ReadingTimeMachine/rtm-sgt-ocr-v1 · Datasets at Hugging Face

We address the linear stability problem with the small disturbances depending on time as exp(-tewr).

We address the linear stability problem with the small disturbances depending on time as $\mathrm{exp}(-\mathrm{i}\omega t)$.

À positive imaginary part of the eigenvalue co means an instability.

A positive imaginary part of the eigenvalue $\omega$ means an instability.

The radial scales of the disturbances are assumed small compared to the stellar radius while the equations are global in both the horizontal dimensions.

The dependencies on radius and longitude ὁ are taken as Fourier modes exp+kr).

The dependencies on radius and longitude $\phi$ are taken as Fourier modes $\mathrm{exp}(\mathrm{i}m\phi + \mathrm{i}kr)$.

The linear equations for small perturbations in differentially rotating fluids with toroidal magnetic fields are already given by Kitchatinov Riiddiger (2008)).

The linear equations for small perturbations in differentially rotating fluids with toroidal magnetic fields are already given by Kitchatinov Rüddiger \cite{KR08}) ).

Here. the nonmagnetic version of the equations 1s considered im detail.

Here, the nonmagnetic version of the equations is considered in detail.

The equations are formulated for normalized parameters (the rules of conversion to physical variables are given below).

The equation for the potential W of the toroidal flow reads where ©=w/Q, 1s the normalized eigenvalue. Q=Q/Qp is the normalized rotation rate. jj=cos8. V is the poloidal flow potential. is the angular part of the Laplacian operator. and is the key parameter for the influence of the stratification.

The equation for the potential $W$ of the toroidal flow reads where $\hat{\omega} = \omega/\Omega_0$ is the normalized eigenvalue, $\hat{\Omega} = \Omega/\Omega_0$ is the normalized rotation rate, $\mu = \cos\theta$, $V$ is the poloidal flow potential, is the angular part of the Laplacian operator, and is the key parameter for the influence of the stratification.

The diffusion terms are characterized by the parameters where v and y are the microscopic viscosity ard the thermal conductivity.

The diffusion terms are characterized by the parameters where $\nu$ and $\chi$ are the microscopic viscosity and the thermal conductivity.

Apart from the Eq. (3))

Apart from the Eq. \ref{3}) )

for the toroidal flow. the complete system of three equations includes the equatior for poloidal flow. and the equation for the normalized entropy S. The equations (3). (7) and (8) form an eigenvalue problem which we solved numerically.

for the toroidal flow, the complete system of three equations includes the equation for poloidal flow, and the equation for the normalized entropy $S$, The equations \ref{3}) ), \ref{7}) ) and \ref{8}) ) form an eigenvalue problem which we solved numerically.

The diffusion parameters (6)) for the upper radiative core of the Sun are e,=107 and &=2x107!° which are used in the calculations.

The diffusion parameters \ref{6}) ) for the upper radiative core of the Sun are $\epsilon_\chi = 10^{-4}$ and $\epsilon_\nu = 2\times 10^{-10}$ which are used in the calculations.

The disturbances in physical units follow from. their normalized values by The velocity field can be restored from the potentials of poloidal (P,,) and toroidal (Τη) flows. (Chandrasekhar 1961)). where e,. ej. and e, are unit vectors in the radial. meridioral and longitudinal directions.

The disturbances in physical units follow from their normalized values by The velocity field can be restored from the potentials of poloidal $P_u$ ) and toroidal $T_u$ ) flows, (Chandrasekhar \cite{C61}) ), where $\vec{e}_r$, $\vec{e}_\theta$, and $\vec{e}_\phi$ are unit vectors in the radial, meridional and longitudinal directions.

Without rotation (Q— 0) and for small diffusion the equations (3)). (7)) and (8)) reproduce the spectrum of g-modes.

Without rotation $\Omega\rightarrow 0$ ) and for small diffusion the equations \ref{3}) ), \ref{7}) ) and \ref{8}) ) reproduce the spectrum of $g$ -modes.

poke More related to the stability problem is the limit of very large 2-parameter (5)) which leads to the following 2D approximation,

More related to the stability problem is the limit of very large $\hat\lambda$ -parameter \ref{5}) ) which leads to the following 2D approximation.

The ratio of ΑΛΩΣ in stars can be so large (~10° in the upper radiative core of the Sun) that (51) can also be large in spite of short-wave approximation in radius. Ar>>I.

