From: lip@s1.gov (Loren I. Petrich)
Newsgroups: sci.astro,sci.physics
Subject: Size of Second-Order GR Effects in Binary Pulsars
Date: 18 Dec 92 00:40:51 GMT
Organization: LLNL


	I have finally succeeded in working out the size of the
second-order effects on the pulse timing for a binary pulsar. I get
some "universal" constants, independent of the orbit size, though
dependent on the eccentricity.

	If we take the eccentric anomaly E, given by

	E - e*sin(E) = nt

	where t is the time and n is the mean motion, the radius is
given by

	r = a*(1 - er*cos(E))

	and the phase angle is given in terms of the true anomaly as:

	[theta] = (1 + g)*v(E,ea)

	where the modified eccentries er and ea are:

	er = e*(1 + (M/a)([gamma] + 1 - f/2))

	ea = e*(1 + (M/a)([gamma] + 1 - 3f/4))

	and the precession factor

	g = (M/(a(1-e^2)))*(-[beta] + 2[gamma] + 2)

	Here, M1 and M2 are the two masses, M is the total mass, and f
is the (reduced mass)/(total mass) ratio, or M1*M2/M^2. As usual, a is
the semimajor axis and e the eccentricity. The true anomaly v(E,e) is
given by:

	v = 2*arctan[sqrt((1+e)/(1-e))*tan(E/2)]

	[gamma] and [beta] are two post-Newtonian parameters;

	Also, G = c = 1; to get back the units, find whichever
combination gives the right dimensions. Thus M/a -> (GM)/(ac^2).

	The projected radius for object 1 is:

	(M2/M)(r - (1/2)*f*d*(1 - r/a))*sin(i)

	where d is the relative mass difference (M2 - M1)/M and i is
the inclination. This is equivalent to a displacement of

	[(M2/M)*a*sin(i)]*(1 - er1*cos(E))

	where

	er1 = e*(1 + (M/a)([gamma] + 1 - f/2 + (1/2)*(M1/M)*d))

	The effective displacement sizes are (M2/M)*a*sin(i) times the
changes in eccentricity and are:

	(M2*([gamma] + 1 - f/2) + (1/2)*f*(M2 - M1))*e*sin(i)

	for the radius and

	(M2*(([gamma] + 1 - 3f/4) + 2*(-[beta] + 2*[gamma] + 2)))*e*sin(i)

	for the angle.

	The quantity (-[beta] + 2*[gamma] + 2) can be estimated
directly from the precession.

	For the Hulse-Taylor binary, both components have masses of
1.4 solar masses. Using [gamma] = 1 (GR value), I find that the
effects have a size of about 12.5 microseconds * [e*sin(i)], or 5.6
microseconds, excluding precession. Including that effect gives 53
microseconds * [e*sin(i)] or 24 microseconds for the angular effect.
The HT pulsar has e = 0.617 and sin(i) = 0.716.

	NOTE: This is all in the isotropic gauge used in MTW and other
places.

	This effect is significant only for significantly elliptic
orbits, such as the HT pulsar,

	To this effect must be added the Shapiro time delay, which is

	t = (1 + [gamma])*M2*log((2L)/(r + n.r))

	where n is the direction to the observer and n.r is the scalar
product with that and the radius vector. L is the distance to the
observer, which is irrelevant here, and which can be replaced by some
constant. The coefficient of the logarithm is, using the parameters
given before, about 14 microseconds.

	For the HT pulsar, the effect's size depends on the
orientation of the orbit. For the orbit's periapsis pointing to the
Earth, the effect's size is 5 microseconds, while for the orbit's
periapsis pointing away from the Earth, the effect's size is 44
microseconds. These are end-to-end, and to be comparable to the
amplitude values listed earlier, they must be divided by 2, yielding
2.5 and 22 microseconds, respectively.

	The result is that these second-order effects can be detected
if one has a timing accuracy of a few microseconds or better, no
matter what the size of the orbit. It was indicated that for some
pulsars, one can do better than that, somewhere in the 100-300
_nanosecond_ range, so one should have at least 30 sigmas of
significance in that case.

	However, the bulk of the contribution is proportional to
M2*([gamma] + 1), with the terms in M1*M2/M^2 contributing only a
microsecond or less. Thus, we have the following observables:

	Order zero in (v/c) {Newtonian}:

	(M2/M)*a*sin(i)

	a^3/M

	e

	{Precession: (M/a)(-[beta] + 2*[gamma] + 2)}

	Order one in (v/c) {gravitational redshift}:

	M2*(M1 + 2*M2)/(M*a)

	Order two in (v/c) {time-delay and post-Newtonian}:

	M2*([gamma] + 1)*sin(i) {the largest effect}

	M2*f*sin(i)

	(M2 - M1)*f*sin(i)


	Orders 0 and 1 suffice to determine M1, M2, a, and sin(i) if
one assumes the truth of GR (4 obs -> 4 var). If one allows the
post-Newtonian parameters to vary, then one does not have an
unambiguous parameter determination (4 obs -> 5 var). Observing
order-2 effects allow for testing consistency with GR (5 obs -> 4
var), though the largest of them still does not allow for an
unambiguous parameter determination (5 obs -> 6 var). If one can get
to better than a microsecond, one will have enough data points for an
unambiguous determination of the various parameters, including the
post-Newtonian ones [gamma] and [beta] (6 or 7 obs -> 6 var).

	For some pulsars, the accuracy is reportedly 100-300
nanoseconds, which should be more than enough for the biggest of the
second-order effects, but which will be pretty close for the smaller
ones, the other two terms, which are proportional to f.

	Any prospects for the known binary pulsars, like the HT
pulsar?
-- 
/Loren Petrich, the Master Blaster
/lip@s1.gov
Newsgroups: sci.astro