Munroe-Question: Moving the Earth to Reduce Solar Radiation? Rep Gohmert asked one of the funnier questions in Congress about a year ago --- could the US forestry service move the orbit of the earth?
In the best tradition of Randall Munroe's "What If?", has anyone worked out how much energy it would require to change the orbit of the earth so as to change the solar constant?  Right now, the planet gets about 1,370 W/m^2.  If I could strap a rocket to the rear of the earth and fire it at the right moment with the goal of reducing this solar radiation by, say, 1 W/m^2,  how much energy would my rocket have to impart on the earth to get it into such an orbit?
In one hour, the earth receives about 173 PWh solar energy.  I wonder how many hours of (perfectly converted) solar energy it would take.
 A: The solar constant is $S=L_\odot/4\pi a^2$ where $L_\odot$ is luminosity and $a$ is Earth's orbital semimajor axis. So if you want to reduce it by $\Delta S$ you need to increase the semimajor axis to $$a'=\frac{a}{\sqrt{1-\Delta S/S}}.$$
Lifting earth straight from $a$ to $a'$ would take $GM_\odot M_\oplus(1/a-1/a')$ energy, but presumably we also want it to remain in a nice circular orbit so we need to adjust the kinetic energy to match.
The total energy of Earth is $$E=(1/2)M_\oplus v^2 - \frac{GM_\oplus M_\odot}{a}$$ where $v=\sqrt{GM_\odot/a}$ (from the vis-viva equation for a circular orbit). Plugging in, we get $-(1/2)GM_\odot M_\oplus/a$. So moving it from $a$ to $a'$ while adjusting the velocity downward takes $$\Delta E = (1/2)GM_\odot M_\oplus\left(\frac{1}{a}-\frac{1}{a'}\right).$$  It is actually a bit cheaper than a straight lift... assuming we can convert the kinetic energy into potential energy in a safe way.
So if the solar constant change is small $\Delta S \ll S$ so it is nearly constant the time to get lift is $$ t =(1/2)\left(\frac{1}{\pi R^2_\oplus S}\right)GM_\oplus M_\odot \left(\frac{1}{a} - \frac{1}{a'}\right) = (1/2)\left(\frac{1}{\pi R^2_\oplus S}\right)GM_\oplus M_\odot \left(\frac{1 - \sqrt{1-\Delta S/S}}{a}\right).$$
In reality there will be a bit less solar input later on, so this will be a mild underestimate. Inefficiencies in your orbital mechanics will move it towards the double, corresponding to just lifting the planet.
So, plugging in, I get the time to reduce the solar constant by 1 W/m^2 to be $\approx 175000$ years.
If we are escaping a brightening sun one would want to keep the solar constant constant, of course. So the actual rate of expansion would be significantly slower.
