Pulsation modes for increasingly relativistic polytropes

Nils Andersson1* and Kostas D. Kokkotas2

1 Department of Physics, Washington University, St Louis MO 63130, USA2 Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece

Accepted 1998 February 2. Received 1997 November 27; in original form 1997 June 11

A B S T R A C TWe present the results of a numerical study of the fluid f, p and the gravitational w modes forincreasingly relativistic non-rotating polytropes. The results for f and w modes are in goodagreement with previous data for uniform density stars, which supports an understanding ofthe nature of the gravitational wave modes based on the uniform density data. We show that thep modes can become extremely long-lived for some relativistic stars. This effect is attributed tothe change in the perturbed density distribution as the star becomes more compact.

Key words: radiation mechanisms: non-thermal – stars: neutron.

1 I N T RO D U C T I O N

In a recent paper (Andersson, Kojima & Kokkotas 1996a) wepresented a detailed survey of the pulsation modes of the simplestconceivable stellar model: a non-rotating star with uniform density.The reasons for chosing this, admittedly unrealistic, model weretwofold. First of all, the analytic solution to the TOV equations foruniform density is well-known (Schutz 1985). This means thatcalculations could readily be performed for many different valuesof stellar compactness and average density. Secondly, more realisticmodels for a neutron star are ‘almost constant density’ so the resultsfor this simple model may not be too different from what one wouldfind for realistic equations of state.

The main conclusions of our study were as follows.

(i) Gravitational-wave (w) modes (Kokkotas & Schutz 1992)exist for all stellar models, even though the modes become veryshort lived for less relativistic stellar models. Moreover, the w modespectra are qualitatively similar for axial and polar perturbations(for a description of the two classes of relativistic perturbations –also referred to as odd and even parity perturbations – see Thorne &Campolattaro 1967 or Kojima 1992). Since axial perturbations donot couple to pulsations in the stellar fluid (Thorne & Campolattaro1967) we concluded that the w modes are pure ‘space–time’ modesthat do not depend on the particulars of the fluid for their existence.This notion is supported by results obtained for various modelscenarios (Andersson, Kokkotas & Schutz 1996b; Andersson1996).

(ii) The behaviour of the various pulsation modes changesdramatically as the star is made more compact than R < 3M (weuse geometrized units c ¼ G ¼ 1). For such ultracompact stars theexistence of a peak in the effective curvature potential outside thesurface of the star will affect the various modes. Some modes canbasically be considered as ‘trapped in a potential well’, cf.

Chandrasekhar & Ferrari (1991) and Kokkotas (1994). Thistrapping has the effect that some w modes become extremelylong-lived.

(iii) There are ‘avoided crossings’ between the fluid f mode andthe various polar w modes for extremely compact stars. Thissuggests that the f mode should be considered as the first in thesequence of trapped modes for these stars.

The results of the mode-survey for uniform density modelshelped improve our understanding of the origin of, and the relationbetween, various pulsation modes of relativistic stars. We wouldexpect the main conclusions to hold also for more realistic equa-tions of state (at least qualitatively), but the motivation for con-firming these expectations by actual calculations is strong. Forexample, axial w modes have so far only been calculated foruniform density stars. Although our present understanding of theorigin of the w modes indicates that axial modes must exist for allstellar models it is important to verify that this is, indeed, the case.Furthermore, more realistic stellar models support other families ofmodes. Of particular interest for gravitational-wave physics are thep modes (that correspond to pressure waves in the stellar fluid). Thefact that the properties of the f and w modes change dramatically asthe star becomes increasingly relativistic leads to questions whetherthe p modes are also affected in an interesting way. With this shortpaper we address these issues.

