Issue 
A&A
Volume 520, SeptemberOctober 2010



Article Number  A16  
Number of page(s)  5  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201015513  
Published online  23 September 2010 
Approximate analytic expressions for
circular orbits around rapidly rotating compact stars
(Research Note)
M. Bejger  J. L. Zdunik  P. Haensel
N. Copernicus Astronomical Center, Bartycka 18, 00716 Warszawa, Poland
Received 2 August 2010 / Accepted 12 August 2010
Abstract
Aims. We calculate stationary configurations of
rapidly
rotating compact stars in general relativity, to study the properties
of circular orbits of test particles in the equatorial plane. We search
for simple, but precise, analytical formulae for the orbital frequency,
specific angular momentum, and binding energy of a test particle that
are valid for any equation of state and any rotation frequency of the
rigidly rotating compact star, up to the massshedding limit.
Methods. Numerical calculations are performed using
precise 2D
codes based on multidomain spectral methods. Models of rigidly
rotating neutron stars and the spacetime outside them are calculated
for several equations of state of dense matter. Calculations are also
performed for quark stars consisting of selfbound quark matter.
Results. At the mass shedding limit, the rotational
frequency
converges to a Schwarzschildian orbital frequency at the equator. We
show that the orbital frequency of any orbit outside equator can also
be approximated by a Schwarzschildian formula. Using a simple
approximation for the framedragging term, we obtain approximate
expressions for the specific angular momentum and specific energy on
the corotating circular orbits in the equatorial plane of neutron star,
which are valid to the stellar equator. The formulae recover reference
numerical values with typically 1% of accuracy for neutron stars with
.
These values are less precise for quark stars consisting of selfbound
quark matter.
Key words: dense matter  equation of state  stars: neutron  stars: rotation
1 Introduction
Circular orbits of test particles moving freely along geodesics in a neutron star equatorial plane represent, to a very good approximation, orbits of gas elements in a lowmass Xray binary (LMXB) thin accretion disk. The simplest model of these orbits is obtained for a Schwarzschild spacetime produced by a static neutron star (Shapiro & Teukolsky 1983). However, accretion in LMXBs is associated with neutron star spinup, and is a commonly accepted mechanism for producing millisecond pulsars (Bhattacharya & van den Heuvel 1991; Alpar et al. 1982). This scenario is corroborated by the discovery of rapid pulsations with frequencies up to 619 Hz, in more than a dozen LMXBs. List of such bursters is given in Table 1 of Kiziltan & Thorsett (2009), which already needs an update: according to Galloway et al. (2009), the transient burster EXO 0748676 pulsates actually at 552 Hz instead of 45 Hz, and a new source Swift J1749.42807, pulsating at 518 Hz, was reported by Altamirano et al. (2010). Nine neutron stars in LMXBs spin at frequencies ranging from 524 Hz to 619 Hz, and their rotation significantly affects the spacetime and thus the orbits of test particles in their vicinity.
At a given distance from the star center, a test particle moves at an orbital frequency on a circular orbit in an equatorial plane, with the energy and angular momentum per unit rest mass and , respectively. Axial symmetry is assumed, and all quantities are defined in a chosen reference system, using suitable spacetime coordinates. The star is assumed to be rigidly rotating, its gravitational mass and angular momentum being M and J, respectively. Analytical formulae expressing , , and as functions of radial coordinates were obtained in the slowrotation approximation, keeping only linear terms in the star's angular momentum J, by Kluzniak & Wagoner (1985). Their expressions coincide, as they should, after an appropriate change of coordinates, with those obtained in the lowest order in J for the Kerr metric of a rotating black hole of the same M and J (see, e.g., Shapiro & Teukolsky 1983). However, exact 2D calculations show that the slowrotation approximation is not valid for orbits in the vicinity of compact stars (neutron stars and quark stars) rotating at frequency higher than 400 Hz (Zdunik et al. 2002; Berti et al. 2005; Shibata & Sasaki 1998; Miller et al. 1998).
The present research was motivated by a specific astrophysical project: modeling of the spinup process of an old ``radiodead pulsar'', via accretion within a LMXB, into a millisecond pulsar (Bhattacharya & van den Heuvel 1991). During this recycling process, the neutronstar magnetic field pushes the inner radius of thin Keplerian disk to r_{0}, which is usually significantly larger than the radius of the innermost stable circular orbit around an idealized zero magnetic field star. To perform efficiently a large number of simulations corresponding to different astrophysical scenarios, one needs reliable analytic expressions for and , valid for any rotation frequency up to the massshedding limit , and for r_{0} ranging from the stellar radius to a few thousand of radii. This range of r_{0} is required to model the complete process of the recycling of an old ``dead'' pulsar rotating at 0.1 Hz, with polar magnetic field 10^{12} G, to a millisecond pulsar rotating at 5001000 Hz, and also a pulsar with a polar magnetic field 10^{8}10^{9} G (Lorimer 2008). The inspiration for the present note came from our previous puzzling result on the implications of (still unconfirmed) detection of neutron star rotation at 1122 Hz for the equation of state (EOS) of dense matter (Bejger et al. 2007). At a given gravitational mass M, we found remarkably that the orbital frequency in the Schwarzschild spacetime produced by M, calculated at the actual massshedding equatorial circumferential radius, coincided (within a fraction of a percent) with a true massshedding frequency for this M, which is equal to the true orbital frequency of a test particle at the equator (Haensel et al. 2008; Bejger et al. 2007). The Schwarzschildian formula yielded true , when one inserted into it an actual equatorial radius. As we show explicitly in the present note, this is due to the mutual cancelation of the effects of both the dragging of the inertial frames and the neutron star oblateness.
In the present note, we show that using a Schwarzschildian formula for , combined with simple approximations for basic quantities (metric functions, orbital velocity) one can reproduce very precisely (within a fraction of a percent) orbital parameters outside a rotating neutron star, down to the star surface, and for rotation frequencies up to the massshedding limit. We explain this remarkable property using existing expansions of , , and in the powers of the specific stellar angular momentum (Abramowicz et al. 2003). In addition, using exact 2D calculations of spacetime of a rotating neutron star, we study how Schwarzschildian at the equator converges, in the massshedding limit, to the star rigidrotation frequency, a feature pointed out already in Bejger et al. (2007).
Notations, spacetime coordinates, and exact expressions for the metric functions and for quantities associated with circular equatorial orbits are introduced in Sect. 2. Two wellknown analytic models of equatorial orbits are summarized in Sect. 3. Our approximation for is described in Sect. 4. Approximations for and are presented in Sect. 5. Approximations for , , and are compared with the exact values of these quantities, and their high precision is explained using systematic expansions derived by Abramowicz et al. (2003). The formula for the specific energy of test particle, given in Abramowicz et al. (2003), contains a misprint that we correct. Finally, Sect. 6 contains a discussion of our results and conclusions. Details concerning equations of state, used in our calculations, are collected in the Appendix.
Figure 1: Frequency of stellar rigid rotation (solid line) compared with the frequency of a test particle orbiting a point mass, equal to the mass of the rotating star (Schwarzschildian frequency), at a distance of stellar circumferential radius (dotted line). Both frequencies approximately coincide at the massshedding configuration radius (marked with a circle). Stellar model based on the DH EOS of Douchin & Haensel (2001) described in the Appendix. 

