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Suppose a hot Jupiter is collided into parent star due to tidal force, does the life of the star becomes longer due to extra hydrogen, or becomes shorter due to extra mass?

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    $\begingroup$ Is the life of massive stars in general longer due to the extra hydrogen, or shorter due to the extra mass? $\endgroup$ – pela Jul 14 '16 at 8:43
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    $\begingroup$ If they're roughly the same metallicity then I would say the extra mass causes the star to burn slightly quicker, but I would add that this is probably negligible over the lifetime of the star itself. $\endgroup$ – Dean Jul 14 '16 at 9:21
  • $\begingroup$ Quicker would be my answer too. More mass = shorter stellar lifespan. That's a general rule with stars. $\endgroup$ – userLTK Jul 14 '16 at 12:51
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The short answer is pela's comment: more mass equals shorter life.

For a bit of a longer explanation...

The only way the life of the star could be prolonged is if the new material can be transported to the core. This requires convection. Most stars are not fully convective, having distinct layers of either convective or radiative zones. The net effect of the collision in this case is to simply add more mass to the star as a whole while adding no new fuel to the core (or a burning shell); ergo, shorter lifespan.

But some stars do (or may) have fully convective stages, such as the main sequences of very small (aka red dwarf) stars and very large stars with enough metals for a dominant CNO cycle. So in such a situation the material of the colliding body would be transported to (and from) the core, giving more fuel to burn. However, the star still has the same, increased mass, regardless of where the "new mass" currently is inside the star, resulting in higher core temperature. The net effect of which is that the increase to the fusion rate exceeds the proportional mass increase. So, shorter lifespan.

The basic rule here being that core temperatures are dictated by the gravity well of the star, which is dictated by the mass (and a few other GR contributions, but usually predominantly mass). The more mass, the deeper the well. So adding more mass just means the core (and areas near it) are in a deeper gravity well, meaning the temperatures go up. Stellar fusion reaction rates are approximately non-linear polynomials in the temperature; some go as far as being proportional to $T^{40}$, the 40-th power of the temperature! It's this non-linearity which results in the fundamental relation of the short answer: more mass equals shorter life. Increasing the mass by a given proportion increases the fusion rates by a (much) larger proportion.

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