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In my previous Blog entry, the pretty chaotic orbit around a pair of binary black holes was shown. Below is a more sedate orbit, perhaps useful in the far, far future. It may offer spacecraft an "easy escape route" from binary black holes.
Consider orbiting black holes, each of mass M, separated by distance 50M in geometric units[1], oriented as shown at time t=0, with constant orbital speed VM = 0.1c (orbit period T=1571M). Let a particle fall from rest at a coordinate distance x=75M. Since the gravitational field is not isotropic, one can expect the particle to fall in with a curved path, as shown in the next figure.
The particle is initially dragged towards the top hole and as the bottom hole comes closer, the path bends downward. When the particle crosses the orbital radius of the holes (25M), it reaches a speed of 0.252c. This is in fact slightly less than what it would have been in a pure Newtonian case, as will be explained later. The normalized 'geometric time' of t=494M can be converted to seconds by multiplying it by G/c3, where M is the mass of the holes in kg.
At t=575M, the particle has taken a 'slingshot' around the hole and is crossing the hole orbit radius again, with a significant gain in velocity (now 0.374c). This is the relativistic 'gravity assist' or 'flyby' maneuver, providing considerably more 'delta-V' (almost 50%) than what a 'Newtonian flyby' would have suggested. The reasons for the extra gain in the relativistic case will be discussed later.
Finally, at t=754M, the particle reaches its original distance (75M) from the center again, but with enough velocity (0.27c) to escape from the black holes (the escape velocity at distance 75M is 0.233c). As can be deduced from the diagrams, the 'test' lasted for slightly less than half a full orbit of the black holes. So, starting from rest in the coordinate system, the particle gained enough energy from the moving black holes to exceed escape energy - quite impressive!
The principle of gravity assist is simple: pass behind an orbiting body (without entering a closed orbit) and you will gain energy from the body; pass in front of it and you will lose energy to the body. If you have to pass on both sides, like in this example and want to gain energy, pass closer behind the body than what you pass in front of it. This is a 'juggle' that NASA has to do almost every time with their interplanetary probes - in order to save fuel and enable more payload to be carried.
I will post some equations in the next Blog entry and attempt to explain the differences between the Newtonian and relativistic slingshot effects. BTW, this does not cause the "flyby anomaly" experienced by many spacecraft. In the weak field, low velocity limit of solar system exploration, the relativistic corrections to Newton's equations are many orders of magnitude smaller than the measured anomaly.
Jorrie
Notes:
[1] Geometric units normalize c and G to unity, so that mass, energy, distance and time are all expressed in meters (or cm in older books and papers) and velocity is dimensionless. This simplifies equations and calculations considerably and it is very easy to convert the results back to SI units. The conversion factors are: G/c2 = 7.41 x 10-28 m/kg and G/c3 = 2.47 x 10-36 s/kg, or the inverses if you need to convert the other way around. As an exercise, you may want to plug in one solar mass (~2 x 1030 kg, or ~1480 meter) for each black hole and check the real times of the experiment.
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