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You’re in orbit. What now? Part 3 of a series on spaceflight.

Today, we will dip our toes into the somewhat mind-bending world of orbital mechanics.  In today’s post, we will learn why the shuttle’s launch windows are so short, why rendezvous and docking can be so tricky, and why the shuttle that visited the Hubble needed its own standby rescue mission.

You can do this in any orbit

So, we’ve already discussed just how difficult it is just to get here, and for many purposes, that in itself is enough.  There’s plenty of stuff you can do in orbit — any orbit, and many experiments that we’ve done in space are not dependent on where we are in space.

But what if we want to visit something that’s already there?  Since the Columbia disaster, every manned mission we’ve done has been to visit something already in space.  Be it a satellite, another space craft, a space station, the moon, or even another planet, if we cannot control our orbit precisely enough to meet up with another object, then our options in space are extremely limited.

All kinds of stuff in earth orbit -- many, many different earth orbits.

So how do we do this?  How do we create our orbit such that we can rendezvous with another spacecraft? Space is big.  Colossally big.  Even the space in low-earth orbit is huge relative to the things we want to visit, and there are infinitely many possible orbits in which we can be.  So, to start with, we need to be able to figure out just where we are, and where we want to go.  Let’s start then with how we can describe where we are.

To pinpoint a location on the surface of the earth, you only need two pieces of information: latitude and longitude.  Because you’re restricted to the earth’s surface, and because you can stay in one point on the earth’s surface, this is the only information you need.  To pinpoint a location in space, we need a third location, say, the altitude.  The problem, of course, is that we aren’t looking for a location in space, we are looking for an object, and as we saw in the first part of this series, any object at a particular location in space will not stay there for long.  So we need even more information — we need to know how fast we’re going, and in which direction.  That’s three more bits of information we need to consider.  Suddenly, determining where we are in space is no longer a three-dimensional problem, it’s a six-dimensional problem.

There’s good news, however.  We have freedom to choose what these six dimensions are.  Based on my description of the problem, it would seem that the most obvious way to choose these dimensions would be exactly what I said — position and velocity.  Together, these bits of information can be represented as two three-dimensional vectors, the orbital position vector [ \; x \quad y \quad z\;] and the orbital velocity vector [ \; v_x \quad v_y \quad v_z \; ].  Together, these are called the orbital state vectors, and they are enough to completely describe an orbit.

Kepler discovered the elliptical shape of planetary orbits.

For practical purposes, however, this is not a very useful way of describing an orbit.  Yes, knowing where we are and how we are moving gives us enough information to completely describe what we are doing, but it would be far more useful to find a set of parameters that more directly describes the shape of the orbit.  The traditional set is known as the Keplerian orbital elements.

Named for Johannes Kepler, who discovered that all of the planets move in elliptical orbits, these orbital elements describe the size, shape, and orientation of the orbit.  For a basic two-body problem, such as a spacecraft orbiting a spherical planet, neglecting things like the gravitational influence of other bodies, the orbit will always be a conic section — a circle, an ellipse, a parabola, or a hyperbola, with the body to be orbited at one of the foci of the section.  For our purposes, we will not consider parabolas and hyperbolas, which are for escape trajectories.  Instead, we will focus only on circles and ellipses.  For simplicity’s sake, let’s say that the planet is located at the point F_2 on the diagram below.

Let’s go through the six parameters now:

Properties of an ellipse

The first parameter we will look at is eccentricity.  Eccentricity is a single number that describes the shape of our ellipse.  On the diagram at right, the eccentricity is defined by the ratio of the distance between the foci (F_1 and F_2) and the major axis (the widest distance across the ellipse — from a to -a).  An eccentricity of zero makes a perfect circle.  As the eccentricity increases towards 1, the ellipse becomes flatter and flatter.  Once the eccentricity reaches or surpasses 1, it’s no longer an ellipse — rather, a parabola or a hyperbola, i.e., an escape trajectory.

