Astrodynamics

Three months ago I wrote about Orbital Mechanics, focusing on the ins and outs of how spaceships and satellites navigate their way around a planet’s orbital space. We got into the ins and outs of velocity vectors, inclination angles, and prograde vs retrograde thrust.

But what about when a spaceship leaves Earth — or whatever planet it happens to be in orbit around; Kepler 62f, perhaps? — and makes its way toward another planet in the Solar System. Mars, for instance. We’re all astute enough to realize that it’s not as simple as pointing the ship at Mars and firing the rockets. For one thing, of course, Mars is moving, as it is orbiting around the Sun just like Earth is, so even if a straight-line course were possible, we would need to “lead the target,” i.e. aim for the point in Mars’ orbit where Mars will be at the time we arrive.

Sphere of Influence

And, if that’s all it took, this would be a short post. But of course, it’s not as simple as that, as I’m sure you suspected by now. Once our spaceship leaves Earth’s sphere of influence, and until such time as we arrive in Mars’ sphere of influence, we aren’t just in some flat space with no gravitational pull on us. By sphere of influence, or SOI, I mean the region of space where the primary gravitational pull is from whatever celestial body “owns” that SOI. Close to Earth, or in orbit around Earth (even a very high orbit), the bulk of the gravity we feel is from Earth. And, if we don’t maintain sufficient velocity for the altitude of our orbit, then it is Earth into which we will fall. Likewise for Mars.

But in between Earth and Mars, we are still in an SOI — that of the Sun. So, once our spaceship leaves Earth’s SOI, we are primarily impacted by the Sun’s gravity, just as Earth and Mars are.

In fact, once we leave Earth’s SOI, we are still in orbit. It’s just that it’s no longer Earth that we’re orbiting, it’s the Sun.

Up, Down, and the Plane of the Ecliptic

If you’re like me, you probably grew up thinking of the Solar System as basically a flat plane (the plane of the ecliptic), with the Sun in the center and the planets in their various orbits sliding around that plane at a given distance from the Sun. Each of the planets has a north pole that points roughly in the same directly, “up” from the plane. Well, each of the planets except Uranus; Uranus is completely lying on its side, with an axial tilt of 98°, rolling through its orbit like a ball whereas the others are more like spinning tops (apologies to Fraser Cain of Universe Today for borrowing his imagery; I can’t think of a better way to describe it). Anyway, Uranus aside, this way of thinking about the Solar System and the plane of the ecliptic leads to the idea that the Solar System has something of an “up” that all the planets’ north poles somewhat point to, and a “down” that the various south poles point to.

This is not a useful way to think about space. Thinking like this is what leads to engineers constructing the Death Star, with such obvious design flaws that both the first and second versions suffered catastrophic failure with just a tiny nudge. Don’t be one of those engineers.

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image credit: Andy Langager / flickr.com via cc by-nc 2.0

Astronauts onboard the ISS don’t think of the general direction of Earth’s north pole as “up.” If anything, it’s sort of sideways. The effects of free fall notwithstanding, the ISS and everyone in it remain subject to Earth’s gravity at about 90% of what we feel on the ground — they just don’t notice it because they are falling all the time, just as I described in my previous post on this topic. If there’s a “down,” it’s toward the Earth’s surface, and “up” is away from Earth.

So likewise, when we are orbiting the Sun, and outside any planet’s SOI, it’s more useful to think of “down” as being toward the Sun, and “up” as away from the Sun. On a more technical level, the inner planets (Mercury and Venus) are in lower orbits, and the outer planets (Mars and beyond, though the belters of The Expanse might have a thing or two to say about referring to Mars as an “outer” planet) are in higher orbits.

Delta-V, Prograde and Retrograde

So, referring back to the earlier post, you’ll recall that it’s not possible (or at least not feasible without a very powerful torchship, but that’s beyond the current discussion) to travel from a lower orbit to a higher one by simply aiming the spaceship “up” and firing the rockets, and that likewise we cannot travel from higher orbit to lower orbit by aiming the spaceship “down” and firing rockets. Well, we could, but it wouldn’t give us the desired result.

Instead, we need to fire our rockets either prograde (to raise our orbit) or retrograde (to reduce or lower our orbit). Either way, what we want to achieve is the appropriate delta-v, or change in velocity (often expressed as Δv), required to match orbits with our target.

To get to Mars, once we have left Earth’s SOI (achieved an Earth orbit high enough that the Sun’s gravitational influence takes over), we don’t point our spaceship toward Mars at all. Instead, we point it forward along the direction of Earth’s orbit around the Sun, i.e. in a prograde direction, and fire the rockets. We are already orbiting the Sun at the same velocity as Earth, which just like the orbital velocity of a satellite around Earth is defined by the same equation, √(GM/r), that you surely recall from Orbital Mechanics. Without going into the equation’s details, the Earth moves along its orbit at a rate of approximately 108,000 kilometers per hour, which is fast.

So our spaceship already has this much velocity around the Sun. Mars, in contrast, orbits at an average of 86,760 km/hr (you’ll recall that objects in higher orbit move slower). What we need to do is accelerate prograde, expending delta-v to raise our orbit until it matches that of Mars.

Hohmann Transfer

In Orbital Mechanics I didn’t go quite far enough into the details of just how raising or lowering an orbit works, other than discussing about thrusting prograde or retrograde. However, the principle remains exactly the same, whether in orbit around the Earth or orbit around the Sun.

