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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?

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  • $\begingroup$ I'm sure you understand that all motion is relative. What object they push against will impact the amount of needed energy. That could be propellant ejected into space, Mars itself, or something else. Without that, this question doesn't have just one answer. $\endgroup$ – Alan Rominger Feb 17 '14 at 19:53
  • $\begingroup$ Apply: $E=-G\frac{Mm}{r^2}$ $\endgroup$ – jinawee Feb 17 '14 at 20:00
  • $\begingroup$ @jinawee: well, that's the force formula, but yes. AlanSE: there's a natural-enough answer to the question. The solar system with earth in earth's orbit and the solar system with earth in mars's orbit are certainly two different systems with different potential energies an all reference frames, especially in the Newtonian picture, which is sufficient for this question. $\endgroup$ – Jerry Schirmer Feb 17 '14 at 20:07
  • $\begingroup$ @jinwee. ok. using wikipedia values: $\endgroup$ – Charles Teague Feb 17 '14 at 20:18
  • $\begingroup$ @jinwee. ok. using wikipedia values: <br>G = 6.67x10-11 <br>M sun = 1.9891 x 10^30 <br> m earth = 5.97x10^24 <br> r1 = earth-sun dist = 149598261 km. <br> r2 = mars-sun dist = 227939100 km <br> I got E earth = -3.542 x 10^ 22 and E mars = -1.52 x 10^ 22. so, roughly 2 x 10^ 22 joules? <br> over 1 billions years at 365 / 24 / 3600, I converted that down to 634195 j/sec <br> But what is 634 kj per sec? seems like a small amount to move a PLANET ? $\endgroup$ – Charles Teague Feb 17 '14 at 20:24
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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.

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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.

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