A main sequence star has a mass of 2.2\({M_ \odot }\) where \({M_ \odot }\)=  1 solar mass. The lifetime of a star on the main sequence is proportional to \(\frac{M}{L}\) where M is the mass and L is the luminosity of the star.

Using the mass–luminosity relation L∝M^3.5 show that the

(i) luminosity of the star is 16\({L_ \odot }\) where \({L_ \odot }\)=1 solar luminosity.

(ii) lifetime of this star on the main sequence will be approximately \(\frac{1}{7}\) of the lifetime of the Sun.

The star in (a) will evolve to become a white dwarf. The diagram represents the stages in the evolution of the star.

(i) On the diagram, label the two intermediate stages.

(ii) State what may be deduced about the mass of this star when it is in the white dwarf stage.

(i) L/luminosity=2.2^3.5 L_{\( \odot \)};
(L≈16 L_{\( \odot \)})
Some explanatory algebra required, not just 2.2^3.5.

(ii) \(\left( {{\rm{since }}T \propto \frac{M}{L}} \right)\frac{T}{{{T_ \odot }}} = {\left[ {\frac{{{M_ \odot }}}{M}} \right]^{2.5}}\) or \(\frac{T}{{{T_ \odot }}} = {\left[ {\frac{{{M_ \odot }}}{M}} \right]^{ - 2.5}}\);
\(\frac{T}{{{T_ \odot }}} = \left( {\frac{1}{{{{2.2}^{2.5}}}} = } \right)\frac{1}{{7.2}}\);
(the lifetime will be approximately 7 times less than that of the Sun)

or

\(\frac{T}{{{T_ \odot }}} = \frac{M}{L} \times \frac{{{L_ \odot }}}{{{M_ \odot }}}\);
\( = \frac{{2.2}}{{16}} = 0.14\);
(the lifetime will be approximately 7 times less than that of the Sun)

Be careful to check that the working is valid for the value obtained.

(i)

(ii) less than the Chandrasekhar limit / less than 1.4\({M_ \odot }\);