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2 Notations, metric, and orbits
The metric for a rapidly rotating object, assuming that
the spacetime is axisymmetric, stationary, asymptotically flat, and
free of
meridional currents can be written by means of the 3+1 formalism of GR,
in the
socalled maximalslicing quasiisotropic coordinates
where A and B are factors of the 3metric ( in spherical symmetry), N is called the lapse, and is the only nonzero component of the shift vector. The integrals of motion can be obtained by constructing the Lagrangian and using the variational principle (see e.g., Shapiro & Teukolsky 1983). There are two integrals of motion of a test particle of rest mass m, corresponding to the t and coordinates: E, the energy of a test particle (in units of mc^{2}), and l, the specific angular momentum (in units of mc). One can also define V, the effective potential
(2) 
Conditions for circular orbits are
The radius of the circular orbit in the equatorial plane of the star (circumferential radius) is related to the coordinate radius by the relation
The angular velocity of the matter, as measured by a distant observer, can be described by the components of the 4velocity vectors u ( ) and equals . Following Bardeen (1970), the proper velocity of matter in the equatorial plane is
The specific (per unit mass) energy of a particle and its specific angular momentum can be expressed in terms of the velocity v(Bardeen 1972):
From both expresions in Eq. (3), one obtains the corotating test particle velocity v
v  =  
(7) 
and, from Eq. (5), the orbital frequency in the equatorial plane
3 Approximate analytic solutions
3.1 Schwarzschild solution
Simplest approximation consists of neglecting neutron star
rotation, and dealing with the exterior spacetime of stationary,
sphericallysymmetric object. The metric in the
Schwarzschild coordinates is then
where is the Schwarzschild radius and M is the total gravitational mass of the object. The coordinate radius r is here equal to the cicumferencial radius defined in Eq. (4). Specific energy and specific angular momentum are given by the formulae
(10) 
The corresponding orbital frequency is
3.2 Slowrotation approximation
Another popular approximation can be easily obtained from the Kerr
solution (Kerr 1963) in
BoyerLindquist coordinates; this
slowrotation approximation retains only the
firstorder
terms in the angular momentum J (or equivalently,
in a=J/Mc).
The formulae for ,
and
are
4 Approximation of orbital frequency by the Schwarzschildian formula
The seemingly surprising property of angular frequency being
approximately equal to
can be explained by
studying the higherorder expansion terms of axisymmetric metrics, such
as
HartleThorne metric, presented by Abramowicz
et al. (2003)
(hereafter referred to as AAKT). AAKT give the expansion to the second
order in angular momentum and include the massquadrupole moment, Q,
term, which
plays a decisive rôle in a correct
description of the orbital parameters of a test particle
(Berti
et al. 2005; Shibata & Sasaki 1998;
Laarakkers
& Poisson 1999).
The orbital angular velocity of a corotating
test particle can be approximated, according to AAKT, as
where G=c=1, j=a/M, q=Q/M^{3} and the functions F_{1}(r) and F_{2}(r)are defined therein.
As an illustration, we present^{} in Fig. 2 a comparison of the frequency , for the massshedding configuration calculated for realistic EOS of Douchin & Haensel (2001) (details in the Appendix), with , , and (Eqs. (8), (11), and (14), respectively). As expected, reproduces the true, numericallyobtained values of far more precisely than , especially near the stellar surface. One has to keep in mind however that in Eq. (15) we have used the massquadrupole moment obtained directly from our numerical calculations instead of an approximation to q, which may affect the behavior of . The value of is also quite close to . We sought for an explanation of this phenomenon by analyzing the terms in Eq. (15): in addition to them being small in comparison to the leading term, the jterm is approximately equal, but with the opposite sign, to the massquadrupole moment qterm, so that they effectively cancel each other (see Fig. 2 for an typical comparison; the j^{2}term is much smaller than both of the other terms, two orders of magnitude in this example, and therefore insignificant). This feature is qualitatively and quantitatively present for different polytropic and realistic EOSs as well as bare strangequark matter stars (details concerning the EOSs that we used can be found in the Appendix).
The quadrupole term in Eq. (15) is related to the rotational oblateness of the star and, in contrast to jterms, is of Newtonian nature. One can study the maximum deviation from the Schwarzschildian (and Newtonian) test particle orbital frequency resulting from this effect, assuming a dense matter disk (which corresponds to an ``extreme oblateness'') as a source of gravitational field. The gravitational pull is described by hypergeometric functions (Zdunik & Gourgoulhon 2001) that indicate the maximum frequency deviation % for orbital radii larger than that of the innermost stable circular orbit  the existence of unstable circular orbits is a result of the oblateness of the gravitational field source, treated in the Newtonian theory.
Figure 2: Upper panel: percentage differences between exact numerical values of test particle orbital frequencies, , and approximations presented in the text, plotted against the radius of a circular orbit; , where , or . Lower panel: values of various terms in Eq. (15) versus orbital radius. For comparison, relative frequency difference in the case of the Schwarzschildian formula is also shown (massshedding configuration of DH EOS, details in Appendix). 