The ratio of $N^2/\Omega^2$ in stars can be so large $\sim 10^5$ in the upper radiative core of the Sun) that $\hat{\lambda}^2$ \ref{5}) ) can also be large in spite of short-wave approximation in radius, $kr \gg 1$.

In the limit of large 2? the above equation system reduces to its 2D approximation.

In the limit of large $\hat{\lambda}^2$ the above equation system reduces to its 2D approximation.

In leading order of this parameter Eq. (7))

In leading order of this parameter Eq. \ref{7}) )

gives S=0.

gives $S = 0$.

Then it follows from (8)) that V=0 and Eq. (3))

Then it follows from \ref{8}) ) that $V = 0$ and Eq. \ref{3}) )

reduces to the standard equation of 2D theory of Watson (1981)). describing toroidal flows on spherical surfaces.

reduces to the standard equation of 2D theory of Watson \cite{W81}) ), describing toroidal flows on spherical surfaces.

The2D approximation 15 justified for stable oscillations with not too short radial scales so that 2 remains large.

The2D approximation is justified for stable oscillations with not too short radial scales so that $\hat{\lambda}$ remains large.

Its validity for stability problem 15 less certain because the radial scales of most rapidly growing modes are not know in advance and the value of kr for those modes is normally so large that | (Kitehatinov Rüddiger 2008)).

Its validity for stability problem is less certain because the radial scales of most rapidly growing modes are not know in advance and the value of $kr$ for those modes is normally so large that $} 1$ (Kitchatinov Rüddiger \cite{KR08}) ).

For rigid rotation Eq. (12))

For rigid rotation Eq. \ref{12}) )

4 jd.

provides the eigenvalue spectrum of the r-modes (Papaloizou Pringle \cite{PP78}) ).

chan

Instabilities can emerge with nonuniform rotation.

ges its sign (Watson 1981)).

The condition for instability is that the second derivative, $\mathrm{d}^2((1-\mu^2)\hat{\Omega})/\mathrm{d}\mu^2$ , changes its sign (Watson \cite{W81}) ).

For the angular velocity profile

Many statistics have been put forward to characterize the 2lem emission from the ΕΟΚ. the power spectrum probably being the most frequently studied (222?.tochoosesomerecent examples)...

Many statistics have been put forward to characterize the 21cm emission from the EoR, the power spectrum probably being the most frequently studied \citep[][to choose some recent examples]{BAR08,LID08,PRI08,SET08}.

We have suggested that higher-order statistics may be useful not only to characterize a CS cube that has been cleaned of oregrounds. noise and instrumental effects. but also to extract the signature of reionization from these corrupting influences in the first place.

We have suggested that higher-order statistics may be useful not only to characterize a CS cube that has been cleaned of foregrounds, noise and instrumental effects, but also to extract the signature of reionization from these corrupting influences in the first place.

The skewness of the one-point distribution of brightness emperature. measured as a function of observed frequency (or equivalently as a function of redshift). is one such promising statistic.

The skewness of the one-point distribution of brightness temperature, measured as a function of observed frequency (or equivalently as a function of redshift), is one such promising statistic.

The three detailed simulations of reionization which we have studied show a strong evolution of the skewness with redshift.

Some of the features of this evolution appear to be generic and can be readily understood: in the early stages of reionization the skewness drops below that of the underlying density field as the first ionized bubbles. from which the emission is negligible. are formed.

Some of the features of this evolution appear to be generic and can be readily understood: in the early stages of reionization the skewness drops below that of the underlying density field as the first ionized bubbles, from which the emission is negligible, are formed.

As reionization progresses. the majority of the volume becomes ionized and the skewness increases again. becoming very large at low redshift when the distributionof brightness temperature is peaked at zero. with a tail extending to large values.

As reionization progresses, the majority of the volume becomes ionized and the skewness increases again, becoming very large at low redshift when the distribution of brightness temperature is peaked at zero, with a tail extending to large values.

In simulation f250C there is a well defined dip in the skewness with a width Am].

In simulation f250C there is a well defined dip in the skewness with a width $\Delta z\approx 1$.

Our other simulation in which the Universe is reionized entirely by stars CI-star) shows a more gradual change. with the epoch of reionization extending throughout the redshift range probed by LOFAR.

Our other simulation in which the Universe is reionized entirely by stars (T-star) shows a more gradual change, with the epoch of reionization extending throughout the redshift range probed by LOFAR.

A third simulation. T-QSO. in which QSOs reionize the Universe. shows an intermediate behaviour.

A third simulation, T-QSO, in which QSOs reionize the Universe, shows an intermediate behaviour.

