Astronomers believe that quasars are powered by supermassive black holes at the

Question

Astronomers believe that quasars are powered by supermassive black holes at the centre of the distant galaxies, and it is thought that such a black hole lurks at the centre of our own Galaxy. In the same way that the mass of the Sun may be determined by the speed at which planets orbit it, the mass of this black hole may be determined by how quickly stars orbit it.

To do this we can use the following equations:

Where F is the force acting on the star, G is Newton’s universal constant of gravitation, M is the mass of the black hole, m is the mass of the orbiting star, r is the radius at which the star orbits and v is the velocity at which the star orbits the central black hole.

(ii) Rearrange Equation 5 to make M the subject of the equation. Show your working at each stage.

(b)Astronomers have measured the velocity of a star orbiting the central black hole of our own Galaxy at a distance of 2.2 × 10^10 km and found it to be5.0 × 10^3 kms^-1 . Using your answer from part (a)(ii) calculate the mass of the black hole, expressing it in terms of multiples of the mass of the Sun(2.0 × 10^30 kg) .

(c)Would the orbital motion of the star affect its spectrum when observed from the Earth? Briefly explain your reasoning. (We expect you will be able to answer this in no more than 80 words.)

Explanation / Answer

under   circular motion The gravitational attraction   force is   equals   to the cetripetal force GMm   / r^2   =   mv^2 /r    GM / r   = v^2   Then   mass of the black hole   M =   v^2 *r / G   ............(1) (b)  Given    v = velocity of the star   =    5.0*10^6 m /s r = ditance between black hole and the star = 2.2*10^13 m G = gravitational constant   =   6.67*10^-11   N.m^2 / kg^2 plug all values    in (1) we get mass of the black hole M   = 8.245 *10^36  kg interms of mass of the sun   Ms   =   2.0*10^30 kg M   = 8.245 *10^36 kg     ( Ms / 2.0*10^30 kg )   = 4.1225 Ms (c) let us consider the orbital motion of    stars     The   doppler   effect   can effect   the spectrum    of the stars   due to   its orbital motion The doppler effect only show   the radial components of stars motion that is the motion is towards or   away from us . The plane of revolution   is perpenducular to the line of sight from the earth there is no doppler shift during the radial components of orbital motion of the stars   can cause the change in the actual spectrum   By konwing   the temperature of the stars   we can extract the actual spectrum   of stars The change in the actual spectrum   and apparent spectrum can give   doppler shift   

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