The next parameter we’ll explore is the semimajor axis.  This parameter controls the size of the orbit.  This is basically the distance between the point a and the point C on the diagram.  it’s also the average between the apoapsis (farthest distance from the planet center) and the periapsis (closest distance to the planet center), or a representation of the average height of the orbit.  Semimajor axis and eccentricity together completely describe the shape and size of the orbit.  All we need now is a way to describe the orbit’s orientation in space.

Orbital plane orientation parameters

The next three parameters give us just that.  We have three angles: the inclination, the longitude of the ascending node, and the argument of periapsis.  Now, these have scary sounding names, but it’s really quite simple.

We begin by picking a frame of reference — any arbitrary orientation will do.  We just need a plane and a direction.  For simplicity’s sake, let’s just pick the plane that runs through the earth’s equator, and it doesn’t really matter what direction along that plane we choose for our reference direction — we just need to pick one.

Once we’ve done that, we look at our parameters. Inclination is the angle between the plane of the orbit and the reference plane.  An inclination of zero degrees is one that stays right above the equator, moving eastward.  An inclination of 90 degrees goes over the north and south poles.  An inclination of 180 degrees hugs the equator, but in a reverse motion, moving to the west.  The space shuttle can launch to inclinations between 28 and 58 degrees.

For any nonzero inclination, there are two points where the orbit intersects the equatorial plane.  We call these the ascending and descending nodes.  The ascending node is when the orbit crosses the equator moving to the north, and the descending node crosses moving to the south.  We complete our orientation of the orbital plane by assigning the longitude of the ascending node, which is just the angle between our reference direction and the ascending node.

We’ve now completely defined our orbital plane, but we have not yet determine how our orbit fits in it.  This is where the argument of periapsis comes in.  This tells us how our orbit is oriented within the plan we defined with the last two parameters.  It’s defined as the angle formed between the periapsis — the closest approach to earth, or whichever object we’re orbiting — and the ascending node.

These five parameters tell us everything about the shape and orientation of an orbit, but there’s one item missing.  We need to know where we are in this orbit.  We use the term anomaly to define where our spacecraft is along our orbit at a particular point in time.

Once we know these six parameters, we have completely defined our orbit around a body like Earth.  In the next entry, I will discuss how you go about changing these parameters — or, how you get from one orbit to another.

What does it take? Part II.

In my last entry, I talked about the difference between simply getting into space and getting there for any length of time, i.e., reaching orbit.  Today, we’re going to visit why getting into orbit is such a big stinking deal.

This is what it took to get three guys to the moon.

Recall from last time, we discovered just to get into space and stay there, you’ve got to reach a speed of roughly 17,500 miles per hour.  That’s a huge amount of energy to spend to get there, even for just a tiny little bit of mass.  Take a look at this picture to the right.  Do you see those little black specks just to the left of the left set of treads on the transporter?  Those are people, and just to the left of them is a car.  Now take a look at the very top of the rocket.  The part of the rocket that actually carried people to the moon and brought them back wasn’t a whole lot bigger than that car.  The rest of that thing?  That’s what it took to heave that up there.  It took a skyscraper-sized rocket to get a car-sized capsule to the moon and back.

Now why is that thing so big?  In a word, fuel.  Size-wise, not very much of that rocket is responsible for actually doing the work to get them to the moon.  The bulk of that vehicle is basically a set of giant fuel tanks.  The problem is, of course, that for every bit of fuel you intend to burn at altitude, you need to burn more fuel to lift that fuel up.  If you do the calculus, you’ll arrive at one of rocket science’s holiest of holy equations, the Tsiolkovsky Rocket Equation.

\displaystyle{\Delta v = v_e \ln \frac{m_0}{m_1}}

This equation applies for any rocket maneuver.  Getting into orbit, getting out of orbit, going from low-Earth orbit to the moon — doesn’t matter. So what does this equation mean?  Well, let’s start from the left and work our way right.

\Delta v is the change in velocity.  Any rocket maneuver is going to be based on such a change.  Getting into orbit is a 17,000 mph change, for example.  Getting from low-Earth orbit to escape velocity (if you wanted to, say, visit Jupiter or Mars) is going to be another 8,000 mph.  These aren’t small changes we’re talking about.

v_e is the effective exhaust velocity.  It’s related — though not exactly equal to —  the speed at which the rocket engines can expel the combustion products.  In a way, it’s a measure of the fuel-efficiency of a rocket engine.  For the space shuttle, this value varies (sticking with miles-per hour) from just under 8,000 mph at sea level to just over 9,900 mph.

