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.
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.
- 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.