We describe results obtained for quadrupole (l ¼ 2) modes of thepolytropic equation of state

p ¼ krG;

where k ¼ 100 km2. Our main study was for G ¼ 2, but we alsoconsidered other (especially larger) values of G. Our results arebasically an extension of those presented by Andersson, Kokkotas& Schutz (1995). A detailed description of the way that we extractthe complex frequencies of the various pulsation modes can befound in that paper. In order to keep the present paper short, we willdiscuss neither this numerical approach nor the various perturbationequations here. The equations governing polar perturbations are

Mon. Not. R. Astron. Soc. 297, 493–496 (1998)

q 1998 RAS

*Present address: Institut fur Astronomie und Astrophysik, UniversitatTubingen, Auf der Morgenstelle 10, D-72076 Tubingen, Germany.

described in detail by Lindblom & Detweiler (1981), while the axialequations are given by Chandrasekhar & Ferrari (1991).

2 N U M E R I C A L R E S U LT S

The results of our investigation for increasingly relativistic poly-tropes can be summarized in a single figure. In Fig. 1 we show theinverse damping rate (represented by Im qM) of each quadrupole(l ¼ 2) mode as a function of the pulsation frequency (Re qM) for aG ¼ 2 polytrope. As the central density of the stellar model is variedeach mode traces out a curve in the complex q-plane. In the figurewe have indicated (by diamonds) the densest polytropic model thatis stable to radial perturbations. That is, the specific model forwhich the mass (M) reaches a maximum as a function of the centraldensity (rc). For the G ¼ 2 polytrope used to calculate the data in

Fig. 1 the marginally stable model has central densityrc ¼ 5:7 × 1015 g cm¹3. This corresponds to a compactnessR=M < 3:77.

It is clear that Fig. 1 contains a considerable amount of informa-tion. We now proceed to discuss the results for each separate class ofmodes in more detail.

2.1 Gravitational wave modes

As far as the w modes are concerned our investigation does not addmuch to the study for uniform density stars (Andersson et al.1996a). The results for polytropes are, both qualitatively andquantitatively, in excellent agreement with those for the uniformdensity model. From the upper panel in Fig. 1 it is clear that both theaxial and the polar w modes can be divided into two separatefamilies. The two families are distinguished by the behaviour of themode frequencies as the star becomes very relativistic. For onefamily of modes – the interface w modes (Andersson et al. 1996a;Leins, Nollert & Soffel 1993) – the frequencies approach constant(complex) values for large rc. For the specific polytrope underconsideration we find that the mode frequencies hardly change at allfor rc > 1:9 × 1016 g cm¹3. The modes in the second family – thecurvature w modes – are characterized by an increase in jqMj withrc for less compact stars, but before the star becomes radiallyunstable jqMj reaches a maximum value. The modes in this familythen become extremely slowly damped as rc increases further. Theslowest damped w modes become long-lived as R < 3:5M or so. Thedecreased damping rate is most likely because of the increasedinfluence of the curvature potential barrier (the peak of which is atR < 3M). However, it is important to realize that the G ¼ 2polytrope that we consider is always less compact thanR ¼ 3:2M. Hence, the surface of these stars is never locatedinside the peak of the curvature potential. That the modes stillbecome extremely slowly damped shows that the exterior potentialcan ‘trap’ gravitational waves effectively even though there is no‘potential well inside a barrier’ for these stars, cf. the discussion byDetweiler (1975).

As already mentioned, the present results are very similar to thosefor uniform density stars. Especially worth noticing are the simila-rities between the axial and the polar w modes in Fig. 1. Since theaxial modes cannot induce oscillations in the stellar fluid, thisindicates that the character of the w mode depends solely on theproperties of the curved space–time. In other words: the w modesare ‘space–time’ modes the details of which do not depend on thedynamics of the stellar fluid (Andersson et al. 1996a; Andersson etal. 1996b).

2.2 Fluid modes

The present results for the fluid f mode are also in good agreementwith the uniform density study (Andersson et al. 1996a). For lesscompact stars the damping rate of the f mode increases with rc. Thissimply means that the star radiates gravitational waves moreefficiently as it becomes more compact. However, in a similarway to the curvature w modes, the f mode becomes slower dampedonce the star approaches R ¼ 3M. Again, this can be understood asthe increasing influence of the curvature potential (Detweiler 1975;Andersson et al. 1996a).

