Consider an M-dwarf star of mass 0.1M⊙ and luminosity 10−3L⊙. When the star joins the main sequence 75% of its mass is hydrogen and 25% is helium. The star is fully convective and hence the interior is always fully mixed. Fusion reactions provide all of the luminosity of the star, and each reaction converts 4 hydrogen nuclei into 1 helium nucleus. The combined mass of 4 hydrogen nuclei is 6.690×10−27kg and the mass of one helium nucleus is 6.643×10−27kg. Estimate the main sequence life time of this star, assuming that the luminosity is constant throughout the star's life time. Express your answer in Gyr. The solar mass M⊙=2×1030kg and the solar luminosity L⊙=3.83×1026W.

Consider an M-dwarf star of mass 0.1M⊙ and luminosity 10−3L⊙. When the star joins the main sequence 75% of its mass is hydrogen and 25% is helium. The star is fully convective and hence the interior is always fully mixed. Fusion reactions provide all of the luminosity of the star, and each reaction converts 4 hydrogen nuclei into 1 helium nucleus. The combined mass of 4 hydrogen nuclei is 6.690×10−27kg and the mass of one helium nucleus is 6.643×10−27kg. Estimate the main sequence life time of this star, assuming that the luminosity is constant throughout the star's life time. Express your answer in Gyr. The solar mass M⊙=2×1030kg and the solar luminosity L⊙=3.83×1026W.