In September 2015, LIGO detected gravitational waves from the merger of a ~36 so
ID: 1770305 • Letter: I
Question
In September 2015, LIGO detected gravitational waves from the merger of a ~36 solar mass black hole with a ~29 solar mass black hole. The final black hole had a mass of 62 solar masses with 3 solar masses of energy radiated in gravity waves. (Solar mass means the mass of our sun which is 1.989 × 1030 kg.) a) How much energy was radiated? b) It has been stated that the peak power radiated was more than all of the stars in the entire universe. Let’s estimate. Estimate the power if the energy was radiated in ~0.05 s. There are something like 100 billion galaxies in the observable universe. Our sun radiates about 4 x 1026 W, and the Milky Way contains about 200 billion stars. Taking the Milky Way as typical how does the power compare to the gravity wave power? c) Another fun estimate: The peak strain detected was around h = DL/L = 1 x 10-21. The black hole merger occurred about 1.25 x 1025 m from Earth and h scales inversely with distance (1/d). Assuming a 2-m tall person, if you were 1 A.U. = 1.5 x 1011 m (the distance of the Earth from the Sun) away from the black hole merger how much would you be squished or stretched? In September 2015, LIGO detected gravitational waves from the merger of a ~36 solar mass black hole with a ~29 solar mass black hole. The final black hole had a mass of 62 solar masses with 3 solar masses of energy radiated in gravity waves. (Solar mass means the mass of our sun which is 1.989 × 1030 kg.) a) How much energy was radiated? b) It has been stated that the peak power radiated was more than all of the stars in the entire universe. Let’s estimate. Estimate the power if the energy was radiated in ~0.05 s. There are something like 100 billion galaxies in the observable universe. Our sun radiates about 4 x 1026 W, and the Milky Way contains about 200 billion stars. Taking the Milky Way as typical how does the power compare to the gravity wave power? c) Another fun estimate: The peak strain detected was around h = DL/L = 1 x 10-21. The black hole merger occurred about 1.25 x 1025 m from Earth and h scales inversely with distance (1/d). Assuming a 2-m tall person, if you were 1 A.U. = 1.5 x 1011 m (the distance of the Earth from the Sun) away from the black hole merger how much would you be squished or stretched?
Explanation / Answer
Total mass before collison = 36+29 = 65 Ms, where Ms is one solar mass.
Final mass after collison = 65 Ms
Difference in mass (3 Ms) has been released as energy. It is given that Ms = 1.989 x 1030 kg
c is speed of light = 3 x 108 m/s
a) Energy released = 3Msc2 = 5.37 x 1047 J (Joule is the SI unit of energy)
b) Power released in black hole merger = (Energy/Time) = (5.37 x 1047)/0.05 = 1.074 x 1049 J/s
Given, power of sun = 4 x 1026 W.
Number of stars in milky way = 2 x 1011
Power in Milky way galaxy = Power of sun x number of stars = 8 x 1037 W
Number of galaxies = 1 x 1011
Assuming all galaxies have similar power like Mily way, total power of stars in the universe
= number of galaxies x power of one galaxy = 8 x 1046 W
Total power is much less than power released in the black hole merger (more than 2 orders of magnitudes less)
c) If k is a constant, strain h = k/d, from given proprtionality relation.
Given, h = 1 x 10-21 and d = 1.25 x 1025 m
So k = h x d =1.25 x 103 m
In case of sun distance d' = 1.5 x 1011 m, and so h' = k/d' = 8.33 x 10-8
Given, L' = 2 m, and we have calculated h' (which is equal to dL'/L')
Therefore, dL' = L' x h' = 1.67 x 10-7
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