proportional to the number of times the measurements yielded the same coordinate value for a particular epoch (extra small, small, medium, large, and extra large sizes indicate 1 through 5 measurements per epoch, respectively). |
The best-fit straight lines [rom linear regression analyses on the data are also Relative positious in declination (Ad) es. epoch of observation for the studied fields. | The best-fit straight lines from linear regression analyses on the data are also Relative positions in declination $\Delta\delta$ ) $vs.$ epoch of observation for the studied fields. |
The values of Ad represent the individual positious of the QSO on different CCD frames relative to the barycenter of the SRE. | The values of $\Delta\delta$ represent the individual positions of the QSO on different CCD frames relative to the barycenter of the SRF. |
Symbol sizes aud best-fit straight lines as described in Fig | Symbol sizes and best-fit straight lines as described in Fig |
HORIZONS system (Ciorginietal.1997). | HORIZONS system \citep{yeoman}. |
.. As mentioned earlier in section 2. we have mace observations through various apertures for sampling the comet. | As mentioned earlier in section 2, we have made observations through various apertures for sampling the comet. |
However. 267 aperture is used more often and if not mentioned specifically. we will use this aperture for further discussion. | However, 26" aperture is used more often and if not mentioned specifically, we will use this aperture for further discussion. |
This aperture corresponds to à projected. diameters 730550km. during November 5-7 ancl 733670 km on December 13. 2008. | This aperture corresponds to a projected diameters $\sim$ 30550km during November 5-7 and $\sim$ 33670 km on December 13, 2008. |
The sampled area is large and while small scale inhomogeneities are expected to average oul. laree structures in coma might still show up. | The sampled area is large and while small scale inhomogeneities are expected to average out, large structures in coma might still show up. |
The polarisation values in the blue narrow band are associated. with larger errors due to poor S/N ratio. | The polarisation values in the blue narrow band are associated with larger errors due to poor $S/N$ ratio. |
To address the possibility. of continuum. DC€ band contamination by molecular emission. which might allect the degree of polarisation. we used the spectra provided. by 3uil (private. | To address the possibility of continuum BC band contamination by molecular emission, which might affect the degree of polarisation, we used the spectra provided by Buil (private. |
. Examining the spectrum of November 1. a very weak emission feature (<3% of the continuum) appears to partly. overlap with the DC band. | Examining the spectrum of November 1, a very weak emission feature $<3\%$ of the continuum) appears to partly overlap with the BC band. |
Even if we take the upper limit of 3X. contamination of the BC band. by this feature. it will change the degree of polarisation by <0.05%. | Even if we take the upper limit of $3\%$ contamination of the BC band by this feature, it will change the degree of polarisation by $<0.05\%$. |
In fact in the present case. any such correction will increase the absolute value of the polarisation that will improve the fit. | In fact in the present case, any such correction will increase the absolute value of the polarisation that will improve the fit. |
Molecular production rate in τοΠοιος is reported to. decay exponentially with time (Schleicher2009).. the contribution of molecular emission will further reduce at later date (e.g. November 5-7 and December 13) compared. to November 1. 2007 when the spectrum. was taken. | Molecular production rate in 17P/Holmes is reported to decay exponentially with time \citep{schleicher2009}, the contribution of molecular emission will further reduce at later date (e.g. November 5-7 and December 13) compared to November 1, 2007 when the spectrum was taken. |
Hence we have ignored. the contamination of the BC band by molecular During our observing run. the phase angle remained less than 207 and observed polarisation is negative. | Hence we have ignored the contamination of the BC band by molecular During our observing run, the phase angle remained less than $20^\circ$ and observed polarisation is negative. |
In the following we discuss the wavelength and. phase angle dependence of polarisation ancl compare the present results with the results of Itosenbushetal.(2009). | In the following we discuss the wavelength and phase angle dependence of polarisation and compare the present results with the results of \citet{rosenbush2009}. |
. Figure 2. presents vs a curve for a«25 for the comets in the blue. band (4845A)) and the red band (6000«A τοῦ). | Figure \ref{PP6840} presents vs $\alpha$ curve for $\alpha <
25^\circ$ for the comets in the blue band ) and the red band $6000<\lambda<7000\AA$ ). |
Since our observations are made at low phase angles where the polarisation is negative. we have limited the plots to phase angle <25°. | Since our observations are made at low phase angles where the polarisation is negative, we have limited the plots to phase angle $< 25^{\circ}$. |
Also plotted are the data from Ixiselev's catalogue (Ixiselevetal.2005). for the comets which have been observed by various. researchers in blue and. rec-bandstall narrow anc broad bands). | Also plotted are the data from Kiselev's catalogue \citep{kiselevCat} for the comets which have been observed by various researchers in blue and red-bands(all narrow and broad bands). |
