According to the WIKI Ref, gravitational wave power radiated by a binary pair of black holes can be computed from the equation:
The WIKI Gravitational Radiation Power REF is listed on my BH_Equations page. It is dimensionally correct. I have computed, listed in the Table above, the energy loss rate for a 10SM pair of black holes at various orbital radii normalized in units of Schwarzschild Radii.
The WIKI equation was derived from a treatment of (GR) and is said to hold true for all non-relativistic circular orbit (e=0) binary black holes. Results are Tabulated above for CG separations [Col-1] from 5000 to 0.2(Rs). Binary rotation rate in radians/sec are in [Col-2]. Binary (Spin Vector) revolutions/sec is in [Col-4]. The tangential velocity normalized to 'c' is in [Col-5]. Total Newtonian kinetic energy is in [Col-6]. The Gravitationally Radiated power ([Ft-Lb]/sec) is in [Col-7]; and Power in (watts) is in [Col-10]. [Col-9] is important; it shows the radiated energy per revolution divided by orbital energy at that radius. This provides a easy measure of the fraction of energy lost to radiation per orbit. At 200(Rs) the radiated energy loss/rev drops by 0.00063% of total energy. Notice at 10(Rs) the energy loss is still only 1.12% of the orbital orbital kinetic energy, still relatively small. However, at R/Rs=1 the radiated energy per orbit would be 3.55 times the available energy; binary merger would be instantly at hand.
The radiated power is inversely proportional to the fifth power of CG separation. The radiated power is low for large separations of the CG's but increases exponentially (inverse 5th power of R) upon mutual approach. So states the Radiated Power formula.
I checked the WIKI example for radiated power in our Sun-Earth system of about 200 watts; I got 182 watts. I also checked the 2E+06 SM galactic BH example with a 1-SM BH orbiting at 1.89E+10 m. This orbit is 3.19 times the combined Rs and does produce 1000 seconds/orbit. The power radiated is 5.49E36 watts and the tangential velocity of the 1-SM is 0.395c. I compute in Ft-Lb-sec units which yields power in (Ft-Lb/s). Metric units of course yields power in (joules/sec) which is watts.
The effect of radiated power is to reduce the velocity and energy of the binary that would normally be conserved by a Newtonian orbit; this loss in energy causes de-orbit of mutually approaching masses. This radiated energy loss is the reason that it is said that "matter entering the event horizon can never escape". If the WIKI equation is correct, equal mass binary black holes below 200Rs would quickly de-orbit and merge. This would make the likelihood of my explanation for "jets" highly improbable since the close binary would be extremely short lived.
The WIKI article Power Formula is stated to be true for only Newtonian orbits; therefore the formula would be good for binaries separated by more than about 200(Rs). Look at Table-1; This does leave open the question whether the Gravitational Radiation Power increases more slowly or more severe as the orbital speed becomes relativistic for v/c>~0.025 at R/Rs<200. Presumably the radiated power would be less than the formula value since radiated power would decrease the energy and velocity of the masses and thus decrease acceleration that would be predicted by a Newtonian orbit at similar separations.
It is more than interesting that for R=200*Rs that the radiated power from the 10+10 SM binary would be 1.42E+39 watts. That is 3.67 million million (3.67E+12) times the power output of the Sun. A 20SM binary would be rather small as black hole binaries go. This leads to the question: "why in our galaxy (full of black holes, 100E+06 of them?) have they not already detected gravity waves using multiple detection schemes?
Think of the power that would be radiated by the 4E+06 SM black hole at the galactic center if it were A BINARY. I should think that gravity wave power in our galaxy would be so intense as to be impossible to 'not detect'; not to mention sloshing java from our coffee mugs.
A single formula for radiated power involving both (M1 and M2) is offered. Nothing is said about the proportion of power emitted from each body. If (M1=M2), symmetry would cause equal (Tot_power/2) loss from each body. This would not be the case where one BH is larger than the other.
Since acceleration of mass is said to cause gravitational radiation, the energy may split in inverse proportion to the radial spin distances barycenter-to-D1 and barycenter-to-D2; acceleration = w2r. If M1>>M2 then M1 would be nearly motionless so most all the gravity work done on infall would be dissipated as radiated energy from the relatively massless M2, methinks.
