General relativistic study of the neutrino path and calculation of the minimum photosphere fordifferent stars

Ritam Mallick1,2,* and Sarbani Majumder1,2

1Department of Physics, Bose Institute, 93/1, A.P.C Road, Kolkata - 700009, India2Centre for Astroparticle Physics and Space Science, Bose Institute, 93/1, A.P.C Road, Kolkata - 700009, India

(Received 25 November 2008; published 6 January 2009)

A detailed general relativistic (GR) calculation of the neutrino path for a general metric describing a

rotating star is studied. We have calculated the neutrino path along the equatorial and polar plane. The

expression for the minimum photosphere radius (MPR) is obtained and matched with the Schwarzschild

limit. The MPR is calculated for the stars with two different equations of state (EOS) each rotating with

two different velocities, to analyze the dependence of EOS and rotational velocity on the MPR. The results

show that the MPR for the hadronic star is much greater than the quark star and the MPR increases as the

rotational velocity of the star decreases. The MPR along the polar plane is larger than that along the

equatorial plane.

DOI: 10.1103/PhysRevD.79.023001 PACS numbers: 95.30.Sf

I. INTRODUCTION

Gamma ray bursts (GRBs), the possible engines forGRBs, and their connection with the neutrino productionis a field of high current interest. It was proposed that theneutrino-antineutrino annihilation to electron-positronpairs in compact stars is a possible and important candidateto explain the energy source of GRBs. The previous cal-culations of the reaction ! eþe in the vicinity of aneutron star have been based on Newtonian gravity [1,2],i.e. ð2GM=c2RÞ 1. The effect of gravity was first in-corporated in Refs. [3,4], but only for a static star.

Neutron stars are objects formed in the aftermath ofsupernova. The central density of these stars can be ashigh as 10 times that of normal nuclear matter. At suchhigh density, any small perturbation, e.g. spin down of thestar, may trigger the phase transition from the nuclear toquark matter system. As a result, the neutron star may fullyconvert to a quark star or a hybrid star with a quark core. Ithas been shown [5] that such a phase transition [6] pro-duces a large amount of high energy neutrinos. Theseneutrinos (and antineutrinos) could annihilate and giverise to electron-positron pairs through the reaction !eþe. These eþe pairs may further give rise to gammarays which may provide a possible explanation of theobserved GRB. Furthermore, the rotating neutron star hasbeen shown [7] to produce the observed beaming effect. Atpresent, it is necessary to have a better understanding of theenergy deposition in the neutrino annihilation to eþe inthe realistic neutron star environment.

We would like to study the þ ! eþ þ e energydeposition rate near a rotating compact star. This reactionis important for the study of gamma ray bursts. The generalrelativistic (GR) effect has been studied by [3,4,8]. Resultsin Refs. [3,4] argued it to increase immensely, while that by

[8] claim it to be much smaller. We need to study suchdeposition rate not only incorporating the GR effect butalso the rotational effect. The geodesic of neutrinos are alsoimportant in the study of pulse shapes and accretion discillumination [9] to name a few. Therefore, the path of theneutrino (or generally of the massless particle) is of im-mense importance and needs a detailed study. The neutrinopath for the Schwarzschild metric along the equatorialplane can be found in text books [10] and different papers[3,4]. Asano and Fukuyama [11,12] did the same calcula-tion near a thin accretion disc using the Kerr metric.Prasanna and Goswami [13] studied it for a slowly rotatingstar. In this paper, we present a detailed GR study of theneutrino path for a most general metric describing a rotat-ing star [14] along the equatorial and polar plane. We havemade our calculation using two different equations of state(EOS), one quark and the other hadronic.In this paper, we will first discuss the metric, the EOS,

and the star structure. Next, wewill present the detailed GRcalculation of the neutrino path and minimum photosphere.Finally, we will present our results for the two EOS andhave a brief discussion.

