Kerr geodesics

The first step in parameterizing solutions of the Papapetrou-Dixon equations using orbital parame- ters is to solve the geodesic case. The traditional method for specifying a geodesic in terms of con- served quantities uses the energyE, thez angular momentumLz, and the Carter constant Q[11].

In order to use the orbital parameters, we adopt a mapping from (rp, e, ι) to (E, Lz, Q) based on

unpublished notes supplied by Teviet Creighton and Scott Hughes (and implemented in Hughes’s Kerr geodesic integrator [14]).

In order to use the more intuitive orbital parameters, we must determine the set (E, Lz, Q) given

the set (rp, e, ι). We obtain two of the necessary equations by noting that the radial velocitydr/dτ

vanishes at pericenter and apocenter, since these radii correspond to turning points in the radial motion. The equation for the time-evolution of the Boyer-Lindquist radiusris [11]:

Σ2 _dr dτ 2 =R(r), (3.16) where R(r) = [E(r2+a2)−aLz]2−∆[r2+ (Lz−aE)2+Q], (3.17)

and we use the standard auxiliary variables

Σ =r2+a2cos2θ (3.18)

and

∆ =r2−2M r+a2. (3.19)

The quantityais the Kerr spin parameterJ/M, i.e., the central black hole’s spin angular momentum per unit mass, which is dimensionless in our normalized units. From Eq. (3.17) we see thatdr/dτ = 0 implies thatR(r) = 0, so we obtain one equation at each turning point:1

R(rp) = 0 (3.20)

and

R(ra) = 0, (3.21)

where the apocenter is defined by

ra =

_{1 +e}

1−e

rp. (3.22)

1_{In this chapter we never consider exactly circular orbits, but we note that our prescription fails in this case: the}

conditions (3.20) and (3.21) are identical whene= 0, sincera =rp. For exactly circular orbits one must use the

additional conditionR0_(r

The final equation required to complete the mapping is [15]

Q=L2z tan2ι. (3.23)

The value ofι resulting from this definition agrees closely with the maximum value of|π/2−θ|for a numerically determined solution to the equations of motion, i.e., it faithfully captures a geometric property of the orbit.

Eqs. (3.20)–(3.23) give three equations in three unknowns, which are easy to solve using a non- linear root finder as long as good initial guesses for the energy, angular momentum, and Carter constant can be found. The approach we adopt uses the degenerate cases of circular equatorial orbits to provide the raw material for analytical guesses. The energies for prograde and retrograde circular orbits in the equatorial plane are

Epro(r) = 1−2v 2_{+ ˜av}3 √ 1−3v2_{+ 2˜av}3 (3.24) and Eret(r) = 1−2v 2_−˜av3 √ 1−3v2_−2˜av3, (3.25) where we writev≡p

M/rand ˜a=a/M for notational simplicity. The initial guess for the energy is then an average of these energies, weighted using the inclination angle,2 _{with “radius” given by} the semimajor axis of an ellipse with pericenterrp and eccentricitye:

Eguess =1₂[α+Epro(rsemi) +α−Eret(rsemi)], (3.26) where α±= 1±cosι, (3.27) and rsemi= rp 1−e. (3.28)

It is possible (though rare) for Eq. (3.26) to yield an energy guess greater than 1; in this case, we simply setEguess_{= 1.}

Once we have a guess for the energy, we can find the corresponding guess for the angular momen- tum. Using the value from Eq. (3.26) and the expression for the angular momentum for a circular equatorial orbit gives

Lguess_z = cosι

s

1−e2

2(1−Eguess₎ (3.29)

2_{We adopt the convention thata≥0, so thatιindicates whether the orbit is prograde (0≤ι <}π

2) or retrograde

as an initial guess for the angular momentum. Finally, the guess for the Carter constant is

Qguess= (Lguessz )

2

tan2ι. (3.30)

Plugging Eqs. (3.26), (3.29), and (3.30) into the nonlinear root finder yields the actual values ofE,

Lz, andQto within machine precision in fewer than 10 iterations.

One caveat about our parameterization method is worth mentioning: some values of (rp, e, ι)

correspond to unstable Kerr orbits, and in this case the method returns a set (E, Lz, Q) corresponding

to an orbit with a different pericenter from the one requested. We can illustrate this behavior by factoring Eq. (3.17), which is a quartic function inr:

R(r) = (1−E2)(r1−r)(r−r2)(r−r3)(r−r4). (3.31)

The roots are ordered so that r1 ≥ r2 ≥ r3 ≥ r4. Bound motion occurs for r1 ≥ r ≥ r2, which implies thatr1 =ra and r2=rp, but this works only for stable orbits. In the case that the orbit

requested is unstable, the set (rp, e, ι) returned by the algorithm instead corresponds to a nearby,

stable orbit. In this case, the numerically calculated roots of R(r) satisfy r3 = rp,requested, but the parameterization method returnsr2as the pericenter, i.e., it returns a nearby stable orbit with pericenterrp=r2> rp,requested. As a result, theactual pericenter is larger than the value requested. We use routines from [14] to identify the boundary between stable and unstable orbits (so that the latter may be excluded), but the code has a few minor bugs, and the identification procedure is not infallible. As a result, some such orbits appear in the results below (Sec. 3.5) and can be identified by having pericenters different from those requested. It is essential to understand, however, that the orbits returned by the parameterization algorithm are never unstable, and represent perfectly valid (stable) solutions to the equations of motion.