Think In Terms of LUMINOSITY of Stars for Equivalent Energy Radiated in The Electromagnetic Spectrum. The Apparent Magnitude of the Sun viewed from Earth is -26.7; viewed from 10 parsec or 32.6 LY the Sun's Apparent and Absolute Magnitudes are +4.83. Were the 20SM BBH located at 32.6 LY it would 'shine' in Gravity Wave Energy' with Apparent Magnitude -26.58 which is as 'bright' as the Sun viewed from Earth (remember, the BBH is 32.6 LY from Earth!). According to the Wiki deorbit time formula the (R/Rs=200 separation distance) binary would continue to de-orbit and merge in 2.28 days.
So, Now move the 20SM BBH (26,000 LY) to the center of the Milky Way Galaxy: This 'tiny' BBH still 'shines' at Apparent Magnitude -12.07, a bit less than the Full Moon. The MY-Galaxy must be brimming with Binaries much larger than the 20-SM BBH and yet Gravitational Waves have not been detected. Perhaps the power algorithm is suspect or maybe even question existence of 'gravity wave radiation'? Investigative astrophysics over time will likely reveal the reality.
As far as I knew, no other set of exponents (of the arguments) would produce units of joules/sec (watts). Therefore, I have searched for alternative Gravitational-Radiation-Power equations that are dimensionally correct. The idea being, could (joules/sec) be maintained with other exponent values of the terms G, (M1*M2), (M1+M2) c, and r in the energy equation? The 20 results from 34,300 iterations are shown in the table below. 19 other sets of exponents that produce dimensionally correct units of [Ft-Lb]/sec (or joules/sec = watts) were found. Try No. 16,840 yielded the Wiki Formulation.
In the Tabulation above, the 5 columns on the left are the combination of the 5 exponents that produce the dimensionally correct units of power shown in the 3 right columns: Ft1-Lb1 s-1.
In the above Graphic, Gravity Wave Power (based on the Wiki equation) is converted to Schwarzschild Space. In these writings, (since 2011) I express all Binary Black Hole computations with separation distance (d) inputs expressed as a constant (n) times 2*RSCHWARZSCHILD. Converting to "Schwarzschild quantized space" presents a unified format even with very large changes of BH mass. What I noticed when computing Gravity Wave Power for various Binarys is that changing the Binary mass changed the spin rate but did not affect Radiated Power. This result of "power independent of mass" seems to be screaming something. Something to think on.
This led to manipulating the Wiki Power equation by algebraic substitution of distance (d) by (2*n*RS). In addition, this graphic assumes M1=M2=M to further simplify the result for the special case 'equal mass' binary. I will later add the M1<>M2 generalized solution for un-equal mass Binarys.
The Gravitational Radiation Power equation is said to only be valid for non-relativistic orbits where (v/c)<0.025. Gravitational Radiation power is sourced by (mutually induced) acceleration of the binary masses. The acceleration of the masses can be determined by their mass and relative proximity. This therefore suggests that Gravitationally Radiated Power can be computed well into relativistic (v/c) situations.
Realizing this, I checked the ratio of acceleration/radiation_power for non-relativistic binary black holes where the Gravitational Radiation Power equation does apply. To make the (acceleration/radiated_power) ratio equal to unity, acceleration had to be raised to the (5/2) power.
Gravitationally radiated power, however, is proportional to acceleration 2.5. I find that interesting. Knowing this, The Gravitational Radiation Power can be determined well into relativistic solutions for binary black holes up to the point where their event horizons begin to merge by, not using the Power Equation, but by evaluating the mutual acceleration of the masses at those closer proximities and then scaling the radiated power by accel2.5 times the valad non-relativistic case.
This line of thought of tracking radiated power by acceleration suggests a sanity test for the Gravitational Radiation Power Equation. Conservation would suggest that the limit of energy (integral of power x dt) that can be radiated would be (is) bounded by the available gravity energy as the bodies approach.
Two dimensionally consistent formulas for Gravitational power that would radiate proportionally to acceleration2 emerged from the 'Dimensionallity' Solutions (Tru No's. 11,484 AND 16,448 in Table Above). They are:
It is more than curious, if not inconsistent, that the Gravity Wave Radiated Power formula would produce radiated power proportional to acceleration(5/2) rather than to acceleration2 as may be expected. Energy in a capacitor OR of a mass go to CV2 AND mV2 respectively. Charged particles radiate power to acceleration2. An Exponent of '2' in each case.
There is more to come on the Gravitational Radiation topic. I can include the WIKI Equation in my 3-body numerical solver to project de-orbit rates for close binaries using the WIKI algorithm; then present animations from computational results of binary de-orbit.
Sure hope you enjoy this as much as I! More physicsstuff on this topic coming soon. dac