II. THE STAR

The structure of the star is described by the Cook-Shapiro-Teukolsky (CST) metric [14]

ds2 ¼ eþdt2 þ e2ðdr2 þ r2d2Þþ er2sin2ðd!dtÞ2: (1)

Accurate models of rotating neutron stars for tabulatedEOS can be computed numerically using the RNS code[15–17]. This computer code computes the metric func-tions , , , and! appearing in the axisymmetric metric,and these metric functions depend only on the coordinates and r. The metric function ! is the term responsible forthe frame dragging effect and would vanish if the rotational*ritam@bosemain.boseinst.ac.in

PHYSICAL REVIEW D 79, 023001 (2009)

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velocity () is zero. The coordinate r is related to thestandard radial coordinate that appears in the

Schwarzschild metric, rs, by rs ¼ reðÞ=2 [18]. In thelimit of zero rotation, the combination of metric functionsare

Lim !!0re ¼ rsffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2Mrs

q ; (2)

Lim !!0eðþÞ=2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2M

rs

s; (3)

Lim !!0e½ðþÞ=2dr ¼ drs

1 2Mrs

: (4)

We have previously mentioned that tabulated EOS areneeded to compute the code numerically. In this paper, wehave computed for two different EOS, the quark EOS andthe hadronic EOS. The hadronic EOS has been evaluatedusing the nonlinear Walecka model [19]. The Lagrangiandensity in this model is given by

L ¼ Xi

c iði@ mi þ giþ g!i!

gia

TaÞc i 1

4!! þ 1

2m2

!!!

þ 1

2ð@@m2

2Þ 1

4a

a

þ 1

2m2

a

a 1

3bmNðgNÞ3 1

4cðgNÞ4

þ c eði@ meÞc e: (5)

The Lagrangian in Eq. (5) includes nucleons (neutronsand protons), electrons, isoscalar scalar, isoscalar vector,and isovector vector mesons denoted by c i, c e, , !

,and a;, respectively. The Lagrangian also includes cubicand quartic self-interaction terms of the field. The pa-rameters of the nonlinear Walecka model are meson-baryon coupling constants, meson masses, and the coeffi-cient of the cubic and quartic self-interaction of the mesons (b and c, respectively). The meson fields interactwith the baryons through linear coupling. The ! and meson masses have been chosen to be their physicalmasses. The rest of the parameters, namely, nucleon-meson

coupling constants ( gm

, g!m!, and

gm

) and the coefficients of

cubic and quartic terms of the meson self-interaction (band c, respectively) are determined by fitting the nuclearmatter saturation properties, namely, the binding energy/nucleon ( 16 MeV), baryon density (0 ¼ 0:17 fm3),symmetry energy coefficient (32.5 MeV), Landau mass(0:83 mn), and nuclear matter incompressibility(300 MeV). We have used a stable three-flavor quarkmatter EOS, obtained from the standard Bag model with

B1=4 ¼ 145 MeV.

The shape of a fast rotating neutron star becomes oblatespheroid [14]. The star gets compressed along the z axis,and along the x and y axes it bulges by equal amounts; thepolar radius is thus smaller than the equatorial radius.

III. GR CALCULATION

The coordinate system is oriented such that the equato-rial plane lies along ¼

2 and the polar plane along ¼ 0.

The calculation is done along these two planes. The metricis independent of t and , and the coordinates are cyclic.Hence, the corresponding covariant generalized momentais constant [10], i.e.

pt ¼ p0 ¼ const ¼ E; p ¼ p3 ¼ const ¼ L:

(6)

The magnitude of the four-vector energy momentum isgiven by [10],

gpp þm

2 ¼ 0; (7)

where m is the rest mass of the particle and p ¼ dx

d ,

being the affine parameter. Writing it explicitly we have

g00p02 þ g11p

12 þ g22p22 þ g33p

32 þ g30p3p0

þ g03p0p3 þm

2 ¼ 0: (8)

The contravariant momenta are defined as

p0 ¼ g00p0 þ g03p3; p3 ¼ g30p0 þ g33p3:

