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Spin Effects

0 In the simple case where the spin vectors of the bodies are aligned (or antialigned) with the orbital angular momentum axis, the GRASP chirp-generating functions have the built-in capability of computing the leading order spin-orbit and spin-spin corrections to the inspiral chirp. To use this feature no modification of the chirp-generating routines [phase_frequency() or chirp_filters()] is necessary; simply pass nonzero values of the spin parameters to the functions. This can easily be done by editing the example programs phase_evoltn.c and/or filters.c to pass nonzero values of the variables spin1 and spin2. [See below for definitions and allowed ranges of spin1 and spin2.]

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

$\displaystyle f(t)$ $\textstyle =$ $\displaystyle {M_\odot \over 16 \pi T_\odot m_{\rm tot}}
\biggl\{ \Theta^{-3/8}...
...} [\chi_s + (\delta m /m) \chi_a] -{19\over40}\eta \chi_s \right)
    $\displaystyle \qquad\qquad\qquad
-\left( {237\over 512} \eta [(\chi_s)^2 -(\chi_a)^2] \right) \Theta^{-7/8}
\;\biggr\} \;,$ (6.9.43)

$\displaystyle \phi (t)$ $\textstyle =$ $\displaystyle \phi_c -{1\over\eta} \biggl\{ \Theta^{5/8} + \; {\rm ...} \;
...4} [\chi_s + (\delta m /m) \chi_a] -{19\over16}\eta \chi_s \right)
    $\displaystyle \qquad\qquad\quad
-\left( {1185\over 512} \eta [(\chi_s)^2 -(\chi_a)^2] \right) \Theta^{1/8}
\; \biggr\}\;.$ (6.9.44)

Here $\Theta$ is the dimensionless time variable given by Eq. ([*]). The ellipses represent the nonspin (post)$^n$-Newtonian terms already given in Eqs. ([*]) and ([*]). The quantities $\chi_s$ and $\chi_a$ are dimensionless quantities related to the angular momentum of the bodies by
$\displaystyle \chi_s={1\over 2}\left({ S_1 \over m_1^2}+{ S_2 \over m_2^2} \right) \; ,$     (6.9.45)
$\displaystyle \chi_a={1\over 2}\left({ S_1 \over m_1^2}-{ S_2 \over m_2^2} \right) \; ,$     (6.9.46)

where $S_{1(2)}$ is the signed magnitude of the angular momentum vector of each body expressed in geometrized units (cm$^2$), and $m_i$ 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 $\chi_s$ and $\chi_a$) simply modify the (post)$^{3/2}$-Newtonian contributions and the spin-spin corrections (term quadratic in $\chi_s$ and $\chi_a$) modify the (post)$^{2}$-Newtonian contributions.

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

$\displaystyle {\tt spin1} = \pm { \vert{\bf S_1} \vert \over m_1^2 } \; ,$     (6.9.47)
$\displaystyle {\tt spin2} = \pm { \vert{\bf S_2} \vert \over m_2^2 } \; ,$     (6.9.48)

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 $m_1$ 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 $\bf S_i$ related to angular momentum of the bodies in cgs units? Let $L_i$ denote the spin angular momentum of the i-th body in cgs units (i.e. gram cm$^2$/sec). Then $L_i$ is related to $S_i$ by

$\displaystyle S_i \; [{\rm in\;geometrized\;units}, i.e. \; {\tt cm^2} ]$ $\textstyle =$ $\displaystyle \;
\left({G \over c^3 }\right)
L_i\;({\rm in \; gram\;cm^2/sec}) \;$  
  $\textstyle =$ $\displaystyle 2.477 \times 10^{-39}({\rm sec /gram})
\; L_i\;({\rm in \; gram\;cm^2/sec}) \; .$ (6.9.49)

The conversion of angular momentum in cgs units to the dimensionless variable spin1(2) (the variable actually sent to the routine) is
{\tt spini} = \left( {c \over G m_{i}^2 } \right) L_i
= \lef...
...c / (gram \; cm^2) })
\left( {M_\odot \over m_i} \right)^2 L_i
\end{displaymath} (6.9.50)

where $L_i$ is the magnitude of the spin angular momentum of the i-th body in standard cgs units (i.e. gram cm$^2$/sec), and $m_i$ is the mass in grams.

What is the allowable range for the spin parameters spin1 and spin2? For Kerr black holes, we know $\vert{\tt spin1(2)}\vert = (\vert{\bf S_{1(2)}} \vert / m_{1(2)}^2 ) \leq 1$. For spinning neutron stars, stability studies (based on relativistic numerical hydrodynamic simulations) show that the spin parameter must satisfy $\vert{\tt spin1}({\tt 2})\vert = (\vert{\bf S_{1(2)}} \vert / m_{1(2)}^2 )
\mathrel{\raise.3ex\hbox{ $<$\ } \mkern-14mu \lower0.6ex\hbox{$\sim$\ } } 0.6$. 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 $\mathrel{\raise.3ex\hbox{$<$}\mkern-14mu \lower0.6ex\hbox{$\sim$}}
6.5\times 10^{-3}$. (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 $m \approx 1.4M_\odot$, a radius $R \approx 10{\rm km}$, 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, $I$, as that of a sphere with uniform density, we obtain

$\displaystyle L_{\rm Hul-Tay}$ $\textstyle \sim$ $\displaystyle 2 \pi I f_{\rm spin}$  
  $\textstyle \sim$ $\displaystyle {4 \pi \over 5} f_{\rm spin} M R^2$  
  $\textstyle \sim$ $\displaystyle 1.2 \times 10^{47} ({\rm gram\;cm^2/sec})$  

Using Eq.([*]) to convert this to the dimensionless quantity we have
{\tt spin}_{\tt Hul-Tay} \sim 6.8 \times 10^{-3} \; .
\end{displaymath} (6.9.51)

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

S_{\rm Hul-Tay} \approx 3\times10^8 \, cm^2 \approx 7 {\rm acres} \; .
\end{displaymath} (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 ($0.1M_\odot$,$10M_\odot$) and spins ($-0.2$,$+0.2$) in the frequency band $60{\rm Hz} \leq f_{orb} \leq 2000{\rm Hz}$; 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.

next up previous contents
Next: 2.5 Post-Newtonian corrections to Up: Additional contributions to the Previous: Additional contributions to the   Contents
Bruce Allen 2000-11-19