If humans happened to colonize Mars some time in the future and increase its mass such that its gravity increases from 3.721 m/s2 to 9.8 m/s2, would it not fall into the Sun?
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3$\begingroup$ With large enough MAGIC to increase a whole planet's mass, it should be trivial to also keep the orbit steady at the same time, doing so is a much, much smaller magnitude of impossibility than fiddling with the mass. $\endgroup$– CuteKItty_pleaseStopBArkingAug 2, 2021 at 17:00
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2 Answers
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Firstly, orbital trajectories don't really depend on the mass of the orbiting object as long as the body they orbit (in this case the Sun) doesn't move.
The force of gravity is:
$$F = G\frac{m_1 m_2}{r²}$$
And the acceleration experienced by the orbiting object is:
$$a = \frac{F}{m_2}$$
But combining them, we see that the acceleration is independent of the mass of the orbiting object:
$$a = \frac{F}{m_2} = \frac{G\frac{m_1 m_2}{r²}}{m_2} = G\frac{m_1}{r²}$$
Secondly, it would be entirely depended on the mechanism used to add the mass.
To increase the surface gravity by such a large amount, the new planet would consist of very little of the original mass of Mars, and mostly the added mass.
This mass would have to come from somewhere (even the entire asteroid belt is not enough).
What we know about this "somewhere" is that it isn't at the current trajectory of Mars, so if we had the capability to change the trajectory of this amount of mass (we do not), it would imply we could pick pretty much any new orbit for Mars, including falling into the Sun if desired.
answered Aug 2, 2021 at 11:33
SE - stop firing the good guysSE - stop firing the good guys
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The numbers you've quoted describe the pull of gravity Mars and Earth have on objects at their surfaces. This is based on both their mass and their volume. If you somehow squashed Mars down from its current size to half its radius (one eighth of its volume), that would increase its surface gravity by a factor of four, because its surface would be closer to its center. If you did this without changing its mass, its orbit around the sun wouldn't change at all, no matter how much you changed its surface gravity by expanding or contracting its volume.
SE's answer assumes an approximation that is valid for artificial satellites, but makes measurable errors for planets. In particular, the mass that ought to appear in two-body formulas like Kepler's laws is the "reduced mass"
$$\frac{m_1 m_2}{m_1 + m_2}$$
and both bodies orbit around their mutual center of mass, or barycenter, which is the balance point on the line between them. For Jupiter, this is a significant difference, as its barycenter with respect to the Sun is slightly outside the Sun. Changing the mass of only the orbiting body can move the orbit significantly, if the object and the change are large enough with respect to the central body.
The more significant factor is conservation of angular momentum. Mars and Earth do not crash into the Sun because they are not moving towards the Sun. Instead, they are moving (nearly) at right angles to the direction of the Sun's pull, so the Sun swings their direction around in a (near) circle. They are constantly falling, but falling in such a way that they don't actually get any closer (on average). If the mass of Mars suddenly increased by a large factor, its orbit would change, but it would not crash into the Sun unless the angular momentum of the mass you added was carefully arranged to cancel the angular momentum it currently has, so that its direction of falling would change to directly into the Sun, rather than its many billions of years of falling around the Sun.