To find the contravariant momenta we need the inversematrix g, which is given by

eðþÞ 0 0 weðþÞ0 e2 0 00 0 1

r2e2 0

weðþÞ 0 0 ðweðþÞ eðÞr2sin2

:

Therefore, the contravariant momenta are

p0 ¼ g00p0 þ g03p3 ¼ eðþÞðE!LÞ¼ eðþÞBð!Þ;

p3 ¼ g30p0 þ g33p3 ¼ eðþÞ½!Bð!Þ þ e2

r2sin2L:(9)

The Lagrangian of the system we are considering isgiven by

L0 ¼ 1

2g _x

_x; (10)

where _x ¼ dx

d . Therefore, writing it explicitly, we get

2L0 ¼ ðeþ þ e!2r2sin2Þ _t2 þ e2 _r2 þ e2r2 _2

þ er2sin2 _2 2e!r2sin2 _ _t : (11)

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From the Lagrangian, calculating the equation of motionfor , it is very easy to find that it is zero along theequatorial and polar plane. Therefore, the particle has atstart, and continues to have p ¼ p2 ¼ 0 in the givenplanes.

Finally, substituting these values in the above Eq. (8) weget

eðþÞB2 þ L2

r2sin2e þ e2

dr

d

2 þm

2 ¼ 0:

(12)

¼ m

, i.e. propertime per unit rest mass

dr

d¼ dr

d:d

d¼ m

dr

d;

dr

d¼ m

dr

d:d

d

¼ m

dr

d

d

d

d

d

;

dr

d¼

dr

d

:p:

(13)

We define E ¼ Em

and L ¼ Lm

. As the particle is massless

(neutrino), we define

Limm!0

LE¼ b; (14)

where b is the impact parameter. Substituting this in Eq.(12), we have

e2dr

d

2!ð1!bÞ þ be2

r2sin2

2 eðþÞð1!bÞ2

þ b2

r2sin2eþ3 ¼ 0: (15)

Using the above Lagrangian we can write the covariantmomenta, and from Eq. (6), we have

p0 ¼ pt ¼ @L0

@ _t

¼ ðeþ þ e!2r2sin2Þ _t e!r2sin2 _

¼ E;

p3 ¼ p ¼ @L0

@ _¼ er2sin2 _ e!r2sin2 _t ¼ L:

(16)

Having written the momenta in terms of total energy andtotal angular momentum it is quite simple to solve for _t and_. They are given by

_t ¼ E!L

eþ ; _ ¼ L

er2sin2þ!ðE!LÞ

eþ :

(17)

The angle r between the particle trajectory and thetangent vector to the orbit can be derived by constructingthe local Lorentz tetrad k for our metric

eðþÞ=2 0 0 00 e 0 00 0 re 0

eðÞ=2wr sin 0 0 eðÞ=2r sin

:

The angle r is given by

tanr ¼ V1

V2¼ k1rV

r

k3tVt þ k3V

; (18)

where Vr ¼ _r_t and V ¼ _

_t are the local velocities. Using

Eq. (17) V can be written as

V ¼ L

E!L:

e2

r2sin2þ! ¼ Aðr; Þ þ!: (19)

Therefore, the angle r is

tanr ¼ e

eðÞ=2r sin:

V

V !

dr

d

¼ e

eðÞ=2r sin:Aðr; Þ þ!

Aðr; Þdr

d

: (20)

Squaring the above equation and writing as

dr

d

2 ¼

A

Aþ!

2 eðÞr2sin2

e2tan2r; (21)

we get the final form of drd . Substituting this value in Eq.

(15), we get

e2

Aðr;ÞAðr;Þþ!

