What are the effects of Increasing the gravity on Mars?

Question

I'm interested in the Physics around increasing the gravity of Mars to make it habitable. I was unsure how to work out what gravity would be needed to retain an atmosphere thick enough to hold liquid water, so was thinking on basing things on matching Earth

What would be the effect on the orbit of Mars if it's mass/gravity were able to be instantaneously increased to that of Earths?

Understanding this is a very impractical idea, and particularly ignoring that impacts of the size needed would raise the temperature to insane levels, are there any interesting aspects to explore? Particularly involving calculations

So far I've just calculated the difference in mass and gravity between Earth and Mars, and tried to calculate the energy needed to move that mass

I used this equation: $g={GM\over r^2}$

To get the ratio of gravities: $g_M \over g_E$$=$$M_m \over M_E$$($$r_E \over r_M $$)^2$$={0.64169×10^{24}kg \over 5.9722×10^{24}kg}{6371.0×10^3m \over 3389.5×10^3m}= 0.2$

And this: $KE = {1 \over 2} mv^2$

To try and calcualte the energy needed to move the extra mass needed: $KE = {1 \over 2} (5.9722-0.64169)×10^{24}kg×({24.07×10^3m s^{-1}})^2 =1.5441552×10^{33}J$

I used the mass and radius of Mars/Earth and the mean orbital speed of Mars as velocity, taken from the NASA fact sheet https://nssdc.gsfc.nasa.gov/planetary/factsheet/marsfact.html

Answer 1

tl;dr, the orbit of Mars doesn't change

So, since this mass increase isn't going to significantly affect the motion of the Sun (if mars suddenly became the mass of the sun it obviously wouldn't stay in the stay orbit because the Sun would now be moving too and that complicates things) we are able to look to Kepler's third law to tell us that the mass of a body in the system isn't really part of its orbital path. But that might not be too intuitive, so we can derive it from just some basic physics to make it clearer why Mars' mass doesn't really affect its orbit around the sun.

Let's assume the orbit of Mars is pretty circular (this is a good approximation), we know that the acceleration felt by an object going in a circular path (aka the centripetal acceleration) is $$a = \frac{v_T^2}{r}$$ where $v_T$ is the tangential velocity (the velocity of the object that is not towards the center) and $r$ is the distance from the object in circular motion to the center of the circle. Since $F = ma$, this times the mass of Mars $m_{mars}$ is the force felt by Mars. We also know the equation for the force of gravity is

$$F_{gravity} = \frac{GMm}{r^2}$$

so setting these equal to each other we get

$$m_{mars} \frac{v_T^2}{r} = \frac{GM_{sun}m_{mars}}{r^2} $$ we can then solve for $v_T$, noting that the mass of mars cancels out on each side $$v_T = \sqrt{\frac{GM_{sun}}{r}} $$ we then note that, for a circle, the tangential velocity is the distance ($2\pi r$ since we're talking about a circle here) over the period $P$ $$v_T \approx \frac{2\pi r}{P} $$ substituting this in, we find $$ P^2 \propto r^3 $$ also known as Kepler's third law, showing us that the mass of Mars doesn't affect how long it takes for Mars to make an orbit, nor how far it is from us, and essentially meaning that nothing noticeably changes in the orbit.

Note

This is assuming more or less that the mass of Mars had always been the mass of Earth as opposed to it increasing suddenly; if it increased suddenly, a mechanism would need to exist for the mass to increase suddenly, and the most realistic scenario for this is a collision, which would most certainly change the orbit of Mars. Even if an advanced civilization had the capacity to bring mass somehow to Marss to increase its size, you could essentially think of a delivery of this amount of mass as functionally a series of smaller collisions that would ultimately add up.

Edit: Let’s further consider the idea of an instantaneous mass increase; while a collision would be more or less an instantaneous mass increase, the direction and momentum of the colliding object would matter a great deal, and so let’s suspend some disbelief and say, somehow, the mass increased without any outside influence (in other words, we treat this as a closed system where momentum is conserved).

