# Artificial Satellites

Steven Dutch, Professor Emeritus, Natural and Applied Sciences, University of Wisconsin - Green Bay

## How Rockets Work

Newton's Laws of Motion are:

• An object at rest tends to remain at rest and an object in motion tends to remain in motion.
• Force equals mass times acceleration
• For every action there is an equal and opposite reaction; force equals mass times acceleration

Newton's Laws are all contained in a more general principle called conservation of momentum. Momentum is mass times velocity, and in a system that is not disturbed from outside, the total momentum stays constant. Thus:

• If velocity is zero, momentum is zero (Newton's First Law)
• If velocity is not zero, momentum has some value, but if the mass doesn't change, and momentum is conserved, then velocity doesn't change (Newton's Second Law)
• If mass changes somehow, then so does velocity. If a moving object splits in two, each part has its own velocity and the total momentum of the two pieces equals the momentum of the original piece.
• If an object is stationary, and flings off mass at high speed, the rest of the mass moves in the opposite direction at some speed. The original momentum is zero. If velocity in one direction is considered positive, velocity in the other direction is considered negative. The flung off mass has positive momentum, the rest has negative momentum, and the total momentum remains zero. Force equals mass times acceleration (Newton's Third Law).

Suppose you are standing on very slick ice. You weigh 50 kg. You fire a 10 gram (.01 kg) bullet at 500 m/sec. Its momentum is .01 x 500 = 5 (the units are kg-m/sec, if you're curious). To keep the total momentum of the original system zero, you have to acquire -5 momentum. Since you weigh 50 kg, your velocity will be -5/50 or -0.1 m/sec. You will start sliding backward on the ice at 10 centimeters per second. This is why a rifle has a kick.

As an aside, what matters are instantaneous changes. Once the bullet leaves the gun, it's no longer part of your system, and what happens to it doesn't affect you. You don't feel a momentum change when the bullet strikes its target. Likewise, when friction eventually slows your slide on the ice, that doesn't affect the bullet. Okay, go back to what you were doing.

Rockets and jets work according to Newton's Third Law. They fire mass out at high speed and acquire velocity in the opposite direction. Thus, we can dispel one common myth about rockets and jets: they do not need something to push against. A rocket does not take off because it is pushing against the ground, nor does a jet fly because it is pushing against the air. They move because they are expelling exhaust gases at high speeds. If you like, the rocket or jet is pushing mass away, and the mass is pushing back (equal and opposite reaction.)

Rockets and jets expel mass by burning fuel. A rocket differs from a jet in that a jet gets the oxygen for combustion from the atmosphere, and a rocket carries oxygen in some form with it. Thus rockets can function outside the Earth's atmosphere; jets can't.

When a rocket or jet takes off, it has to carry all its remaining fuel with it. Most of the mass of the Space Shuttle is fuel, and most of that is used to get the remaining fuel off the ground. The miles-per-gallon fuel efficiency of the Space Shuttle in its first foot off the ground is pretty terrible!

Satellites travel elliptical paths with the center of the Earth at one focus (A below - Kepler's First Law, again). Anything shot from the surface of the Earth, a baseball, say, or a cannonball, travels an elliptical path, but the ellipse soon intersects the surface of the Earth again (we often say it's a parabola, and it is to very high precision, but technically it's the outer end of a very long ellipse.) Ballistic missiles do the same thing except their ellipses intersect the surface of the Earth thousands of kilometers away. Nothing shot directly from the surface of the Earth can go into orbit; it will either fall back to Earth again or, if it's moving fast enough, escape completely.

Incidentally, if we could somehow magically let the object pass through the earth's interior, it would not travel in an ellipse. One of the cool things about gravity is that, for a spherical object, the gravity is the same as if all the mass were at a single point in the center. That's if you're outside the planet. If you're inside the planet, the mass above you has no gravitational effect. Only the mass between you and the center counts. If the earth were perfectly uniform, gravity would decrease linearly toward the center and would be zero at the center - all the mass of the earth around you would be pulling in all directions equally. In the real earth, because mass is concentrated in the core, gravity actually increases with depth and is a few per cent higher at the core boundary than on the surface.

However, on the real earth, if we throw something up, it follows an elliptical path until it intersects the surface again.

Objects stay in orbit because of a balance between inertia, that would cause them to keep moving in a straight line, and gravity, that would pull them down. Isaac Newton conceived of artificial satellites (B below). He pointed out that a cannon on a high enough mountain and firing ever faster cannonballs could fire them to greater and greater distances. If fired with a great enough velocity, the curvature of the cannonball's path would be equal to that of the Earth and the cannonball would circle the Earth.

To get into orbit, you have to climb Newton's mountain first (C). Rockets are launched into orbit by launching them vertically to get them above the atmosphere, then accelerating them horizontally to reach orbital velocity. It takes 29,000 km/hour to do this in low Earth orbit. You get 1670 km/hour of this for free thanks to the Earth's rotation. That's why most satellites are launched eastward.