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5 Approximation of the specific orbital energy and angular momentum
We approximate the specific energy and angular momentum of a test particle, in a manner similar to Shibata & Sasaki (1998). As one can see in an example shown in Fig. 2, the difference between the exact orbital frequency and the Schwarzschildian formula is smaller than 1% even for massshedding, compact stellar configurations. Near the surface of the star, is far more accurate than the slowrotation approximation, and as good as .
Motivated by this result, we aim to provide formulae that are
simpler
than those of AAKT and which could serve the ambition of being
useful in practical calculations. To this aim, we propose to substitute
the
exact values for the test particle velocity in the Eq. (5)
by their approximations of the angular frequency
by
from Eq. (11),
the azimuthal shift
component
by the firstorder term in the slowrotation
approximation of a framedragging term 2GJ/r^{3}c^{2},
and the metric function
N by its Schwarzschildian equivalent, :
where for brevity. We then apply , as well as the approximations described above, to Eqs. (6), obtaining and . In Figs. 3, and 4 we compare^{} our approximation with the Schwarzschild, slowrotation, and AAKT (Eqs. (21) and (26) therein) formulae for and . For different EOSs, masses, and rates of rotation, the behaviour of these deviations is quantitatively and qualitatively similar.
Figure 3: Comparison of specific orbital angular momenta as a function of the radius of the circular orbit. The exact value is denoted by , while is an approximation of it. The relative deviations (in percent) are , where or . Results for DH EOS massshedding configuration and for the currentlyhighest pulsar frequency, 716 Hz (details in Appendix). 

