Spin Effects

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 [8],
and are given by

and

Here is the dimensionless time variable given by Eq. (). The ellipses represent the nonspin (post)-Newtonian terms already given in Eqs. () and (). The quantities and are dimensionless quantities related to the angular momentum of the bodies by

where is the signed magnitude of the angular momentum vector of each body expressed in geometrized units (cm), and is the mass in geometrized units (cm). [Below we show how to covert from geometrized units to cgs units.] The sign is positive (negative) for spins aligned (antialigned) with the the angular momentum axis. By comparing the nonspin phase and frequency evolution in Eqs. () and () with the spin corrections in Eqs. () and (), we see that the spin-orbit corrections (terms linear in and ) simply modify the (post)-Newtonian contributions and the spin-spin corrections (term quadratic in and ) modify the (post)-Newtonian contributions.

Specifically, the spin quantities passed to the chirp generation routines
are the signed, dimensionless (Kerr-like) parameters of each body

where the sign is chosen if the spin is aligned (antialigned) with the orbital angular momentum axis. [Note: only in Eqs.()-() is mass expressed in geometrized units.]

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

(6.9.49) |

The conversion of angular momentum in cgs units to the

where is the magnitude of the spin angular momentum of the i-th body in standard cgs units (

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 [12] 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
with
, 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

Using Eq.() to convert this to the dimensionless quantity we have

This is reasonable agreement with the numbers given above.

We can also use the above conversions to give the
angular momentum of the Hulse-Taylor pulsar in
geometrized units

(6.9.52) |

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.