Energy to move The Earth to mars' orbit?

Question

How much energy would it take to add to the earth's orbital velocity to push the earth into mars' orbit?

My thoughts: So, in 1-2 billion years, the sun will gradually get brighter.

Suppose a future civilization wanted to gradually move the earth farther from the sun. And for, now, please ignore that Mars is already in Mars' orbit. let's pretend there is room there.

I'm wondering what the answer looks like, in power terms, over the course of a billion years? are we talking more power than modern civilization, or if we have a billion years can we do it with just a small generator's worth of power output?

Answer 1

Ok, I'm going to summarize the info from the comments.

Assuming: (1) E = - G Mm / r^2 is the energy of an orbiting mass.

from wikipedia G = 6.6.7 x10^-11 M = in this case, mass of sun = 1.9891 x 10^30 kg m = in this case, mass of earth, 5.97 x 10^22 r (earth) = radius of earth orbit = 149,598,261 km r (mars) = radius of mars orbit = 227,939,100 km

then E (earth orbit) calculates out to about -3.542 x 10 ^22 joules. and E (mars orbit) calcs out to about -1.652 x 10 ^22 joules.

Which gives a potential energy difference of about 2 x 10^ 22 joules

Now, to consider the rate of energy needed, over 1 billion years, I divide down.

2 x 10^22 joules / 1x10^9 / 365 / 24 / 3600 == about 634,195 joules per second, or roughly 634 Kjoules per second.

Using google, 1 Kj/sec = 1.34 horsepower, so 634 Kj/sec == almost exactly 850 horsepower.

So, if you have 1 billion years, and you have an 850 Hp motor that can add energy to the earth's orbit magically with 100% efficiency, and you can keep it running non-stop for 1 billion years, then you can move the earth.

It's actually surprising to me, how little power is needed, but I guess that just goes to show how incredibly long 1 billion years really is.

ps. if you aren't into automotive engines, the average high performance car might have 200-250 horsepower, and the world's most powerful car (Bugati Veyron) has 1000 hp. Very large industrial trucks or military vehicles might have 1000 hp, and jet planes would have far far more.

Note: - of course, this doesn't explain how our motor is able to add energy directly to the earth orbit, or where we are going to find a gas tank big enough to last 1 billion years. But as a thought exercise, I'm quite happy that the numbers came out to something understandable in human terms.

Answer 2

From a purely energy point of view, calculate the kinetic and potential (relative to the sun) energies of earth at Mars' orbit minus that in its current orbit.

The kinetic energy is very straight forward, 1/2 m v². Velocity (approximating to circular orbit) is only a function of the mass of the sun and the orbital radius. Or, you can simply look up the values for Earth and Mars.

Potential energy is more complicated than the mgh approximation we use in everyday lives here on the surface of Earth, because g is no longer a constant since the distance to the sun changes substantially. You have to solve the integral or look up the formula.

However, how to start with power here on Earth and actually cause the sustained force it would take is totally another matter. What I describe above is just that absolute minimum energy it would require without violating conservation of energy. There is a long way between that and a working system.

Answer 3

A very different answer is given at https://astronomy.stackexchange.com/questions/38298/can-we-change-earths-orbit-with-modern-technology because they used the correct formula: total orbital energy is -GMm/r, not -GMm/r^2. So the number of joules is in the 10^32, not 10^22 range, and you'd need about 10^10 more power than you calculated above: i.e, roughly 100 times the current electrical power output of earth, applied for a billion years. So maybe a global civilization could do this in thousands of years, but not today. There's certainly enough potential power available, as the earth receives 122*10^24 J/year in solar energy alone, so if we used 1% of that we could move the earth to Mars in roughly a million years. But it would be a major project even then. If you're trying to cool the earth, it's a lot cheaper to set up a solar shade at the L1 point.