A notable difference between the results for the G ¼ 2 polytropeand the uniform density star is the absence of ‘avoided crossings’between the polar w modes and the f mode, cf. fig. 2 of Andersson etal. (1996a). The reason for this is that the G ¼ 2 polytrope never

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Figure 1. The change in the complex mode frequencies as the central densityof the stellar model varies (rc increases in the direction of the arrows). Thedata are for a G ¼ 2 polytrope. The gravitational w modes are shown in theupper panel, while the much longer lived fluid f and p modes are in the thelower panel (The f mode is the leftmost mode in the lower panel.). Axial wmodes are represented by dashed lines while polar modes are shown as solidlines. The diamonds on each curve indicates the densest stellar model that isstable to radial perturbations. Arrows indicate the direction of increasingstellar compactness.

becomes sufficiently compact for such avoided crossings to occur.For the uniform density model the first avoided crossing occured forR < 2:3M, and for the G ¼ 2 polytrope we always have R > 3:2M.One would, however, expect avoided mode-crossings to exist forpolytropes that can be made very compact. We have verified theexistense of avoided crossings for a G ¼ 5 model (which can bemade as compact as R < 2:39M). The existence of avoided cross-ings thus seems to be a generic feature of extremely compactrelativistic stars.

The results we present for the l ¼ 2 p modes are both new andremarkable. Fluid p modes have not previously been calculated forextremely relativistic stellar models. As can be seen in Fig. 1 thesemodes behave in a peculiar ways as rc increases. While thebehaviour for less-relativistic stars is analogous to the f and the wmodes – the damping rate increases when the star becomes morecompact – it is different for the very relativistic models. Once thestar has become radially unstable jqMj attains a maximum. Thenthe pulsation frequency Re qM typically decreases monotonically.At the same time the damping rate decreases drastically and Im qMhas a sharp minimum. For some modes we find several such minimaof Im qM, cf. Fig. 1. The corresponding values of the imaginarypart of the mode frequency are actually so small that our code doesnot have sufficient precision to distinguish them from zero. It is, ofcourse, important to establish that the sign of Im qM does notchange. A change in sign would indicate that the mode becomeslinearly unstable. We believe that our calculations were sufficentlyaccurate to establish that this does not happen – the p modes are allstable. However, it is interesting to note that they become extremelylong-lived (very poor radiators of gravitational waves) for somevalues of the central density.

How can we understand the peculiar behaviour of the p-modefrequencies? Let us first consider the pulsation frequency Re qM,and compare our results to ones for simpler stellar models. Thewell-known result for the f mode, established for incompressiblefluid spheres by Kelvin in 1863 (Tassoul 1978) , suggests that ourmode should approach

q2f ¼

2lðl ¹ 1Þ

2l þ 1M

R3

� �; ð1Þ

for less relativistic stars. A similar result, for compressible homo-geneous spheres (Tassoul 1978), shows that the p modes ought toapproach

q2p ¼ ½Dn þ

��������������������������D2

n þ lðl þ 1Þ

qÿ

M

R3

� �; ð2Þ

where

2Dn ¼ ½2l þ 3 þ nð2n þ 2l þ 5ÞÿG ¹ 4 ; n ¼ 0; 1; 2; . . . ð3Þ

and G is the polytropic index. As can be seen from Fig. 2 ournumerical results agree quite well with these approximations (forl ¼ 2). That is, the pulsation frequency of each fluid mode changesin the anticipated way as the star become more relativistic. Alter-natively, the results in Fig. 2 can be taken as justification for usingthe approximate results also for relativistic stars. They clearlyprovide useful estimates also for objects of neutron star compact-ness, R=M < 5.