When we accelerate prograde, the orbit as a whole doesn’t just expand outward to the higher altitude. Instead, it takes our current mostly circular orbit around the Sun and reshapes it into an elliptical orbit instead, with the perihelion, or closest point to the Sun, being the distance (altitude) of our starting orbit, i.e. Earth’s orbit. As we apply thrust, either more powerfully or for a longer period of time, our orbit becomes more and more elliptical, with the aphelion raising higher and higher, or farther and farther out.

990px-Orbital_Hohmann_Transfer.svg
image credit: user:MenteMagica / wikimedia.org

The perihelion of our orbit doesn’t change, however. Eventually, after we have accelerated for a long enough time, our aphelion matches the orbit of Mars, but our perihelion remains at the orbit of Earth. We are now in what’s known as a Hohmann transfer orbit, and if we do nothing else, our spaceship will cycle endlessly back and forth between Mars’ orbit and Earth’s orbit. That, of course, could be quite useful if our intent is to set up some kind of shuttle or cargo transport back and forth, assuming we can match up the times of aphelion and perihelion to when Mars and Earth will be in the same space as our transport craft.

But we aren’t setting up a cargo shuttle. We want to get to Mars, and we want to stay there (for now). We could, upon arriving at aphelion (and thus the orbit of Mars), once again burn prograde. By burning at aphelion instead of perihelion, we won’t be further raising our aphelion; instead we’ll be raising our perihelion. This has the effect of (slowly) reducing the elliptical eccentricity of our orbit, i.e. of circularizing it. And indeed, this is precisely how we would set a satellite into, say, geosynchronous orbit around the Earth.

But, this would negate all the efficiencies of the Hohmann transfer, and require a lot of burning, a lot of propellant, and a lot of time (unless we use a continuous thrust engine, such as an ion drive; more about this later).

Instead, if we are clever and time our launch window so that our arrival at Mars is close to when it will be 180° around the Sun from where Earth was at launch time, then we arrive with least travel time (and least delta-v), and the last thing we want to do is now waste all that by raising our perihelion. Instead, by arriving at just the right time, we will insert our spacecraft into Mars’ SOI (sphere of influence, you’ll recall) and decelerate to slow the craft down enough that it is captured by Mars’ gravity.

MAVEN_orbital_path_rev-fi
image credit: NASA/JPL

Now we’re in Mars orbit! From this point, all the same rules apply about adjusting our orbit around Mars as did for adjusting it around Earth, or for that matter around the Sun. Or around Kepler 62f if that’s what we’re talking about.

Return to Earth

So now it’s time to go home. How is it different? It isn’t, really. In fact, it’s just the same as any orbital maneuver aimed at reducing altitude. We need to reduce our altitude from 228 million kilometers to 150 million kilometers. To do this, we burn retrograde, against the direction of Mars’ orbit. This has the reverse effect from before, in that it reduces the altitude (from the Sun) of our perihelion until it matches Earth’s orbit. We aren’t actually orbiting the Sun in the reverse direction now — we aren’t burning nearly hard enough for that to happen — we’re just pushing back against our orbit so that we start falling in toward the Sun.

Again, we want to time it with a launch window, one in which it will take us just about one-half of a revolution around the Sun to arrive at Earth’s orbital altitude just as Earth arrives to the same spot. And, upon reaching Earth’s SOI, we need to decelerate so that we are captured by Earth’s gravity, and just like that, we’re home! Well, in orbit around home, anyway, but we know what to do from here.

Interplanetary Transfer Summary

When rockets launch from the surface of Earth to reach orbit, or beyond, they usually do so in an easterly direction, and from as close to the equator as practicable, in order to take advantage of the velocity already imparted upon them by the Earth’s rotation. In other words, to reduce how much delta-v needs to be expended to obtain the velocity required for orbit.

When spacecraft leave Earth to go to Mars, they take advantage of the velocity already imparted upon them by Earth’s revolution around the Sun. Burn prograde, and you ascend to higher orbit, or to the outer planets. Burn retrograde, and you descend to lower orbit, or to the inner planets. It’s really no different, whether in Earth orbit, or interplanetary.

But what happens if our spacecraft leaves the Solar System entirely?

Interstellar Orbits

Leaving out for now questions of faster-than-light travel, Alcubierre warp engines, and other such exotic drives, interstellar maneuvers are no different from interplanetary maneuvers. It’s just a question of scale. Just as the Earth and the other planets in our Solar System revolve around the Sun, the Sun and the hundreds of billions of other stars that make up the Milky Way galaxy all revolve around the center of the galaxy. Just as each planet has an orbit around the Sun, each star has an orbit around the galactic center.

And just as there is a path, and a sequence of burns, that describe an efficient Hohmann transfer route between two planets, there is likewise a Hohmann transfer route between any two stars within the galaxy. There are even more or less ideal launch windows, though given the timescale of stellar orbits, we don’t have as much choice about them; we pretty much have to go when we’re ready to go and accept the less-than-ideal configuration.

Interplanetary and interstellar movement is really the same thing; it’s just a matter of scale. A rather large matter of scale, to be sure, but conceptually no different.

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image credit: NASA/Goddard

So when we’re ready to travel between the stars, if our means remains conventional — whether with a short-burn, high-thrust reaction drive, or a continuous-burn, low-thrust drive, or a continuous-burn, high-thrust torch drive — then the way in which we plot our course doesn’t change. We burn prograde or retrograde about the galactic center, and we utilize our home star’s orbital velocity to impart a delta-v advantage upon ourselves.

There are no straight lines. And our spacecraft is always, always in orbit around something.


header image credit: NASA/JPL

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