When the Sun converts mass to energy, do the orbits of the planets change?
If I'm correct, fusion reactions convert some mass into energy. Shouldn't this conversion reduce the gravitational "pull" (or warping) of the object undergoing the reaction? So, in the case of our Sun shouldn't the planets' orbits be slightly different over time since the mass of the Sun is gradually being reduced by fusion? I understand that the effect would be very slight over observable time and might be swamped by the angular momentum of the orbiting bodies.
Yes, the mass of the Sun is indeed being reduced due to nuclear fusion processes in the Sun's core, which convert part of the mass into energy. (This energy is eventually radiated away in the form of light from the Sun's surface.) However, the effect on the orbits of the planets is very small and would not be measurable over any reasonable time period.
One way we can see that this must be a small effect is to look at the main fusion reactions which produce the Sun's energy, in which four hydrogen atoms are transformed into one helium atom. If you look at a periodic table, you will see that one helium atom has about 0.7% less mass than four hydrogen atoms combined -- this "missing mass" is what gets converted into energy. Therefore, at the absolute most, only 0.7% of the Sun's mass can get converted, and this takes place over the entire 10 billion year lifetime of the Sun. So it must be a very small effect. (In actuality, not all of the Sun's mass is hydrogen to start with, and only the mass in the inner core of the Sun gets hot enough to undergo fusion reactions, so we really only expect around 0.07% of the mass to get converted.)
It is also easy to directly calculate the rate at which the Sun converts mass to energy. Start with Einstein's famous formula:
E = M c2
where E is the energy produced, M is the mass that gets converted and c is the speed of light (3 x 108 meters/second). It is easy to extend this formula to find the rate at which energy is produced:
(rate at which E is produced) = (rate at which M disappears) x c2
The rate at which the Sun produces energy is equal to the rate at which it emits energy from its surface (its luminosity), which is around 3.8 x 1026 Watts -- this number can be determined from measurements of how bright the Sun appears from Earth as well as its distance from us. Plugging this into the above formula tells us that the Sun loses around 4,200,000,000 kilograms every second!
This sounds like a lot, but compared to the total mass of the Sun (2 x 1030 kilograms), it actually isn't that much. For example, let's say we want to measure the effect of this mass loss over 100 years. In that time, the Sun will have lost 1.3 x 1019 kilograms due to the fusion reactions, which is still a very tiny fraction of the Sun's total mass (6.6 x 10-12, or about 6.6 parts in a trillion!).
How does this affect the orbits of the planets? Intuitively, if we imagine a planet orbiting the Sun at some speed, as the Sun loses mass its gravitational pull on the planet will weaken, so it will have trouble keeping it in the same orbit. The planet's velocity will therefore take it further away from the Sun, and the orbital separation between the Sun and planet will increase.
The formula that governs this situation turns out to be that the orbital separation is proportional to 1 divided by the Sun's mass -- this can be derived from the fact that the Sun-planet system must conserve its angular momentum as the Sun loses mass. The orbital period of the planet, meanwhile, is proportional to 1 divided by the Sun's mass squared.
For small percentage changes in the Sun's mass (as we are considering here), all the above formulas reduce to a nice simple approximation: For every percentage decrease in the Sun's mass, the orbital separation of the planet will increase by the same percentage, and the orbital period of the planet will increase by twice the percentage.
Above, we said that in 100 years, the Sun's mass will decrease by 6.6 parts in a trillion. Therefore, the orbital separation of the planet will increase by 6.6 parts in a trillion and the orbital period will increase by 13.2 parts in a trillion. If the planet in question is the Earth (whose orbital separation from the Sun is around 150,000,000 kilometers and whose orbital period is 1 year), the Earth-Sun separation will increase by about 1 meter, and the orbital period will increase by about 0.4 milliseconds! Neither of these values is large enough for us to be able to detect.
I'm not sure exactly how long we'd have to wait to see a measurable effect in the Earth-Sun orbit. Probably, there are other effects which overwhelm this one and would make it difficult or impossible to detect, even over very long time periods -- for example, changes in the Earth's orbit due to perturbations from other planets. The Sun's mass is also changing due to other effects (such as the solar wind), but over the long run these are probably smaller than the Sun's mass loss due to fusion (as pointed out in another Ask an Astronomer site's answer to this question).
Overall, I think it is safe to conclude that (a) there will be no noticeable effect on the planets' orbits over anything resembling a human lifetime, and (b) there will be a noticeable effect over timescales approaching the lifetime of the Sun, since the Sun will lose around 0.07% of its mass over that time period, leading to a change in the Earth's orbital period of about half a day.
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