Using only Newtonian gravity and approximate values to get an approximate answer

Question

Using only Newtonian gravity and approximate values to get an approximate answer. Consider two collapsed stars, each with a mass of 10^32 kg. They are orbiting the midpoint between them with a frequency of 100 Hz, and so are radiating gravitational waves at twice this frequency*. Estimate the distance between them. (Newton's constant of universal gravitation is 6.67*10^-11 N.m^2.kg^-2. * They only briefly have this frequency, because the energy of their gravitational radiation comes from their orbital energy, so they are spiralling rapidly inwards Supplementary question: Dr Blair jnr, (an excited Australian physicist working on the project) was interviewed on Radio National this morning and chose to demonstrate them using a cello. Why a cello rather than a violin? Both can play what physicists call a chirp and what musicians call portamento. Can you think why it might be difficult to find frequencies much higher than 100 Hz?

Explanation / Answer

given two stars orbitting mid point between them with frequency 100Hz

let l be distance between them

radius of revolution r =l/2

so gravitational force on each of them = Gm2/(l2) = Gm2/(4r2)

and this will balance the centrifugal force on the star which is mw2r and w = 2*pi*f and f = 100Hz

Gm2/(4r2) = mw2r

r = (Gm/4w2)1/3

l = 2* (Gm/4w2)1/3 = 323294.789 meters

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