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Figure 4: Comparison of specific orbital energies as a function of the radius of the circular orbit. Configurations and notation analogous to Fig. 3. 

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Figure 5: Comparison of orbital frequencies for stellar configurations of DH EOS and bare strange quark star EOS, rotating at 716 Hz. Notations as in Fig. 2, details in Appendix. 

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6 Discussion and conclusions
We have performed calculations of stationary configurations of rotating compact stars, for a set of representative EOSs, both polytropic and realistic ones, as well as for the bag models describing the EOSs of hypothetical bare strange quark stars (as in Haensel et al. 2009). In all these cases, we observed quantitatively and qualitatively similar behaviour of the orbital frequency of a test particle moving on a circular orbit in the equatorial plane. In all cases, the value of the orbital frequency was close to that of a test particle in a Schwarzschildian spacetime around a point mass (equal to that of the actual compact star), at the properly defined radius. This result is valid for any EOS and for any rotation frequency of the compact star, up to the massshedding limit.
We note that the approximations of and , proposed in Sect. 5, reproduce true (numerical) values within about one per cent (being the least accurate in the vicinity of the stellar surface). Their accuracy near the stellar surface decreases to a few percent for rapidly rotating stars with extreme compactnesses ( ), especially in the case of rapidly rotating and oblate bare strangequark matter stars with large quadrupole moments, but as is shown in Fig. 5 the approximations are acceptable for the currently highest observed spin rate of 716 Hz. For comparison, Figs. 3 and 4 show the difference in the precision for the DH EOS in the case of the massshedding limit and the rotation frequency 716 Hz.
Overall, our approximations for and are much better than the slowrotation approximation, and one does not need to compute the massquadrupole moment of the star, the knowledge of which is otherwise crucial in the existing systematic expansions (Berti et al. 2005; Abramowicz et al. 2003). In cases of larger compactness and oblateness, as well as submillisecond rotation periods, the approximation from Sect. 5 is in closer agreement with the results of numerical simulations than the formulae of Abramowicz et al. (2003); in Fig. 6, we present an extreme example of a bare quark star spinning at 1300 Hz.
The analytic formulae for , , and give a very good approximation (typically within a one per cent) of exact values for neutron stars with astrophysically relevant masses, . They are valid for rigidly rotating neutron stars, for rotation rates up to the mass shedding limit, and down to the equator. However, they cannot be used to calculate the properties of the innermost stable circular orbit, because this calculation involves second radial derivatives of metric functions (see, e.g., Cook et al. 1994).
Figure 6: Comparison of , , and for SQM1 EOS star rotating at 1300 Hz. Notation as in previous figures. 