The peculiar behaviour of the p-mode damping rates can beexplained by changes in the matter distribution inside the star.According to the standard quadrupole formula, gravitational wavesresulting from the fluid motion in the star can be estimated by

h ,1r

�R

0r4 ∂2

∂t2 drðr; tÞ dr ¼ ¹q2eiqt

r

�R

0r4drðr;qÞdr ; ð4Þ

where dr is the Eulerian variation in the density. We have assumedthat the perturbation is monochromatic with a harmonic time-dependence drðr; tÞ ¼ drðr;qÞeiqt. In the notation of Lindblom &Detweiler (1981) the required density perturbation follows from(for the quadrupole)

drðr;qÞ ¼ ¹r2 e¹n=2X ¹p þ r

r3 el=2ðM þ 4ppr3ÞWh

¹ ðp þ rÞH0

� r1¹G

kG; ð5Þ

where we have used the fact that the acoustic wave speed in apolytrope is kGrG¹1.

To get to a useful result, we must also figure out how eachcomplex-frequency mode contributes to the relevant physicalquantity (such as the density variation). The reason is obvious:for complex frequencies q the drðr;qÞ that follows from (5) will becomplex, whereas the corresponding physical quantity should bereal valued. The analysis of this problem proceeds exactly as for theanalogous black hole problem (Andersson 1997). The main stepsare (i) note that the modes come in pairs qn and ¹q*

n (where theasterisk denotes complex conjugation). (ii) Since all perturbationequations (see Lindblom & Detweiler 1981) contain only q2, andðq2Þ* ¼ ð¹q*Þ2, we can conclude that

drðr;¹q*nÞ ¼ ½drðr;qnÞÿ

*: ð6Þ

From a specific mode frequency qn we therefore get a contribution(for more details see Andersson 1997)

drðr; tÞ , 2Re drðr;qnÞeiqnt� �

: ð7Þ

Both the real and the imaginary part of the eigenfunction drðr;qnÞ

will thus be important.Let us now assume that the gravitational-wave damping is

sufficiently slow that the mode is essentially undamped for anentire cycle (this is a reasonable assumption for the slowly damped

Pulsation modes for increasingly relativistic polytropes 495

q 1998 RAS, MNRAS 297, 493–496

Figure 2. The pulsation frequencies of the fluid modes as functions of thestellar compactness R=M. We compare our results for f, p0, p1 and p2 (solidlines, from bottom to top) to approximate results for homogeneous stellarmodels (dashed lines). The curves for the p1 and the p2 mode terminate whenthe damping rate of these modes become too small to be computable with ournumerical method. The corresponding ReqM could (although we have notdone this) then be determined from a real-frequency approach. The result ofsuch a calculation should approach the approximate results for large R=M.

fluid modes). Averaging over one period we get

< h2> ,e¹2Imqnt

�R

0r4drðr;qnÞdr

���� ����2¼ e¹2ImqnteðqnÞ : ð8Þ

The value of eðqnÞ provides a measure of how ‘efficient’ a specificmode is as a radiator according to the quadrupole formula. Anassociation between the p mode minima of Im qM and minima ineðqnÞ would indicate that the features seen in Fig. 1 are as a result ofchanges in the perturbed density distribution as rc increases. InFig. 3 we compare Im qM to eðqnÞ. This figure seems to establishthe anticipated correlation between minima in the two functions forthe first few p modes.