Bv combining these simulations with models of the foregrounds. noise and instrumental response. we have generated datacubes which are intended to simulate the output of the LOFAR EoR experiment.

By combining these simulations with models of the foregrounds, noise and instrumental response, we have generated datacubes which are intended to simulate the output of the LOFAR EoR experiment.

We have studied two cases: firstly. one in which we smooth the foregrounds and signal to the resolution of the telescope using a Gaussian kernel. then add uncorrelated Gaussian noise: secondly. one in which we degrade the foregrounds and noise to the resolution of the telescope using a realistic PSF. and add noise which is uncorrelated in the Fourier plane rather than the image plane. producing what we refer to as ‘dirty’ images.

We have studied two cases: firstly, one in which we smooth the foregrounds and signal to the resolution of the telescope using a Gaussian kernel, then add uncorrelated Gaussian noise; secondly, one in which we degrade the foregrounds and noise to the resolution of the telescope using a realistic PSF, and add noise which is uncorrelated in the Fourier plane rather than the image plane, producing what we refer to as `dirty' images.

In the former case. we can see the signature of reionization in the skewness by fitting out the foregrounds to obtain residual images. and then denoising these images with a simple smoothing operation.

In the former case, we can see the signature of reionization in the skewness by fitting out the foregrounds to obtain residual images, and then denoising these images with a simple smoothing operation.

The skewness in these images as a function of redshift shows significant evidence of reionization.

The result is quite robust to the details of the foreground fitting and the smoothing. Under-

or over-fitting the foregrounds affects the recovered skewness less severely than the recovered variance.

Extracting a signal from the dirty cubes requires a more sophisticated denoising scheme.

In an optimistic scenario where the correlation matrices of the original signal and of the noise are known. we can again recover the evolution of the skewness quite cleanly using Wiener deconvolution.

In an optimistic scenario where the correlation matrices of the original signal and of the noise are known, we can again recover the evolution of the skewness quite cleanly using Wiener deconvolution.

We have touched upon some areas for improvement: simulations which remain realistic but extend to larger scales and exhibit an even greater range of reionization histories: taking into account the polarization of the foregrounds and the instrumental response. and incorporating new observational constraints as they arrive: testing the minimal assumptions we must make about the signal in order for our extraction scheme to work. for example whether a poor estimate of the correlation matrix. of the CS seriously affects the extracted skewness: and studying a wider range of statistics beyond the variance and power spectrum.

We have touched upon some areas for improvement: simulations which remain realistic but extend to larger scales and exhibit an even greater range of reionization histories; taking into account the polarization of the foregrounds and the instrumental response, and incorporating new observational constraints as they arrive; testing the minimal assumptions we must make about the signal in order for our extraction scheme to work, for example whether a poor estimate of the correlation matrix of the CS seriously affects the extracted skewness; and studying a wider range of statistics beyond the variance and power spectrum.

All of these will be areas for future work.

Even at this stage. however. our results justify some optimism that the new generation of radio telescopes can detect the signature of reionization using order statistics.

Even at this stage, however, our results justify some optimism that the new generation of radio telescopes can detect the signature of reionization using higher-order statistics.

GH is supported by a grant from the Netherlands Organisation for Scientitic Research (NWO).

GH is supported by a grant from the Netherlands Organisation for Scientific Research (NWO).

As LOFAR members. the authors are partially funded by the European Union. European. Regional Development Fund. and by 'Samenwerkingsverband Noord-Nederland’. EZ/KOMPAS.

As LOFAR members, the authors are partially funded by the European Union, European Regional Development Fund, and by `Samenwerkingsverband Noord-Nederland', EZ/KOMPAS.

GM and IT acknowledge that this study was supported in part by Swiss National Science Foundation grant 200021-116696/1 and Swedish Research Council grant 60336701.

GM and II acknowledge that this study was supported in part by Swiss National Science Foundation grant 200021-116696/1 and Swedish Research Council grant 60336701.

Finally. collecting (3.22)) and (3.23)). we get that there exists a constant C=C(T.Mj) independent of 0<2X1 such that: which gives the announced result.

Finally, collecting \ref{EM:I_1}) ) and \ref{EM:I_2}) ), we get that there exists a constant $C=C(T,M_0)$ independent of $0<\e\le 1$ such that: which gives the announced result.

This Corollary is a straightforward consequence of Remark 2.1.. Theorem 2.2.. estimate (3.21)) and Lemma 3.3.. In this section. we prove (hat (he svstem (1.1))-(1.2)) admits solutions « in the distributional sense.