Finally, m_0 and m_1 are the initial and final masses of the vehicle.  The difference between the two is the propellant (i.e., fuel and oxidizer) that will get expelled at high speed from the engines.  The logarithm there means that as required velocity change increases, the fuel required increases exponentially.

So, as an example, suppose it takes 20,000 lb of fuel to deliver 5,000 mph of \Delta v to a 10,000 lb rocket.  Suppose that we want to double the \Delta v we provide.  By this equation, if we use the same engine, we would have to square the mass-ratio from the original requirement.  So, if 5,000 mph requires a 3:1 mass ratio, 10,000 mph is going to require a 9:1 mass ratio.  This means that to double our velocity change, we need to go from 20,000 lb of fuel to 80,000 lb.  Getting to 15,000 mph would require a 27:1 mass ratio — 260,000 lb of fuel.  Adding \Delta v capability quickly becomes … *ahem* … astronomically expensive.

Now…. the worst part.  That’s not even the whole story.  This doesn’t say ANYTHING about thrust, i.e., how much force you exert.  If you are going to lift a heavy rocket off the ground, you need a lot of force.  Unfortunately, engines that have a high thrust tend to have low v_e — the shuttle’s solid rocket boosters provide 83% of the liftoff thrust, but they have v_e of just slightly higher than half that of the main engines.  This enormous tradeoff between thrust and efficiency makes the first stage of rocket flight incredibly expensive — the shuttle burns through half its fuel in just the first two minutes of its eight-minute ascent.

With all this expense, it’s no wonder that we are only just now seeing private ventures build spacecraft that can reach orbit.  We haven’t even scratched the surface of being able to do something useful once we’re up there.  Next time we’ll talk a bit about orbital mechanics and how we control exactly where in space we go.

So what does it take to get into space anyway?

So you want to get into space? In some ways, that’s actually the easiest part of space travel. Just go up far enough, and voila, you’re in space!

Just how far?  Well, that’s a little fuzzy.

This is not space. Not yet.

This picture isn’t space just yet. This is only 100,000 feet, about 18 miles up. This picture was in fact taken from a balloon. Some fighter jets can reach this altitude. Meteors burn up before they descend to this altitude — a spacecraft here wouldn’t survive long.

The commonly recognized boundary of space is called the Kármán line, more than three times this altitude, at 100km (62 miles) above sea level. While it is difficult to reach this altitude, this barrier was broken by a rocket built by a group of hobbyists in 2004.

So, you don’t need to be a big government agency or defense contractor to reach space. What exactly, then, is so darn hard about space travel?

Short answer: everything else.

Ok, you want the long answer? Let’s start with what you do once you’re up there. The hobbyist rocket reached an altitude of 72 miles, but it reached it in the same manner that a ball thrown straight up might reach 50 feet. It hits that height, and then it has no momentum left. Then it starts to fall.

In a manner of speaking, you could say that it starts to “fall” the minute its engine cuts out. When the air is so thin that you can effectively neglect its resistance, there’s really nothing special physically about the moment you switch from moving up to moving down. In this kind of world, you’re in one of two states: either there’s a force acting on you, or there isn’t.

If you’re standing on the ground, there’s a force acting on you — the ground pushing up on your feet. It’s something so common that we actually consider this the null case. In reality, the only time there is no force acting on you is when you are in free fall. Only then do you get to see Newton’s physics’ true behavior, unadulterated by the constant acceleration of gravity.

“But wait,” you say. “If you’re in free fall, you are under the constant acceleration of gravity.”

Einstein disagrees with you.

Not so fast there.  Einstein showed that gravity is not a force per se, but rather a curvature in space itself.  Without getting into the mind-bending physics of relativity, we can use this way of thinking:

Right now, you’re probably sitting at a desk, on a couch, at a table, on the floor.  Maybe you’re standing up, maybe you’re laying down.  What forces do you feel acting on your body?  The force of gravity pushing you down?  Really?  Are you sure it isn’t the chair, or the floor, or the couch pushing you up?  Close your eyes for a moment.  Imagine that there is no gravity acting on you.  What would make you feel the force you are feeling?

