Time: Upper Bounds
The first major decision comes on setting upper bounds to our merger probability, a time in which the black holes must have successfully merged by in order to be called probable. In an idealized isolated environment, i.e. a computer simulation, every binary black hole system would eventually merge. To set realistic standards for the probability, three time scales have been chosen.
Universal Time ScaleWhat better upper bound that the highest limiting factor that humans know of. Setting an upper bound as the age of the Universe gives us a peak into what we could hope to observe from the early days of the Universe.
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Solar Time ScaleGiven the time it has taken for the Sun to evolve into the Star it is today and for humans to come to thrive on planet Earth, what is possible for two black holes to do in the same time?
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10 Million Years10 million years may seem to be a large time scale from a human perspective, but in relation to the Universe this time scale is trivial. What conditions must be met for black holes to merge in such rapid time.
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Time: The equation
To examine a probability based on initial mass and radius of the binary black hole system, we will need and equation that relates time to both mass and radius.
Starting classical with Newtonian Energy of orbiting systems we have
E = -(G∗M1∗M2)/2r.
Taking the time derivative of both sides yields
dE/dt = (G*u*M)/(2*r^2) * dr/dt
Where u = (M1*M2)/(M1+M2) and M = (M1+M2)
Luminosity, defined as L = -dE/dt, takes the form
L = (64* G^4 * M^3 *u^2)/(5* c^5 * r^5)
for gravitational waves.
Combining the two equations yields and equation for dr/dt which can be integrated to obtain a function for time.
t = (5* c^5 * r^4)/(256* G^3 * u * M^2)
This equation relates time, mass, and radius by constants and is precisely the equation necessary to figure out probabilities.
E = -(G∗M1∗M2)/2r.
Taking the time derivative of both sides yields
dE/dt = (G*u*M)/(2*r^2) * dr/dt
Where u = (M1*M2)/(M1+M2) and M = (M1+M2)
Luminosity, defined as L = -dE/dt, takes the form
L = (64* G^4 * M^3 *u^2)/(5* c^5 * r^5)
for gravitational waves.
Combining the two equations yields and equation for dr/dt which can be integrated to obtain a function for time.
t = (5* c^5 * r^4)/(256* G^3 * u * M^2)
This equation relates time, mass, and radius by constants and is precisely the equation necessary to figure out probabilities.
Time: Equation Put To Use
Before complicating the problem at hand, let us examine how long in general it will take two black holes of equal mass to merge for a few different masses. The plots of the left hand side examine similar black holes of masses 3-5 solar masses and the right hand side uses masses of 1000-2000 solar masses. From top to bottom there are plots of the Universal time scale, Solar time scale, and 10 million years respectively. The dotted line represents the relative time scale that will be used for each.