$\hfill\Box$ This Corollary is a straightforward consequence of Remark \ref{molli_est}, Theorem \ref{EM:theo:exip_regu}, estimate \ref{L1_estimate}) ) and Lemma \ref{EM:lem:etem}.. In this section, we prove that the system \ref{EM:burger}) \ref{EM:initialdata}) ) admits solutions $u$ in the distributional sense.

They are the limits of vw given by Theorem 2.2. when 7— 0. To do this. we will justify the passage to the limit as z tends to 0 in Che svstem (2.14)) by using some compactniess tools that are presented in a first subsection.

They are the limits of $u^{\e}$ given by Theorem \ref{EM:theo:exip_regu} when $\e\rightarrow 0$ To do this, we will justify the passage to the limit as $\e$ tends to $0$ in the system \ref{EM:burgersapp}) ) by using some compactness tools that are presented in a first subsection.

First. for all open interval J of R. wedenote by

First, for all open interval $I$ of $\R$ , wedenote by

27.5 and 22.5 Me remains, but these merge at t£~1.1x105.

27.5 and 22.5 $\mearth$ remains, but these merge at $t\sim 1.1\times 10^5$.

The final state of the system is then a 50 Me planet orbiting at r~1.5.

The final state of the system is then a 50 $\mearth$ planet orbiting at $r\sim 1.5$.

A similar outcome is obtained in Model3 in which the long-term evolution resulted in a 15 Me planet evolving in a high-eccentricity orbit with a semi-major axis of a,~2.

A similar outcome is obtained in Model3 in which the long-term evolution resulted in a 15 $\mearth$ planet evolving in a high-eccentricity orbit with a semi-major axis of $a_p\sim 2$.

At earlier times, the increase in eccentricities following disc dispersal led to a collision between the 12.5 and 22.5 Μο bodies, thereby forming a new 35 Mg planet.

At earlier times, the increase in eccentricities following disc dispersal led to a collision between the 12.5 and 22.5 $\mearth$ bodies, thereby forming a new 35 $\mearth$ planet.

At t~2x10°, the latter is observed to undergo a close encounter with the cental binary, leading to this body being completely ejected from the system.

At $t\sim 2\times 10^5$, the latter is observed to undergo a close encounter with the cental binary, leading to this body being completely ejected from the system.

Interestingly, the three-planet system in Model2 appears to be dynamically stable over long time scales, with the planets maintaining their commensurabilities.

This indicates that multiplanet resonant systems could potentially be found in circumbinary discs, where the existence of the resonance helps to maintain the stability of the system.

In this paper we have presented the results of hydrodynamic simulations aimed at studying the evolution of multiple planets embedded in a circumbinary disc.

We first focused on a system consisting of a pair of planets interacting with each other.

We assumed that one body is trapped at the edge of the inner cavity formed by the binary, while the other migrates inward from outside the orbit of the innermost body.

Our calculations show different outcomes, depending on the planet mass ratio q=m;/mpo.

Our calculations show different outcomes, depending on the planet mass ratio $q=m_i/m_o$

of the core-collapse supernovae (CC-SNe: Woosley(1993))) and mergers of neutron stars with another neutron star (NS-NS) or wilh a companion black hole hole (BII-NS: Paczviski (1991)))the currently. favored astronomical progenitors of cosmological ganmuna-ray bursts (GRBs).

of the core-collapse supernovae (CC-SNe; \cite{woo93}) ) and mergers of neutron stars with another neutron star (NS-NS) or with a companion black hole hole (BH-NS; \cite{pac91}) )–the currently favored astronomical progenitors of cosmological gamma-ray bursts (GRBs).

While all short GRD are probably produced by mergers. the converse need not hold in general.

While all short GRB are probably produced by mergers, the converse need not hold in general.

Black holes with a neutron star companion should have a diversity in spin 1999).

Black holes with a neutron star companion should have a diversity in spin \citep{van99}.

. These mixed binaries can be short and long lived. depending on the angular velocily of the black hole.

These mixed binaries can be short and long lived, depending on the angular velocity of the black hole.

It predicts that some of the short GRBs. produced by slowly spinning black holes. also feature X-ray aftereglows (vanPutten90014) as in GRD050509D (Gehrelsetal.2005). ancl GRBOS50709 (Villasenoretal.2005:Foxal. 2005).

It predicts that some of the short GRBs, produced by slowly spinning black holes, also feature X-ray afterglows \citep{van01a} as in GRB050509B \citep{geh05} and GRB050709 \citep{vil05,fox05,hjo05}.