Since angular momentum must be conserved in this scenario, $$L_{initial} = L_{final}$$ or in other words $$m_{mars}v_{T,i}r_i=m_{earth}v_{T,f}r_f$$ $$m_{mars}\frac{2\pi r_i^2}{P_i}=m_{earth}\frac{2\pi r_f^2}{P_f}$$ Using Newton’s version of Kepler’s 3rd law $$P^2=\frac{4 \pi^2}{G(m_1+m_2)}a^3$$ And using the simplifying assumption $m_{sun}>>m_{earth}>m_{mars}$ $$\frac{m_{mars}^2r_i}{m_{earth}^2}=r_f$$ Seeing as the mars is about 10% the mass of Earth, this means that the distance of Mars from the Sun would decrease 100 fold, at least if no other bodies were concerned. It would probably go into some chaotic trajectory due to its interaction with other planets, and from there it would be anyones guess what would happen, although crashing into the sun seems likely.

It’s worth noting that, while we chose to conserve angular momentum, energy here is not conserved because we are magically synthesizing new mass out of nowhere. This situation may seem counterintuitive, and it should, because this scenario isn’t physical, but also because orbital mechanics are notoriously not intuitive.

Answer 2

I'm interested in the Physics around increasing the gravity of Mars to make it habitable. I was unsure how to work out what gravity would be needed to retain an atmosphere thick enough to hold liquid water, so was thinking on basing things on matching Earth

It is the escape velocity, and not the surface gravity, of a world which is most important in how long it can retain an atmosphere.

Here is a link to a recent question and my answer:

Would 25% less gravity produce dramatic differences in animal morphology?

Earth has an escape velocty of 11.186 kilometers per second. And according to calculations by Dole, 1964, in some cases a world with with an escape velocity of only 6.25 kilometers per second, 0.5587 that of Earth, could retain 0.368 of its original atmospheric oxygen for about 100 million years.

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf

And it is quite possible that your fictional characters terraforming Mars would think that retaining 0.368 of the original oxygen in the atmosphere for about 100 million years would be long enough for their purposes. And maybe they would think that they could replace oxygen in the Martian atmopshere as fast as it was lost - a century ago the Martians had giant air factories to replenish the Martian atmosphere in the stories of Edgar Rice Burroughs.

Mars has a radius of 3,389.5 kilometers (about 0.53202 that of Earth), a volume of 6.13118 times 10 to the 11th power cubic kilometers (0.151 that of Earth), a mass of 6.4171 times 10 to the 23rd power kilograms (0.107 that of Earth), a mean density of 3.9335 grams pwer cubic centimeter (0.71336 that of Earth), a surface gravity of 3.72076 meters per seconod per second (0.3794 that of Earth), and an escape velocity of 5.027 kilometers per second (0.449401 that of Earth).

So possibly the terraformers of Mars in your story might want to tackle the incredibly vast project of increasing the mass of Mars enough to give it an escape velocity of 6.25 kilometers per second, 1.223 kilometers per second more than its present value.

Or maybe they might dare to tackle the even more immense project of increasing the mass enough to give it an escape velocity of 7.5 kilometers per second, 2.473 kilometers per second more than its current value.

But either would be a much smaller project than increasing the mass of Mars enough to give it an escape velocity of 11.186 kilometers per second, which is 2.225 times the current escape velocity, and 6.159 kilometers per second higher.

And perhaps worlds even smaller than Dole calculated might retain their atmospheres for very long periods of time.

My answer to another question, the question I linked to above, discusses a couple of theories in which even smaller worlds could retain atmospheres for long, some of them even less massive than present day Mars.

Would 25% less gravity produce dramatic differences in animal morphology?

And of course an easier and much less massive project to retain the atmosphere of a terraformed Mars would be to simply put an airtight roof over the planet to hold in the air. That would be an incredibly vast megaproject by comtemporary standards, but would involve moving a fantastically less amount of matter than necessary to significantly increase the mass and thus escape velocity of Mars. It would be a fantastically easier project.

And if the terraformers build a roof over Mars to keep the air in, they won't have to worry about creating an artificial magnetosphere to protect the Martian atmosphere from the charged particles in the solar wind.