Question

a. A relatively rare transition of the hydrogen atom emits a radio photon with with ? = 21 cm. This emission line is extremely important to astronomers for two main reasons: it cuts through most gas and dust without being absorbed and, although transition is rare, there is so much hydrogen in space that the 21 cm photons are themselves quite common. Suppose that one such photon from a distant galaxy is measured to have a wavelength of just 11 cm. How fast is that galaxy receding from us?

b. Assuming that the speed you found in problem a is enitrely due to cosmic expansion, how far away is the galaxy from us?

Answer 1

a. According to Hubble law, velocity of a galaxy is increasing as it is moving away from us. The recessional velocity is calculated using the relation;  

Where H_o - is called as Hubble constant = 71 km s^-1 Mpc^-1 (This means, a galaxy at 1 million parsec away from earth is moving away from us at the speed of 71 km per second.) D - distance of that galaxy.

To calculate the recessional velocity, we use the following relation;

Where, is the difference between measured wavelength for aline to wavelength for the same line observed in the spectrum of an object at rest.

In this case, and and c = 3 x 10⁸ m s^-1.

Substituting these values in the above equation, we get

The recessional velocity of the galaxy is 2.72 x 10⁸ m s^-1

b. Substituting for v_r = 2.72 x 10⁸ m s^-1 and H_o = 71 km s^-1 Mpc^-1 in the equation (1), the calcuated distance is,

Thus one such photon from a distant galaxy which is measured to have a wavelength of just 11 cm is at a distance of approximately 3841 million parsec, i.e. 1.2528384 x 10¹⁰ lightyears.