Arcturus Motion

The light from Hayesmark shows a redshift compared to the sun, meaning that the wavelength we receive is less than the emitted wavelength. This is symptomatic for the source and recipient moving away from each other.

The analogous effect on sound waves is more familiar; when a car moves away from us, we hear a lower pitch (longer wavelength) than usual.

Suppose a "crest" of the wave is emitted. During the next period $\tau$ , the wave travels $\lambda_s = c\tau$ toward us. Meanwhile, Hayesmark moves away from us over a distance $v\tau$ before the next "crest" is emitted. Thus the spacing between the "crests" is really $\lambda_o = \lambda_s + v\tau = \lambda_s\left(1 + \frac v c\right) = \lambda_s\frac{c + v}c.$ While this gives qualitatively correct results (and quantitative for small values of $v$ ), it must be adjusted for effects of relativity. This leads to the equation quoted in the problem statement. Note that if $\beta = v/c \ll 1$ , $\sqrt{\frac{c+v}{c-v}} = \sqrt{\frac{1 + \beta}{1 - \beta}} \approx \sqrt{(1+\beta)(1+\beta)} \approx \sqrt{1 + 2\beta} \approx 1 + \beta = \frac{c + v}c,$ which is our non-relativistic result.

Simply because universe is expanding, the distance between stars should also increase

Acc to me the the spectrum of the other satr mentioned has more larger wavelength spectra as compared to that of sun as both should be measured from same place so ........the wavelngth is in general form lambda =v+ct .......and hence considering it as source its moving away

The spectrum is shifted slightly to the right than our sun’s, so light is travelling at a slightly longer wavelength / lower frequency relative to our sun. By the Doppler effect this means that the source (which is the star) should be moving away relative to us.