State Hubble’s law.

The wavelength of a line in the spectrum of atomic hydrogen, as measured in the laboratory, is 656 nm. The same line in the spectrum of light from a distant galaxy is measured to be 790 nm. The galaxy is 940 Mpc from Earth.

(i) Show that the recessional speed of the galaxy is 6.13×10⁴ km s^–1.

(ii) Determine, using your answer to (b)(i), a value for the Hubble constant.

(iii) Show, using your answer to (b)(ii), that the age of the universe is of the order of 10¹⁷ s. (1 pc =3.1×10¹³ km)

the recessional speed of galaxies from Earth is proportional to their distance from Earth;

or

v=Hd; (with symbols defined)

(i) \(v = c\frac{{\Delta \lambda }}{\lambda }\);
\( = 3.0 \times {10^5} \times \frac{{134}}{{656}}\);
\( = 6.13 \times {10^4}{\rm{km}}{{\rm{s}}^{ - 1}}\)

(ii) \(H = \frac{v}{d}\);
\( = \left( {\frac{{6.13 \times {{10}^4}}}{{940}} = } \right)65.1{\rm{km}}{{\rm{s}}^{ - 1}}{\rm{Mp}}{{\rm{c}}^{ - 1}}\);

(iii) \(T = \frac{1}{H}\);
\( = \frac{{3.1 \times {{10}^{19}}}}{{65.1}} = 4.76 \times {10^{17}}{\rm{s}}\);
≈10¹⁷s