Having found the likely explanation for the extremely slowdamping of various p modes for some values of rc, it is worthwhileto discuss how the eigenfunction drðr;qnÞ changes as we vary rc.Let us first consider the f mode. This mode is distinguished by thefact that the eigenfunctions (i) have no nodes inside the star and (ii)grow monotonically towards the surface of the star. The p modes aredifferent; in general there are n nodes in both the real and theimaginary part of drðr;qÞ for the nth p mode. However, the locationof these nodes change as we vary rc. We find that the distribution ofdrðr;qÞ changes drastically with varying rc. Specifically, for thefirst p mode and rc < 1 × 1016 g cm¹3 the single node is close to the

surface of the star and the bulk of drðr;qÞ is located in the outer 50 percent of the star. However, for rc > 3 × 1016 g cm¹3 the node hasmoved close to the centre of the star and the bulk of drðr;qÞ is nowlocated inside half the radius of the star. It is understandable that thesetwo, very different, configurations can be rather different as radiatorsof gravitational waves. For all p modes the trend is that the bulk of theeigenfunctions move towards the centre of the star as we increase rc.In a way, this means that the p modes for the very relativistic star havethe same features as the g modes (Tassoul 1978).

3 F I N A L C O M M E N T S

We have presented numerical results for the pulsation modes ofnon-rotating relativistic polytropes. These results agree well, bothqualitatively and quantitatively, with previously obtained results foruniform density stars (Andersson et al. 1996a). This providesfurther evidence for our understanding of the nature of the variouspulsation modes, especially the notion that the gravitational wave wmodes are mainly a result of the properties of the curved space–time (Andersson et al. 1996b). Our present survey also considers thep modes of a relativistic star. We find that the polytrope p modefrequencies are close to an approximation based on the pulsation ofa compressible homogeneous sphere for less relativistic stars.However, we also show that the p modes change in a peculiarway as the star becomes more compact. Specifically, we find thateach p mode can be very slowly damped (in fact, almost undamped)for some values of the central density. We have shown that thisfeature can be understood in terms of the change in the distributionof the perturbed density with varying central density. Although theresult is not of tremendous astrophysical importance (since theaffected stars are not stable to radial perturbations, anyway) it isinteresting to note that the efficiency with which a p mode radiatesgravitational waves can vary considerably between two stellarmodels that have almost identical central densities.

AC K N OW L E D G M E N T

KDK gratefully acknowledges support from the Greek GSRTthrough the PENED programme No. 797.

REFERENCES

Andersson N., 1996, Gen. Rel. Grav, 28, 1433Andersson N., 1997, Phys. Rev. D, 55, 468Andersson N., Kojima Y., Kokkotas K.D., 1996a, ApJ, 462, 855Andersson N., Kokkotas K. D., Schutz B. F, 1995, MNRAS, 274, 1039Andersson N., Kokkotas K. D., Schutz B. F, 1996b, MNRAS, 280, 1230Chandrasekhar S., Ferrari V., 1991, Proc. R. Soc. Lond., A434, 449Detweiler S. L., 1975, ApJ, 197, 203Kokkotas K. D., Schutz B. F., 1992, MNRAS, 255, 119Kokkotas K. D., 1994, MNRAS, 268, 1015Kojima Y., 1992, Phys. Rev. D, 46, 4289Lindblom L., Detweiler S., 1983, ApJS, 53, 73Leins M., Nollert H-P., Soffel M. H., 1993 Phys. Rev. D, 48, 3467Schutz B. F., 1985, A first course in general relativity, Cambridge Univ. PressTassoul J. L, 1978, Theory of rotating stars, Princeton Univ. PressThorne K. S., Campolattaro A., 1967, ApJ, 149, 591

This paper has been typeset from a TEX=LATEX file prepared by the author.

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Figure 3. We compare the damping rate of the fluid pulsation modes(represented by Im qM and solid curves) to the efficieny measure eðqnÞ

(dashed curves). Both quantities are shown as functions of the centraldensity of the stellar model (in units of 1014 g cm¹3) The scale forIm qM is logarithmic and ranges from 10¹8 to 10¹2 while the scale foreðqnÞ is arbitrary. The data are for (a) the f mode, (b)–(d) the first three pmodes. Of interest here is a possible correlation between minima in the twofunctions. Such a correlation explains the minima in the damping rate of thep modes in terms of changes in the perturbed density distribution in the star.[The f mode is only included for comparison – there should be no minima ineðqnÞ for that mode.]