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In my prior Blog,(a) I wrote on the fact that Newtonian two-body orbits are identical to the orbits of 'light-weight' objects around a heavy central mass, provided that the two masses are summed and placed at the location of any one of the two. Now I've found that in most cases, two-body relativistic orbits are also virtually indistinguishable from such 'one-body' orbits.
It is obviously not the case in very strong gravitational fields, where the interaction between the two gravitational fields cause a type of "empty space friction" that emits gravitational waves. The two orbiting, in-spiraling neutron stars pictured here show such a case(b) (it's actually density and temperature distribution plots in the orbital plane, not the neutron stars themselves).
If they orbit at a relatively large distance, the gravitational radiation quickly becomes negligible, because the energy loss rate diminishes with the fifth power of distance.(c) In such a case, the standard relativistic point particle orbital equation do hold. For circular orbits it is relatively easy to work with. The relative circular orbital speed is actually the same as for the Newtonian case.
The relativistic orbital energy has a slightly different look than what Newton predicted.(d)
The ˜ means it is specific energy, Joules per kg of total mass (m1+m2). The relativistic parameter gtt = 1-2G(m1+m2)/(rc2) is the square of the gravitational time dilation factor and the denominator is hence the combined gravitational and velocity time dilation. Here vt is the relative tangential velocity as seen by a distant observer, i.e. the circular orbital speed of eq. 4.
Amazingly, when eq. (19) is approximated for low speed, weak field conditions (vt << c and r >> 2G(m1+m2)/c2), it reduces to: ≈ ½ v2 - G(m1+m2)/r + c2, which is the Newtonian specific energy plus c2. The c2 stems from the fact that Einstein added the rest energy E = mc2 to the Newtonian mechanical energy, hence he added specific energy E/m= c2.
For losses due to gravitational waves, we can specialize the equation in note (c) below for circular orbits:
This gives the specific energy loss rate in J/kg per second over one orbit with an average semi-major axis r=a. I have calculated the initial value for two point masses of 1.5 Suns each, orbiting at a distance of 885 km (100 times their combined Schwarzschild radius, orbital period ~ 262 ms, hence very relativistic). The result is a staggering orbital energy loss of 7.63 GJ/sec per kg of combined mass. This increases as the orbit shrinks, so the two masses will spiral into each other in well less than two hours.
[Edit] I've updated the Two-body Orbits PDF to include relativistic orbits. You can download the pdf here.
-J
(a) http://cr4.globalspec.com/blogentry/15924/Two-Body-Orbits
(b) http://www.mpa-garching.mpg.de/~mor/nsgrb.html
(c) http://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity (second last equation)
(d) See eq. 6.7 of Relativity-4-Engineers, chapter 6, freely available here: How Orbits Work. It is modeled after Misner, Thorne and Wheeler's Gravitation, eqs. 25.17 and 25.18.
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Re: Relativistic Two-Body Orbits
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