Well, if your chair was accelerating upwards at a rate of exactly 1 g, would that be any different?  To look at this in a slightly more silly way, imagine you’re in a sealed room with no windows.  You have no idea where you are — you might be in deep space, you might be in Spokane, Washington.  All you know is that everything is pulled down at 1 g.  You hold a tennis ball in front of your face and let it go.  It hits the ground in a little over half a second.  You do the same with a hammer.  It falls at exactly the same speed.  So does everything else you drop.  Without harboring any preconceived notions of gravity (hard to do when you’ve grown up with it), which do you think would be more likely, that everything had some attraction to the floor with a force perfectly tuned so that everything falls at exactly the same speed?  Or would it be more likely that everything you let go of is actually at rest, and you are accelerating upward at 1 g?  Does the tennis ball feel any force between the time you let go of it and the time it hits the ground?

This long digression is really meant to underscore a specific concept — the minute a force is removed from an object, it enters a state of free fall.  If you fall off a roof, you enter free fall because the ground stopped pushing you up.  Likewise, when a rocket’s engine shuts down, the rocket enters free fall, even if it is still moving up.

So, back to the whole getting in space thing.  One of the first things you want to do once you get into space is to figure out a way to stay there.  One way to do this is to have enough of a force maintained on you to keep you at that altitude.  You can climb Mount Everest, and when you reach the top, you stay at 29,000 feet because Mount Everest keeps pushing you up.  The problem is, exerting a force with a rocket requires fuel, and lots of it.  You can’t keep it up for long.  There’s got to be a better way.

Fortunately, the earth isn’t flat, and “down” isn’t absolute.  Rather, the earth is nearly spherical, and “down” just points toward its center.  That means “down” where you are is not the same direction as “down” where I am.  It’s not even the same direction as “down” at your next door neighbor’s house.  To illustrate, if you were to hang a plumb line from the top of each of the towers of the Golden Gate bridge so that the bottom just touched the water, the tops would be approximately two inches farther apart than the bottoms would be.

When down changes, things can fall in a curve

We can take advantage of this fluidity of “down” to keep our spacecraft up in space.  Take a ball and throw it as hard as you can, level with the ground.  It curves downward, right?  But remember that “down” isn’t the same everywhere.  Likewise, “level with the ground” isn’t the same direction everywhere either.  As the ball moves away from you, the Earth curves away from its path, even as it falls toward the ground.  At the speed you can throw a ball, this curvature is imperceptible, but if you can go faster, the ball will travel farther, and the earth will curve away even more.  We can take this idea to an extreme, then.  There should be some speed at which you can throw a ball so that the Earth curves away from it just as fast as it is being pulled toward the center.  You can think of this as the ball continuously falling toward — but always just missing — the ground.  Once you can do this in a stable manner, you’re in orbit around the Earth.  You’re not staying up, rather, you’re falling, but the earth is falling away from you fast enough that you never hit it.

So…. the only thing you need to stay up is enough sideways velocity to create an orbit.  Just how much is this velocity?  Well, it depends on your height.  I’ll save the math for a later post, but long story short, the higher you are, the less velocity you need to stay in orbit.  To enter a stable low Earth orbit, like where Space Shuttle Atlantis will be going in a couple weeks, you need to reach a velocity of approximately 17,500 miles per hour.  That’s a mind-bogglingly fast speed — about five miles every second.  To put that in perspective, at that speed, you can travel from New York City to Los Angeles in just slightly more than eight minutes.  In fact, a low Earth orbit has an orbital period — the time it takes to completely circumnavigate the earth — of only an hour and a half.

So, to recap, to get into space and stay there, you have to 1) get at least 60 miles up (you probably want higher so the thin, wispy remnants of the atmosphere don’t drag you down sooner than you want), and 2) you have to get up to 17,500 miles per hour once you’re there.  More on just how you do that tomorrow.