As observations in narrow continuum. bands at the low phase angles are scanty. we considered all the observations mace in red filters (6000<A« 7000.1) to generate an average vs a curve for à«25°. | As observations in narrow continuum bands at the low phase angles are scanty, we considered all the observations made in red filters $6000<\lambda<7000\AA$ ) to generate an average vs $\alpha$ curve for $\alpha < 25^\circ$. |
Unpublished data. as mentioned in the catalogue. and the data for which. PA; 0.0fora«20 and the outliers from the main trend have been ignored. | Unpublished data, as mentioned in the catalogue, and the data for which $> 0.0$ for $\alpha < 20^{\circ}$ and the outliers from the main trend have been ignored. |
The best polvnomial fit thus generated is presented in Figure 2. with solid The present observed: values of the polarisation in 17? are shown in two curves with solid. circles with error. bars (zm). | The best polynomial fit thus generated is presented in Figure \ref{PP6840} with solid The present observed values of the polarisation in 17P are shown in two curves with solid circles with error bars $\pm \sigma$ ). |
For plotting purpose. all the data obtained. through the same filter but with dilferent. apertures are averaged | For plotting purpose, all the data obtained through the same filter but with different apertures are averaged |
icelines considerably interior to Jupiters 5.2 AU orbit. | icelines considerably interior to Jupiter's 5.2 AU orbit. |
Uncertainties in opacities mean that while (he particle mass in most T Tauri disks is al least that of the MSN. it could be larger if mass is hidden in particulates much larger than (he wavelength of the observations. | Uncertainties in opacities mean that while the particle mass in most T Tauri disks is at least that of the MSN, it could be larger if mass is hidden in particulates much larger than the wavelength of the observations. |
While ISO detected warm IH» around T Tauri stars (Thietal.2000).. more detections with higher singal-to-noise are needed (ο give stringent constraints on gas mass. | While ISO detected warm $_2$ around T Tauri stars \citep{thi00}, more detections with higher singal-to-noise are needed to give stringent constraints on gas mass. |
We briefly review Sekiva's (1998) lechuique for deriving dust density profiles before interpreting the singular cusps which appear in these profiles. | We briefly review Sekiya's (1998) technique for deriving dust density profiles before interpreting the singular cusps which appear in these profiles. |
For small particles which are well-coupled to eas motions. when (he particle stopping tme is shorter than the dvnamical and eddy turnover (times. (he gas-solid mixture can be thought of as a single stratified (Iuid. | For small particles which are well-coupled to gas motions, when the particle stopping time is shorter than the dynamical and eddy turnover times, the gas-solid mixture can be thought of as a single stratified fluid. |
This limit applies verv well to mam-sized and smaller particles. however the particles must be large enough so (hat their settling (mes. see (6)). are shorter than the time since global turbulence becomes weak enough to allow particulate settling. | This limit applies very well to mm-sized and smaller particles, however the particles must be large enough so that their settling times, see \ref{set}) ), are shorter than the time since global turbulence becomes weak enough to allow particulate settling. |
The well-coupled limit is Che most conservative assumption in which to demonstrate GI since larger. decoupled solids are stirred less efficiently by gas turbulence. | The well-coupled limit is the most conservative assumption in which to demonstrate GI since larger, decoupled solids are stirred less efficiently by gas turbulence. |
Racial hvdrostatie balance vields an orbital velocity of the combined fluid which depends on the vertical distribution or particles (Nakagawa.Sekiva.&Hayashi1986): To derive the above. we have ignored the variation of (he gas clensity in the vertical direction since (he particulate subcdisk is much thinner than a gas scale height. | Radial hydrostatic balance yields an orbital velocity of the combined fluid which depends on the vertical distribution or particles \citep{nsh86}: To derive the above, we have ignored the variation of the gas density in the vertical direction since the particulate subdisk is much thinner than a gas scale height. |
When solids providemost of the inertia. py>py. the (nid motion is IXeplerian. but when gas dominates we have the usual formula for pressure-supported rotation. ¢=(1—ae. | When solids providemost of the inertia, $\rho_{\rm p} \gg \rho_{\rm g}$, the fluid motion is Keplerian, but when gas dominates we have the usual formula for pressure-supported rotation, $v_{\phi} = (1-\eta)v_{\rm K}$. |
As particles settle towards the midplane. the vertical shear rate (which promotes mixing) and the buovaney (which stabilizes against mixing) both increase. | As particles settle towards the midplane, the vertical shear rate (which promotes mixing) and the buoyancy (which stabilizes against mixing) both increase. |
This competition between destabilizing and stabilizing influences is conventionally characterized bv (the Richardson number: In the above formulation. the Drunt-Vaisala frequency. IN. is a measure of buovancy: | This competition between destabilizing and stabilizing influences is conventionally characterized by the Richardson number: In the above formulation, the Brunt-Vaisala frequency, $N$ , is a measure of buoyancy: |
malch between (he N-rav centroid aud the Tycho Reference Catalog position of (he star (17469309:015.οπω Hog et al. | match between the X-ray centroid and the Tycho Reference Catalog position of the star $17^{\rm h}46^{\rm m}39\fs075,
-28^\circ 53^\prime 51\farcs73$; Hog et al. |
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