In Orbit
A Primer, Part II
This is a continuation of my primer on achieving orbit and orbital maneuvering. The first part is here.
So, we've achieved a stable orbit around the Earth. We've burnt our propellants and guided our spacecraft to an altitude above the Earth's atmosphere and to a velocity of around 7.5 km/s or more. We're falling, but we're never going to hit the ground. Good news: conceptually, the problem of moving around becomes a lot simpler now that we're outside the soupy atmosphere, and have a fair bit of velocity already under our belts.
Moving oneself on Earth is an extremely specific and exceptional case, in the context of our universe. In 99.99999...9% of the universe, there's no atmosphere around you, nor ground beneath your feet. You don't pick a destination, and move yourself in a straight line to it. Generally speaking, travel through space is about changing the shape of your orbit to take you to different places, or to intercept other objects.
As an example, take the manner in which Apollo spacecraft traveled to the Moon. Launch on the Saturn V rocket placed the craft in a low altitude "parking orbit," a more or less circular orbit about 180 kilometers in altitude. With no further maneuvering, the crew could have happily orbited the Earth, remaining at a 180 kilometer altitude, for a very long time (ignoring the minor detail of available food/water/oxygen). But the Moon is about 384,000 kilometers in altitude, and that's where they wanted to go.
To get there they needed to add energy to their orbit. So at a precisely calculated point in their orbit, they oriented the craft straight ahead -- in the direction of their velocity -- and fired their rocket engine. This accelerated the vehicle, adding velocity, or kinetic energy to the orbit. At the point in their orbit opposite where they were performing the maneuver, it was adding potential energy, i.e. altitude. The rocket engine's acceleration of the vehicle stretched the initially circular orbit into an ellipse, with its apogee, the point of the orbit furthest from Earth, in the Moon's vicinity. This maneuver was called the "trans-lunar insertion" or TLI, because it inserted the craft into a trans-lunar trajectory. It may be easier to visualize with this animation, which I adapted from a NASA document. Note that the burn is taking place on the left side of the Earth in this:
Bob Braeunig, who has compiled a fantastic mathematical analysis of the elements of an Apollo spaceflight, created an excellent to-scale visual of the Apollo spacecraft's orbit after a trans-lunar insertion:
Orbital Maneuvers
A crucial part of maneuvering in space is knowing your orientation; you have to know which way you're pointing. Like most things in space, it's not as intuitive as on Earth. If you're flying a plane in Earth's atmosphere, and you begin to point your aircraft in a different direction, the effect will be obvious. Aerodynamic forces will alter your trajectory, pushing your aircraft in the new direction that you're pointing. The direction of your velocity will generally match where the nose is pointing. Airplanes tend to get into trouble when this isn't true for extended periods of time.
In space, though, you have a tremendous amount of velocity to begin with, and the only force acting on you (when you aren't firing your engine) is gravity. If the engine isn't firing, your orientation has no bearing on your velocity whatsoever. A good visualization of this is the Rendezvous Pitch ("Rbar") Maneuver performed by the Space Shuttle prior to docking with the International Space Station. To enable photography of the thermal protection tiles by ISS crew members, the Space Shuttle would perform a full back-flip prior to docking. As you can see by the movement of the surface and clouds below, the Shuttle's velocity never changes in magnitude or direction; it simply rotates itself in space.
The most important orientations have handy names:
Facing "forward" in the direction of your orbit, as the shuttle is in the laughably not-to-scale illustration above, is a prograde (or posigrade, if you're an old grizzled rocket dude) orientation. If the shuttle flips over and faces opposite the direction of its orbit, it is facing retrograde. Perpendicular to your orbit are the normal and anti-normal vectors.
If you point your rocket prograde and burn your engine, as the Apollo spacecraft did in the TLI example above, you're increasing your velocity, and thus adding energy to your orbit. The effect of this is to raise the opposite point of your orbit higher and higher. If you point retrograde and burn your engine, you're decreasing you velocity, and removing energy from your orbit. This lowers the opposite point of your orbit. When the shuttle wanted to return home, for example, it would point retrograde and fire its OMS (Orbital Maneuvering System) engines, which decelerated the vehicle, lowering the opposite point of its orbit until it dipped into the atmosphere substantially.
If your orbit isn't circular,efficiency of the burn comes into play. Without getting too geometric, let's remember an excerpt from Kepler's laws. In a non-circular orbit your velocity isn't constant; your kinetic energy and potential energy are constantly trading off. At apogee, where your altitude/potential energy are highest, your velocity/kinetic energy is lowest. At perigee, it's the opposite: velocity is highest, altitude lowest. For a visual, check out this animation from Wikimedia Commons:
Would-be space travelers can take advantage of this by changing their orbit with prograde or retrograde burns at apogee or perigee. To raise or lower your apogee, burn prograde or retrograde, respectively, at perigee. To raise or lower your perigee, burn prograde or retrograde, respectively, at apogee. Making these changes in velocity is most efficient when your velocity is at its lowest or highest.