2eðÞr2sin2

e2tan2r

!ð1!bÞ

þ e2b

r2sin2

2eðþÞð1!bÞ2þ b2

r2sin2eðþ3Þ ¼0:

(22)

This equation can be solved using the potentials obtainedfrom the RNS code to obtain a minimum radius r ¼ R, theminimum photosphere radius, below which a masslessparticle (neutrino) emitted tangentially to the stellar sur-face (R ¼ 0) would be gravitationally bound.In the limit in which the CST metric reduces to the

Schwarzschild metric, i.e.

Lim !!0re ¼ rsffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2Mrs

q ;

Eq. (22) (for ¼ =2) reduces to the equation for bobtained by Salmonson and Wilson [3], i.e.

b ¼ rs cosrffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2M

rs

q :

GENERAL RELATIVISTIC STUDY OF THE NEUTRINO . . . PHYSICAL REVIEW D 79, 023001 (2009)

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IV. RESULTS

The minimum photosphere is calculated solving Eq.(22) using the potential functions obtained from the RNS

code. Starting our calculation by choosing the centralenergy density of the star to be 1 1015 gm=cm3.Figures 1–5, give the nature of the equations of stateused. In the figures, ¼ cos. Figure 1 shows that thequark matter EOS considered here is much steeper than thehadronic matter EOS. Figure 2 (for quark matter EOS)shows that at the center of the star the pressure is maxi-mum, and as we go outside it falls off and becomes zerooutside the star. Along the pole the pressure falls off in amuch steeper way than along the equator because the polarradius is much smaller. As the rotational velocity decreasesthe equatorial radius of the star decreases but the polarradius increases (although still less than equatorial radius).For the Keplerian velocity the star is maximally deformedand as the rotational velocity of the star decreases the starregains a more spherical shape. Figure 3 show the same nature for a hadronic star. In Figs. 4 and 5, the variation of

energy density is shown for the quark and hadronic starsimultaneously, and its nature is more or less similar to thatof the behavior of pressure discussed above.Using the quark matter EOS on the RNS code, the

Keplerian velocity of the quark star comes out to be 0:89104 s1. For comparison we have also computed the RNS

code with the rotational velocity of 0:5 104 s1. Thesame treatment is done also for the hadronic EOS wherethe Keplerian velocity is 0:61 104 s1 and for compari-son the other rotational velocity was chosen to be 0:4104 s1. The code solves the metric for the given EOS andgives the different potential functions as a function of r and. Solving Eq. (22) with these values of potential functionsfor different , we obtain the value of minimum photo-sphere for different planes. Table I sums up all our resultsin a compact form.

0 5e+14 1e+15 1.5e+15

energy density (gm/cm3)

0

1e+35

2e+35

pres

sure

(dy

ne/c

m2 )

HadronQuark

FIG. 1. Variation of pressure with energy density for quark andhadronic matter EOS.

0 5 10 15Radial distance (Km)

0

5e+34

1e+35

Pres

sure

(dy

ne/c

m2 )

Ω = 0.61, χ = 0Ω = 0.61, χ = 0.99Ω = 0.4, χ = 0Ω = 0.4, χ = 0.99

FIG. 3. Variation of pressure along the radial direction of thestar for two different rotational velocities each with two differentvalues of for the hadronic matter EOS.

0 5 10 15Radial distance (Km)

0

1e+35

2e+35

Pres

sure

(dy

ne/c

m2 )

Ω = 0.89, χ = 0Ω = 0.89, χ = 0.99Ω = 0.5, χ = 0Ω = 0.5, χ = 0.99

FIG. 2. Variation of pressure along the radial direction of thestar for two different rotational velocities each with two differentvalues of for the quark matter EOS.

0 5 10 15

Radial distance (Km)

5e+14

1e+15

Ene

rgy

dens

ity (

gm/c

m3 )

Ω = 0.89, χ = 0Ω = 0.89, χ = 0.99Ω = 0.5, χ = 0Ω = 0.5, χ = 0.99

FIG. 4. Variation of energy density along the radial direction ofthe star for two different rotational velocities each with twodifferent values of for the quark matter EOS.