. Long GRBs. produced by rapidly rotating black holes. ave expected to form in CC-SNe trom short. intra-«dayv. period binaries (Paczviski(1998):vanPutten (2004))). but also [rom mergers wilh a companion neutron star (wanPutten1999). or out of the merger of (wo neutron stars (Daiottietal.2008:vanPutten2009)..

Long GRBs, produced by rapidly rotating black holes, are expected to form in CC-SNe from short, intra-day period binaries \cite{pac98,van04}) ), but also from mergers with a companion neutron star \citep{van99} or out of the merger of two neutron stars \citep{bai08,van09b}.

A diversity in the origin of long GRBs in CC-SNe and mergers naturally accounts for evenis wilh and without supernovae. notably. GIRDOGOGId (vanPutten2008:Caitoetal. 2009).. with and without pronounced X-ray allerelows (vanPullen&Gupta2009) and in wind versus constant densitv host environments. that are relevant to recent studies of extraordinary. and. FermiLAT events (Cenkoetal.2010a.b).

A diversity in the origin of long GRBs in CC-SNe and mergers naturally accounts for events with and without supernovae, notably GRB060614 \citep{van08a,caito09}, with and without pronounced X-ray afterglows \citep{van09} and in wind versus constant density host environments, that are relevant to recent studies of extraordinary and -LAT events \citep{cen10,cen10b}.

. Rapidly rotating black holes can sweep up surrounding matter and induce the formation of mulüpole mass-moments.

Rapidly rotating black holes can sweep up surrounding matter and induce the formation of multipole mass-moments.

La (his process. spin energv is catalviicallv converted to a long duration gravitational wave burst (GWD: vanPutten (2003))).

In this process, spin energy is catalytically converted to a long duration gravitational wave burst (GWB; \cite{van01b,van03a}) ).

For stellar mass black holes. (his output max be detected by advanced gravitational wave detectors for events in the local Universe.

For stellar mass black holes, this output may be detected by advanced gravitational wave detectors for events in the local Universe.

As candidate inner engines to long GRBs. evidence for the associated spin down of the black hole has been found in the normalized light curve of 600 long GRBs in the DATSE catalogue (vanPutten&Gupta2009).

As candidate inner engines to long GRBs, evidence for the associated spin down of the black hole has been found in the normalized light curve of 600 long GRBs in the BATSE catalogue \citep{van09}.

. The high Irequency range of the planned aclvanced detectors LIGO-Vireo (Barish&Weiss1999;Arceseetal. 2004).. the Large-scale Crvogenic Gravitalional-wave Telescope (LCGT. Ixurodaetal. (2010))) and the Einstein Telescope (ET. Hild.Chelkowski (2008))) covers the «quadrupole emission spectrum of orbital motions around stellar mass black holes. thus establishing a window to rigorously probe (the inner most workings of GRBs and some of the CC-SNe.

The high frequency range of the planned advanced detectors LIGO-Virgo \citep{bar99,arc04}, the Large-scale Cryogenic Gravitational-wave Telescope (LCGT, \cite{lcgt}) ) and the Einstein Telescope (ET, \cite{et08}) ) covers the quadrupole emission spectrum of orbital motions around stellar mass black holes, thus establishing a window to rigorously probe the inner most workings of GRBs and some of the CC-SNe.

We anticipate an event rate for long GWDs of 0.4-2 per vear within a distance of LOO Alpe from the local event rate of long GRBs (Guetta&DellaValle2007).

We anticipate an event rate for long GWBs of 0.4-2 per year within a distance of 100 Mpc from the local event rate of long GRBs \citep{gue07}.

. It compares favorably with that of mergers of binary neutron stars (e.g O'Shaughnessy (2010))).

It compares favorably with that of mergers of binary neutron stars (e.g \cite{osh08,aba10}) ).

Since the local event rate of type Ib/c supernovae is ~80 per vear within a cdistauce of 100 Mpe. the branching ratio of Type Ib/c supernovae into long GRBs is therefore rather

Since the local event rate of type Ib/c supernovae is $\sim 80$ per year within a distance of 100 Mpc, the branching ratio of Type Ib/c supernovae into long GRBs is therefore rather

Due to the changing shape the SPL solution in this case requires some artificial viscosity in order to prevent particle interpenetration during the compression phase.

Due to the changing shape the SPH solution in this case requires some artificial viscosity in order to prevent particle interpenetration during the compression phase.

