What would happen if the sun shrunk by 25% in 2000 years?

Question

The big bad vilain is stealing energy from our sun. He has been doing this for 2000 years and the sun has already shrunk in mass by 25%. What are the effects on our earth. From what i found here and here I think earth should stay in orbit. But I was wondering how much this would affect the number of days in a year.

Answer 1

We can calculate the effect on Earth's orbit by applying conservation of angular momentum. Earth's orbital angular momentum is $$\ell=mvr=mr\sqrt{GM/r}=m\sqrt{GMr}$$ with $M$ the mass of the Sun, $m$ the mass of Earth, $v$ Earth's orbital velocity at any given point and $r$ the distance from Earth to the Sun at any given point. As angular momentum is a conserved quantity, and as the mass of Earth doesn't change, then $$Mr=\frac{\ell^2}{m^2G}$$ is also conserved. So before and after the Sun loses mass, $Mr$ is the same. We can then calculate the final semimajor axis of Earth: $$a_f=\frac{M_i}{M_f}a_i=\frac{1M_{\odot}}{0.75M_{\odot}}(1\text{ AU})=1.33\text{ AU}$$ We can also apply Kepler's third law to determine the period: $$P^2=\frac{4\pi^2}{GM}a^3$$ and plugging in the numbers gives us $P\approx1.78\text{ yrs}$, or around 650 days.

Although you didn't ask explicitly about it in the question, we could consider what happens to Earth's surface temperature, which scales like $$T_{\text{eff}}\propto\left(\frac{L}{a^2}\right)^{1/4}$$ where $L$ is the luminosity of the Sun. If $L$ doesn't change, then the temperature should decrease by about 14% - a bit too cold for my comfort. On the other hand, the luminosity certainly will decrease. Main sequence stars similar to the present-day Sun typical obey the mass-luminosity relation $L\propto M^{3.5}$, so naively inserting that above, we find that the new luminosity would be 37% of the Sun's present luminosity, leading to a true temperature drop of 33%, which would be quite awful. This is a bit unrealistic, but you'd certainly see the Sun cool and contract - and by enough to ensure that Earth is no longer in the habitable zone as we know it.

Would there be any major changes, or will the Sun continue to operate as simply a less massive, cooler, smaller version of itself? I think the latter is likely. It would need to lose about 60% of its mass to become fully convective (the core is mostly radiative), and it would need to lose about 70% of its mass to never evolve onto the red giant branch. Its evolution would take longer - the time on the main sequence is roughly $\tau\propto L/M\propto M^{-2.5}$ - but it would follow a similar track. We won't see exotic fusion pathways or an unusual stellar remnant.

Speaking of stellar evolution, the mass-loss part of this scenario is actually similar to what happens when a Sun-like star evolves onto the asymptotic giant branch after spending time as a red giant. It should undergo mass loss on the order of $10^{-8}$ to $10^{-5}M_{\odot}\text{ yr}^{-1}$, thanks to powerful stellar winds. In your case, the Sun loses mass at a rate of $10^{-4}M_{\odot}\text{ yr}^{-1}$, which we do see in some extreme AGB stars. So purely in terms of mass loss, your villain is sort of accelerating the Sun's evolution dramatically - although of course it shouldn't trigger hydrogen shell burning outside the Sun's core, so it's not like the Sun will actually become an AGB star. It's simply going through a mass-loss phase like one.