When spinning bodies are involved, the full gravitational
waveform can be quite complicated; the orbital plane
and the spin vectors of the individual bodies can
precess. The precession causes a modulation of the signal.
However, this GRASP routines only implements the the special case
when the spins are assumed to be aligned (or antialigned) with the
orbital angular momentum axis.
In this case there is no precession and, therefore, no modulation
of the amplitude of the signal.
Also in this case,
the spin-corrections to the orbital frequency and phase
are given by simple modifications to the nonspin phase and frequency
Eqs. () and ().
The necessary terms can be found in Eq.(F22) in Appendix F of ,
and are given by
Specifically, the spin quantities passed to the chirp generation routines
are the signed, dimensionless (Kerr-like) parameters of each body
Some calculations (e.g. those requiring a precise definition of the orbital phase) are sensitive to the index assigned to the bodies. The GRASP convention is that is the smaller of the two masses; therefore spin1 should be the spin assigned to the smaller of the two masses.
How are the dimensionless spin parameters spin1(2)
and the geometrized angular momentum
related to angular momentum of the bodies in cgs units?
Let denote the spin angular momentum
of the i-th body in cgs units
(i.e. gram cm/sec).
Then is related to by
What is the allowable range for the spin parameters spin1 and spin2? For Kerr black holes, we know . For spinning neutron stars, stability studies (based on relativistic numerical hydrodynamic simulations) show that the spin parameter must satisfy . These limits can serve as a hard upper bound for a choice of spin parameters. However, observed pulsars in binaries have spin parameters substantially smaller than this limit, e.g. for the Hulse-Taylor pulsar we have spin1 . (See  for discussion and references.)
As a sanity check and a demonstration of how to calculate
the spin parameters, we verify the numbers quoted above for the
Hulse-Taylor binary pulsar. The pulsar is a neutron star
, a radius
, and a
spin frequency of about 17Hz. [Don't confuse the spin
period (1/17 sec) with the orbital period (8hrs).]
If we model the moment of inertia, , as that of a sphere
with uniform density, we obtain
We can also use the above conversions to give the
angular momentum of the Hulse-Taylor pulsar in
Like all post-Newtonian equations, Eqs. () and () are slow-motion approximations to the fully relativistic equations of motion; therefore they are most accurate - and behave best - for smaller values of the spin parameters. The GRASP routines have been tested for a modest range of masses (,) and spins (,) in the frequency band ; they seem to give reasonable results in this regime.
Finally, the admonitions and suggestions given in Sec. () about setting up banks of filters hold here also: test the chirp-generating functions with the extreme values of masses and spins you intend to use in your search. If the functions give satisfactory results at the ``corners" of the parameter space, they should work on the interior of the parameter space.