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We would like to thank Prof. M. Abramowicz and Prof. P. Jaranowski for useful discussions. This work was partially supported by the Polish MNiSW research grant no. N N203 512838. M.B. acknowledges the support of Marie Curie Fellowship no. ERG2007224793 within the 7th European Community Framework Programme.
Appendix: Numerical implementation and the EOSs chosen for calculations
The calculations have been performed using the rotstar code from the LORENE library^{}. We use EOSs employed in Haensel et al. (2009):
 1.
 Realistic microphysical EOSs of dense matter. Unless marked otherwise, the figures employ the results for DH EOS of Douchin & Haensel (2001), for a stellar configuration corresponding to a nonrotating star of mass (central baryon density ). The massshedding configuration has a gravitational mass of , whereas that rotating at 716 Hz has a mass of .
 2.
 Bag models of bare strange quark stars. Figures 5 and 6 show results for SQM1 EOS. The nonrotating counterpart to the stellar configuration presented in Fig. 5 has the gravitational mass of . The Fig. 6 configuration has a central baryon density of n_{b,c} = 0.63 fm^{3}, nonrotating gravitational mass of , and compactness 2GM/Rc^{2} = 0.47.
 3.
 Relativistic polytropes. We considered the range (Tooper 1965).
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Footnotes
 ... present^{}
 As in Berti et al. (2005) (Sect. 3.1 therein), we use exact numerical values of the gravitational mass M, total angular momentum J, equatorial radius R, but also the massquadrupole moment Q of a given configuration as input to AAKT expansions to reproduce as well as possible the test particle orbital frequencies, angular momenta, and energies and compare them with their analogues from different spacetimes.
 ... compare^{}
 Note the difference in definitions of the specific orbital angular momentum between the article of Abramowicz et al. (2003) and here: . We also notice that there is a misprint (?) in the Eq. (30) of AAKT. Their expression leads to divergence, with increasing j, of approximate from the exact values of this quantity. This could be repaired by replacing 20M^{2}r in their Eq. (30) by +20M^{2}r.
 ... library^{}
 See http://www.lorene.obspm.fr (detailed description of the EOS implementation can be found at http://www.lorene.obspm.fr/Refquide/classEos.html).
All Figures
Figure 1: Frequency of stellar rigid rotation (solid line) compared with the frequency of a test particle orbiting a point mass, equal to the mass of the rotating star (Schwarzschildian frequency), at a distance of stellar circumferential radius (dotted line). Both frequencies approximately coincide at the massshedding configuration radius (marked with a circle). Stellar model based on the DH EOS of Douchin & Haensel (2001) described in the Appendix. 

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In the text 
Figure 2: Upper panel: percentage differences between exact numerical values of test particle orbital frequencies, , and approximations presented in the text, plotted against the radius of a circular orbit; , where , or . Lower panel: values of various terms in Eq. (15) versus orbital radius. For comparison, relative frequency difference in the case of the Schwarzschildian formula is also shown (massshedding configuration of DH EOS, details in Appendix). 

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In the text 
Figure 3: Comparison of specific orbital angular momenta as a function of the radius of the circular orbit. The exact value is denoted by , while is an approximation of it. The relative deviations (in percent) are , where or . Results for DH EOS massshedding configuration and for the currentlyhighest pulsar frequency, 716 Hz (details in Appendix). 

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In the text 
Figure 4: Comparison of specific orbital energies as a function of the radius of the circular orbit. Configurations and notation analogous to Fig. 3. 

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In the text 
Figure 5: Comparison of orbital frequencies for stellar configurations of DH EOS and bare strange quark star EOS, rotating at 716 Hz. Notations as in Fig. 2, details in Appendix. 

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In the text 
Figure 6: Comparison of , , and for SQM1 EOS star rotating at 1300 Hz. Notation as in previous figures. 

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In the text 
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