We apply this using the switch as discussed in relscc:sph..

We apply this using the switch as discussed in \\ref{sec:sph}.

The particle distribution during the evolution of the exact. non-axisymimetric mode is shown in Figure 16.. where initially Vp= Vio=1/2. Ve;=1 and Vo»=1/4.

The particle distribution during the evolution of the exact, non-axisymmetric mode is shown in Figure \ref{fig:teqmnonaxi}, , where initially $V_{11} = 1$, $V_{12}=1/2$, $V_{21}=1$ and $V_{22} = 1/4$.

For comparison with the exact solution we solve. (55))- (62)) using a simple second order modified Euler predictor-corrector method. using the conservation of mass (64)) asa checkon the quality of the integration.

For comparison with the exact solution we solve \ref{eq:dV11dt}) \ref{eq:dBdtnonaxi}) ) using a simple second order modified Euler predictor-corrector method, using the conservation of mass \ref{eq:massnonaxi}) ) as a checkon the quality of the integration.

The solid line shown in Figure 16. is the curve corresponding to the edge of the ‘Tov Star (ic.

The solid line shown in Figure \ref{fig:teqmnonaxi} is the curve corresponding to the edge of the Toy Star (ie.

where p= 0) at the appropriate times.

where $\rho=0$ ) at the appropriate times.

The SPL particles adjust to the changing shape quite well apart from a damping of the amplitude with time caused by the application of artificial viscosity.

The SPH particles adjust to the changing shape quite well apart from a damping of the amplitude with time caused by the application of artificial viscosity.

We have performed a range of simulations using cllferent values of the initial parameters which in general show very similar results.

We have performed a range of simulations using different values of the initial parameters which in general show very similar results.

For simulations with very strong compression the particles can clump together in the manner described in refsee:static due to the force being zero at the origin of the cubic spline kernel used in the caleulations. despite using h=1.2(mp2

For simulations with very strong compression the particles can clump together in the manner described in \\ref{sec:static} due to the force being zero at the origin of the cubic spline kernel used in the calculations, despite using $h=1.2 (m/\rho)^{1/2}$.

This is because a strong compression can push the particles neighbours close enough to be in the region of the kernel where the force decreases towards the origin. causing a clumping instability for positive pressures in compression similar in its clleet to the well known instability for negative stresses in tension(?).

This is because a strong compression can push the particles neighbours close enough to be in the region of the kernel where the force decreases towards the origin, causing a clumping instability for positive pressures in compression similar in its effect to the well known instability for negative stresses in tension.

. As discussed in& ro[secistatie the remedy for this is to use a kernel with non- derivative at the origin. an investigation of which will be performed elsewhere.

As discussed in \\ref{sec:static} the remedy for this is to use a kernel with non-zero derivative at the origin, an investigation of which will be performed elsewhere.

The primary aim of this paper has been to provide a set of benchmarks For simulations of gaseous disks of the kind that arise in star formation.

The primary aim of this paper has been to provide a set of benchmarks for simulations of gaseous disks of the kind that arise in star formation.

Phe svstems. which we call ‘Tov Stars. are similar to their astrophysical counterparts in that they consist of compressible gas. held together by an attractive force which results in the gas having an outer surface where the density ancl pressure fall to zero.

The systems, which we call Toy Stars, are similar to their astrophysical counterparts in that they consist of compressible gas, held together by an attractive force which results in the gas having an outer surface where the density and pressure fall to zero.

They therefore provide. tests which are quite different to the usual tests based. on Dow in periodic rectangular regions.

They therefore provide tests which are quite different to the usual tests based on flow in periodic rectangular regions.

Furthermore. not only can the linear modes be found. in terms of known functions. but non linear solutions can be [ound in terms of a small number of dillerential equations.

Furthermore, not only can the linear modes be found in terms of known functions, but non linear solutions can be found in terms of a small number of differential equations.

The results described. and discussed in this paper can be summarized as follows: DIP is supported by a PPARC postcloctoral research [ellowship.

The results described and discussed in this paper can be summarized as follows: DJP is supported by a PPARC postdoctoral research fellowship.

Lle also acknowledges the support) of the Commonwealth Scholarship Commission and the Cambridge Commonwealth Trust.

He also acknowledges the support of the Commonwealth Scholarship Commission and the Cambridge Commonwealth Trust.

DJP also wishes to thank Monash Universityfor their hospitality during a visit.

lor the axisvnunetric modes s= 0.

For the axisymmetric modes $s=0$ .