Supermassive Black Holes....

Classical Mechanics Level 5

Consider a SBH Binary System free from the influence of the stars....This system can be treated as $2$ Massive Black holes each of Mass $M$ moving under the influence of the other in a circle of radius $a$ ....Consequently Energy is lost due to emission of gravitational waves.The rate of Energy lost is given by $dE/dt=-(1024/5)(G/c^5)(\omega ^3 I)^2$ .where $I$ represents the Quadruple Moment $I=2Ma^2$ of the system.Calculate the time taken by the SBH binary to shrink to a radius $r(Schwarszchild radius<)<a$ .(initial radius). If $t$ can be expressed as $t=\alpha a^\gamma c^\beta /\psi G^\phi M^\zeta$ .Where $(\alpha ,\psi )=1$ .Compute the sum of $\alpha +\beta +\gamma +\psi +\phi +\zeta$ ...We this say the Binary SBH merged.

The answer is 1044.

1 solution

Spandan Senapati
May 28, 2017

Let at an instant radius be $a$ We begin by finding the energy of the system.Using Newtons Law $Mv^2/a=GM^2/4a^2$ .The total energy of the system is $E=Mv^2+U=GM^2/4a-GM^2/2a=-GM^2/4a$ .Also $\omega =v/a$ Now Plugging the required values in $dE/dt$ yields $a^3da/dt=-(256/5)(G^3M^3/c^5)$ .Now integration yields $t=5c^5a^4/1024G^3M^3$ .So the required sum is $\alpha +\beta +\gamma +\psi +\zeta +\phi=1044$

Aren't some APhO problems even tougher than those in IPhO? I find them too tricky to solve.

A Former Brilliant Member - 4 years ago

Ya some are tricky...In APHO one harps to make his own conclusions,deductions supported by mathematical arguments.But ipho mostly the maths isn't quite hard and its mostly physical intuition that works.

Spandan Senapati - 4 years ago

Is this a question of this years APhO ?

A Former Brilliant Member - 4 years ago

@A Former Brilliant Member – Ya its intact.Task 2....I have to solve only the slingshot q and next I will complete the task....You tried that angle of deflection,and force one.(dynamic friction)?

Spandan Senapati - 4 years ago

@Spandan Senapati – No but I am trying them.

A Former Brilliant Member - 4 years ago

@A Former Brilliant Member – This wasn't particularly difficult...A similar question appeared this year in INPHO...Where one had to consider interaction between a star and a black hole....But that one wasn't particularly tough as it clearly mentioned classical mechanics to hold good.

Spandan Senapati - 4 years ago

Yep did it in the same way

Suhas Sheikh - 2 years, 11 months ago

Lol Mathphile.

Spandan Senapati - 2 years, 11 months ago

Supermassive Black Holes....

The answer is 1044.

1 solution

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