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Let us now analyze the table given above. It points outthe fact that as the rotational velocity decreases, the massof the star also decreases. For the same central energydensity, the mass of the quark star is much greater thanthat of the hadronic star but the radius is much smaller. Itsignifies that the quark matter EOS considered in our workis much steeper than that of hadronic matter EOS aspointed out previously in the figures. As the rotationalvelocity decreases, the equatorial radius decreases but thepolar radius increases. It shows that the star is maximallydeformed for the Keplerian velocity and as the rotationalvelocity of the star decreases it tries to regain a morespherical shape. A static star is of spherical shape, wherepolar and equatorial radius are same.

In the above table, we have tabulated the minimumphotosphere radius (MPR) for two different values of ,i.e. for two planes. Along the equator ( ¼ 0) and alongthe pole ( ¼ 0:99). The MPR is much shorter along theequatorial plane than along the polar plane. The MPR is

minimum for the quark star rotating with Keplerian veloc-ity and is maximum for the hadronic star rotating with0:4 104 s1 velocity. The MPR is much greater for thehadronic star than the quark star. As the rotational velocitydecreases, the MPR shifts outward from the center of thestar toward the surface.

V. SUMMARYAND DISCUSSION

In this paper, we have addressed the problem of the pathof the neutrino and the radius of minimum photosphere.We have done a complete GR calculation of the neutrinopath for the most general metric describing a rotating star,and have obtained its geodesic equation along equatorialand polar plane. We have calculated the MPR for fourcases, i.e. stars with two different EOS and both rotatingwith two different velocities. Previous calculations of theneutrino path were either done for a static star [3,4] or for aslowly rotating star [13], and only along the equatorialplane. We have shown that our results also match verywell with the previous findings [3] for the Schwarzschildlimit. We have found that the MPR is greater along thepolar direction than along the equatorial direction. TheMPR is much greater for the hadronic star than that ofthe quark star. As the rotational velocity decreases theMPR increases and is maximum for the static star.Prasanna and Goswami [13] had showed that the MPR isinversely proportional to the rotational velocity of the star.Salmonson andWilson [3] had shown that for the static starthe MPR limit is R ¼ 3M, and that is very close to thesurface. So our results are at par with the previous findingsin those limits, and also give us a glimpse of the rotationaleffect.Finally, we would like to mention that this calculation of

the neutrino path is very important in the sense that thisforms the heart of a different problem like the GRB centralengine, pulse shape, and accretion disc illumination. Themain problem which still lies is the solution of the geodesicequation along any general plane (i.e. other than the equa-torial and polar plane), where p ¼ p2 0. This calcula-tion needs a detailed numerical analysis and cannot besolved analytically. This path is the general path followedby any massless particle (photon) in the vicinity of acompact object. Currently, we are trying to address theseproblems related to the neutrino path.

ACKNOWLEDGMENTS

R.M. and S.M. would like to thank CSIR for financialsupport.

TABLE I. MPR for quark and hadron stars rotating with twodifferent velocities calculated along the equator and the pole.

EOS in 104 s1 Mass in M re, rp in Km MPR in Km

Quark 0.89 2.8 12, 5.5 0 2.74

0.89 2.8 12, 5.5 0.99 3.11

0.5 2.2 9, 8 0 3.6

0.5 2.2 9, 8 0.99 4.96

Hadron 0.61 2 16, 9 0 3.9

0.61 2 16, 9 0.99 5.5

0.4 1.7 12, 10 0 4.85

0.4 1.7 12, 10 0.99 5.85

0 5 10 15Radial distance (Km)

0

5e+14

1e+15E

nerg

y de

nsity

(gm

/cm

3 )Ω = 0.61, χ = 0Ω = 0.61, χ = 0.99Ω = 0.4, χ = 0Ω = 0.4, χ = 0.99

FIG. 5. Variation of energy density along the radial direction ofthe star for two different rotational velocities each with twodifferent values of for the hadronic matter EOS.

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