One element of orbits we've ignored up to this point is inclination. This is the "tilt" of your orbit, with respect to the equator of the body you're orbiting. If your orbit matches the equator it's "equatorial," with an inclination of zero degrees. Inclination increases as your orbit is more tilted with respect to the equator. Here's the illustration above but depicting the shuttle on a more inclined orbit, the 52° inclination at which it rendezvoused with the ISS:
As you can see, the orientations have remained fixed to the orbit of the spacecraft. Inclination is critical for many purposes. If you want to rendezvous with something that's orbiting the same body as you are, you must match its inclination. This placed significant constraints on when Apollo missions could launch. To avoid changing inclinations in orbit, which is costly in terms of propellant, they had to launch when Cape Canaveral was directly under the Moon's orbit, and follow a precise azimuth to end up in an orbit that matched the Moon's inclination with respect to the Earth (between 18 and 28 degrees depending on the timing). If you're attempting to photograph or communicate with something on the surface of the body you're orbiting, you must make sure the inclination of your orbit causes you to pass over its latitude.
Your inclined orbit will cross the equator at two points: the ascending node, labeled above, where it will continue "upwards," and the descending node on the opposite side, where it will continue "downwards." The normal and anti-normal orientations are important here. If you point your spacecraft normal at the ascending node, or anti-normal at the descending node, and fire your engine, it will increase your inclination. You're adding velocity in a direction away from the equator. If you orient your spacecraft normal at the descending node, or anti-normal at the ascending node, and fire your engine, it will decrease your inclination, adding velocity towards the equator. Here are examples of two different maneuvers performed at the ascending node:
[Bane voice]: You'll just have to imagine the descending node
Performing a burn in the normal or anti-normal orientation at any point in your orbit will mess with your inclination; it's just most efficient to change your inclination at the nodes. Even so, inclination changes or plane changes take a lot of propellant to execute, and mission planners tend to launch directly into the desired inclination if possible.
Delta-V
As you've probably noticed, velocity is the star of the show here. Launching into orbit required attaining a particular velocity. Changing the orbit's size and shape required engine burns that changed our velocity in various ways. Change in velocity, or delta-v, is the currency of space travel. To get places, you need to execute maneuvers that cost delta-v. Where your spacecraft can go depends on how much delta-v it is capable of. This is a function of three of the spacecraft's attributes: its mass, the amount of propellant it carries, and how efficient its engine(s) is/are. If you want your spacecraft to have more delta-v, add more propellant, make your engine more efficient, or reduce its mass.
To go from their low Earth parking orbit to the orbit that would bring them to the moon, as in the trans-lunar injection animated above, the Apollo astronauts had to burn the engine prograde to increase their velocity from about 7.7 km/s to about 10.9 km/s, a difference of 3.2 km/s. That difference, 3.2 km/s, is the delta-v cost of the maneuver. The spacecraft had to carry enough propellant to achieve that delta-v, plus a lot more -- enough for a retrograde burn to brake into lunar orbit, enough to land on the Moon and take off again, and enough to escape lunar orbit on a trajectory back to Earth. All tolled, the Apollo spacecraft carried enough propellant, according to its flight plan, for about 2 km/s of additional delta-v after being inserted into a translunar trajectory, totaling 5.2 km/s after launch. And that was for a nominal mission; substantial margins were included in case problems were encountered.
The handy thing about delta-v is that the cost to go somewhere is neutral to your spacecraft's attributes and performance. Want to go from low Earth orbit into a trans-Mars trajectory? Whether you're a 2 ton space probe or a 50 ton crewed vehicle, it'll cost you 3.6 km/s of delta-v. The difference is that the 50 ton vehicle will need to carry a lot more propellant to achieve that amount of delta-v. From a mission planning standpoint, this fixed quantitative standard makes it easy to determine what kind of capability your vehicle has to have in order to carry out its goals. Delta-v allows us to consider space travel from a road map perspective, as is very effectively visualized here. A vast array of corridors to every destination in the solar system, each accessed by discrete changes in velocity, are available to any spacecraft capable of paying the delta-v toll.
In the third and final installment, we'll look at interplanetary transfers.
Image Credits: TLI Animation - Azimuth; TLI to Scale - Bob Braeunig; Orientations - Azimuth; Eccentric orbit animation - Wikimedia Commons; Prograde/retrograde burns - Azimuth; Inclination - Azimuth; Inclination change - Azimuth