Assuming you're far enough out of the earth's atmosphere, you do not have to use fuel to stay in orbit.

Satellites follow Kepler's laws and have elliptical orbits with the center of the earth at the focus. It is impossible to have a satellite orbit over only part of the earth, or to remain fixed above one spot, unless it's on the equator. Even then, the satellite doesn't stand still, it revolves around the earth as fast as the earth rotates.

## Some Special Orbits

Generally we don't have any particular reason to launch a satellite opposite the earth's rotation, so we take advantage of the earth's rotation to save energy. Orbits in the same direction as the earth's rotation are called prograde and those opposing it are retrograde. Orbits over the poles are sometimes slightly retrograde to allow the satellite to track across the earth in certain ways. Very retrograde orbits are really uncommon.

### Orbital Inclination

The angle the plane of the satellite's orbit makes with the earth's equator is called its inclination. Satellites with zero inclination orbit directly along the equator. Satellites with other inclinations can travel as far north and south of the equator as their inclination. A satellite with an inclination of 40 degrees can reach as far as 40 degrees north or south of the equator. A satellite with an orbit of 90 degrees can travel over the poles and is said to be in a polar orbit. Satellites in polar orbits can view the entire earth. An inclination greater than 90 degrees means the orbit is retrograde.

If you launch from a location not on the equator, obviously your satellite will reach that latitude, so the orbital inclination must be at least as great as your latitude. If you want to put a satellite into equatorial orbit, you can launch it and then use fuel to change the orbit once in space, or you can use the fuel on earth and go to the equator. That way you can launch less fuel and more satellite. French Guyana and Kenya are both launch sites for equatorial satellites.

Since what goes up sometimes comes down in the wrong places, the U.S. launches its satellites from the coasts, where accidents won't drop debris onto populated areas. High inclination satellites are usually launched south from Vandenburg Air Force Base in California, where there is clear ocean all the way to Antarctica. Russia launches eastward over sparsely populated Siberia. China has no choice but to launch over populated areas.

### Sun-Synchronous Orbits

All the things that cause planetary orbits to change over time act on satellites, except much faster. In particular the plane of the orbit precesses rapidly because of the gravity of the Sun and Moon. We can use precession to our advantage. One way is to match the precession rate to the earth's motion around the Sun, so that on every pass, the earth is illuminated the same. This is a sun-synchronous orbit and is commonly used in earth observation satellites. Fortunately, sun-synchronous orbits are nearly polar, so the satellite can observe almost the entire earth. We can also design the orbit so that passes repeat precisely over the earth at regular intervals. A nice side effect is that sun-synchronous orbits are nearly polar, making them idea for mapping and environmental monitoring.

### Geosynchronous and Geostationary Orbits

A satellite just above the atmosphere takes about 90 minutes to circle the earth. The Moon takes a month. Somewhere in between, there must be an altitude where satellites take exactly 24 hours to circle the earth. That happens at an altitude of 22,000 miles. Such an orbit is called geosynchronous. A satellite with an inclination would appear to drift north and south over the course of a day, but a satellite with zero inclination would appear to remain stationary in the sky. Such an orbit is called geostationary. In reality, the satellite is moving, but the earth is rotating at the same rate. Your satellite dish is pointing at a satellite 22,000 miles (36,000 km) in space. One downside of a geosynchronous orbit is that, at the equinoxes, the Sun is on the celestial equator, so there is a short interval where geosynchronous satellites pass in front of the Sun. During those windows, radio emissions from the Sun interfere with reception of signals from the satellite.

Old geostationary satellites have to go somewhere to die, lest they clutter the narrow band along the equator where geostationary satellites can orbit. Rather than drop them back to earth, they are nudged into a slightly higher orbit called a graveyard orbit

### Molniya Orbits

Another downside of geosynchronous satellites is that they are below the horizon beyond about 60 degrees latitude. Since much of Russia is at high latitudes, they cannot use standard geosynchronous orbits. A satellite series called Molniya (lightning) employed a very elliptical 12-hour orbit, going out to about 25,000 miles (40,000 km). Thanks to Kepler's Second Law, the satellite appears nearly stationary in the sky for a long time. Other communications satellites and some spy satellites use similar orbits, which are now called Molniya orbits. During one 12-hour orbit the satellite hangs for a long time over Russia, but during the next 12-hour orbit it hangs over North America. This has obvious advantages for both U.S. and Russian spy satellites. The earth's equatorial bulge would cause the near and far points of the orbit to move over time, but for an orbital inclination of 63.4 degrees the drift rate is zero, so that inclination is used.

### GPS Satellites

GPS (Geopositioning system) satellites are placed in orbits with two special characteristics. First, they are circular 12-hour orbits, meaning the satellites orbit at an altitude of 20,000 km. Second, the satellite follows the same track over the earth's surface on every orbit. To do that, the plane of the orbit has to remain constant in orientation relative to the stars. This happens if the orbit has an inclination of 55 degrees. There are six sets of GPS satellites, orbiting 60 degrees apart

Created 20 May 1997, Last Update 20 January 2020