Vfrdtyky

The Expanse books cover a lot of aspects of war in a situation where humans inhabit many planets and moons of the solar system.

The author makes the point that, in a war situation, the outer planets are at an advantage with respect to the inner planets such as Earth. The inner planets are closer to the sun and thus are, in effect, at the bottom of a gravity well. This enables relatively cheap attacks from the outer planets e.g. lobbing debris "down" towards the target planet.

Does this make sense? Can missiles be lobbed from outer orbits to inner orbits or does it require energy to overcome the higher kinetic energy of the outer orbits?

To get to an inner planet you still need to expend delta v (change in velocity, what fuel is used for in rockets). You have to transition from a roughly circular orbit to an elliptical one with yourself at the outer point and your target at the inner point. So you don’t get to smash them “for free”.

However, you do have one big advantage. Your projectiles will be going faster when they hit. As they fall inward on their elliptical orbit they will lose potential energy and gain kinetic energy. The opposite will happen for their projectiles. It may take a similar amount of fuel for both sides projectiles, but the result will be much stronger impacts against the inner planets.

You ask:

Can missiles be lobbed from outer orbits to inner orbits or does it require energy to overcome the higher kinetic energy of the outer orbits? (my emphasis)

but the kinetic energy of planet gets lower not higher as we move away from the star. The orbital velocity at a distance $r$ from a star of mass $M$ is given by:

$$ v = sqrt{frac{GM}{r}} tag{1} $$

So it gets easier to bomb inner planets as you move farther away.

The simplest way to bomb the inner planets would need to be put the bomb into a Hohmann transfer orbit. I say simplest because you could use gravitational slingshots, but this gets very complicated very rapidly. A good example of this is the Parker Solar Probe that is going to use seven slingshots to reduce its speed enough to approach the Sun.

Anyhow suppose you're on one of the outer planets and you want to bomb an inner planet so your orbits look like:

You launch the projectile backwards, i.e. in the opposite direction to your orbital velocity, to reduce the orbital velocity of the projectile and make its orbit elliptical:

This is your Hohmann transfer orbit. You can find the required change in orbital velocity using conservation of energy, or just the vis viva equation (which is derived using conservation of energy):

$$ v^2 = GM left( frac{2}{r} - frac{1}{a} right) tag{2} $$

If we call the radii of the outer and inner orbits $r_o$ and $r_i$ respectively then to find the required orbital velocity at the outer planet we set $r = r_o$ and the semi major axis $a = (r_o + r_i)/2$. Now just subtract off the orbital velocity to get the required launch velocity.

To make this concrete let's tale the example of bombing Earth from Saturn. The radii and orbital velocities are:

$$begin{align}
&& textrm{Earth} && textrm{Saturn} \
textrm{Orbital radius} && 1.50 times 10^{11}~textrm{m} && 1.43 times 10^{12} ~textrm{m}\
textrm{Orbital speed} && 29.9 ~textrm{km/s} && 9.6 ~textrm{km/s}
end{align} $$

And if we use the vis viva equation (2) to calculate the aphelion and perihelion speeds of our bomb that starts at Saturn's orbit and ends at Earth's orbit we get:

$$begin{align}
textrm{aphelion} ,, v_a &= 4.2 ~textrm{km/s} \
textrm{perihelion} ,, v_p &= 40.2 ~textrm{km/s}
end{align}$$

The launch speed from Saturn is the orbital speed minus the aphelion speed of our bomb, so we find the launch speed is about $5.4$ km/sec. This is only the half the escape velocity from Earth so it could easily be achieved for small masses.

The impact speed at Earth will be the perihelion speed of our Hohmann orbit minus Earth's orbital velocity, so it works out at about $10.3$ km/s. That's actually a bit disappointing when it comes to global Armageddon. The speed is low because with the Hohmann orbit the Earth and our bomb are moving in the same direction at the moment of impact. Asteroid impact speed on Earth are more like $20$ km/s. Still, you'd notice if the bomb landed on you.

If you wanted a planet killing impact that's going to require a much larger mass than you're likely to be able to launch from Saturn. In that case you'd need to go much farther out, but it would have to be a lot farther out. Even going out to Pluto only reduces the launch speed to $3.7$ km/s. You'd probably have to go to the Oort cloud where the launch velocities would fall to below $100$ m/s.

For objects in orbit, sending projectiles either up or down the gravity well requires the same amount of energy expenditure.

If the projectiles are explosives or other payload, then there is no advantage to being higher up in the gravity well. However if the projectiles are kinetic energy weapons, then those falling down the gravity well will reach their destinations at higher velocity than those that go up the gravity well. In this case, then the outer-most planets do have the tactical advantage.