Orbital Mechanics 101


Johannes Kepler (1571-1630)

"My aim is to show that the heavenly machine is not a kind of divine, live being, but a kind of clockwork, insofar as nearly all the manifold motions are caused by a most simple, magnetic, and material force, just as all motions of the clock are caused by a simple weight. And I also show how these physical causes are to be given numerical and geometrical expression."

-Johannes Kepler


The Laws of Motion

Humans have ventured into Earth orbit for over 40 years, but mathematicians have been studying orbital mechanics for over 400 years.  In 1609, the astronomer Johannes Kepler published his three laws governing orbital motion. These were based on the work of Tycho Brahe who studied the motion of Mars' orbit around the Sun. 

Kepler's laws provide the basis for modern orbital mechanics, but it was Isaac Newton (left) who, in 1666, proved them by using calculus.  Kepler believed that his laws only governed planets that orbited the Sun. However, Newton proved that Kepler's laws applied to any satellite in orbit around any celestial body.  Interestingly, Newton had lost the papers with the mathematical proof for nearly 5 years, but he finally located them in order to win a bet with Edmond Halley (the discoverer of Halley's comet).

Kepler's laws specifically state that all satellites move in elliptical orbits (with a circle being a special case of an elliptical orbit). Ellipses look like ovals, or squashed circles, and have two imaginary points called foci (plural for focus).  Each foci sits along the major axis which is the line dividing the ellipse lengthwise, and lie an equal distance from the center point. 




The minor axis is the line dividing the ellipse across the shorter path. The distance from any point on the ellipse to both foci always equals the length of the major axis. 

Click here for a simulation of Kepler's First Law: Each planet revolves around the Sun in an elliptical path, with the Sun occupying one of the foci of the ellipse.


An elliptical orbit

If a satellite orbits the Earth, one of the foci is always the center of the Earth and the other is empty.  Engineers measure the amount of squash in an ellipse in terms of its eccentricity.  The act of bringing the foci together so that both lie at the center of the major axis brings the eccentricity to 0 and creates a perfect circle.  The act of spreading the foci apart increases the amount of squash up to a maximum of 1.0.  An ellipse with an eccentricity close to 0 looks like a circle and one that has an eccentricity close to 1 looks like a very long oval. One way to calculate the eccentricity of an ellipse is to divide the distance between the foci by the length of the major axis.

The satellite's distance from the Earth varies along its orbital path, with the closest point to the Earth being the perigee and the farthest point being the apogee. These two points lie on the major axis.

The distance between the apogee (or perigee) and the center of the major axis is called the semi-major axis.  If the eccentricity is high enough, a satellite's orbital path can cross the surface of the Earth. The satellite would then crash into the surface of the Earth. 


A satellite spends the first half of its orbit moving from the perigee to the apogee and the other half of its orbit moving from the apogee to the perigee. To keep track of the satellite or spacecraft, engineers use 360 degrees to measure the orbit.  The first half of the orbit lies from between 0 and 180, and the second half of the orbit lies from between 180 ad 360. This is called the true anomaly.  So, if a satellite has a true anomaly value of between 0 and 180, it is somewhere between the perigee and apogee; and if it is between 180 and 360, it is somewhere between apogee and perigee.

Kepler's Second Law

Kepler's Second Law explains the velocity of the satellite in relation to its point on the orbit.  If you imagine a line from the center of the Earth to the satellite sweeping out an area as it travels, at perigee the satellite must travel farther to cover the same area as it does at apogee.  In other words, the satellite travels faster at the perigee and at the apogee. You can compare this to what happens when you toss a ball into the air.  The ball slows down as it reaches the top of the climb and speeds up as it comes back to you.

The straight line joining the Sun and a planet sweeps out equal areas in equal intervals of time.

For another simulation of Kepler's Second Law, click here.

Kepler's Third Law has to do with the amount of time it takes a satellite to make one orbit around the Earth.  This is called the orbital period.  The orbital period always relates to the altitude of the satellite.  Higher orbits take a longer time to complete than lower orbits. The reason for this is that, in a higher orbit, the satellite must travel a longer distance to complete one revolution, or orbital period around the Earth.

For example, the space shuttle orbits at 300 kilometers (181 miles) above the Earth and takes 90 minutes to complete 1 orbital period.  But communication satellites in orbit are located at 35,786 kilometers (22,236 miles) above the Earth and take 24 hours to complete 1 orbital period. This matches the rotational period of the Earth, keeping these communications satellites over the same spot at all times, which is called a geosynchronous orbit. This is a good orbit to be in if you are a communications satellite!

Click here for a very cool calculator that allows you to change the shuttle's altitude to determine the velocity and orbital period!

In mathematical terms, Kepler's law states that the square of the orbital period is directly proportional to the cube of half the major axis length.  It also states that the time to complete one orbit depends on the size of the orbit, not on its shape.  Therefore, any two orbits of different eccentricity can take the same amount of time to complete if their major axes are the same length.

Click here for a simulation of Kepler's Third Law: The squares of the planets' orbital periods are proportional to the cubes of the semi-major axes of their orbits.

Click here for a tutorial that will allow you to change the radius (R), velocity (V), and inclination (I) of an orbit to visualize it.

Tracking

To determine exactly where a satellite is located, engineers must think in three dimensions. One important reference involves the specific plane of the satellite's orbit. 

Imagine a piece of paper slicing through the center of the Earth in an infinite number of possible ways. An XYZ coordinate system, called J2000, is used by engineers to determine this plane. It is a set of data telling the engineers exactly where the shuttle is in its orbit in space. Various elements of position, velocity, and time produce a state vector that predicts when the satellite will pass over any ground site at any time. 

From a state vector, ground controllers can tell how fast the shuttle is moving and what country it is over. The Z axis points through the North Pole from the center of the Earth, and the X and Y axes point out from the center of the Earth through the equator. The X axis pointing towards a certain star in the constellation Ares on the first day of the year 2000; hence the name J2000. The Z axis points through the North Pole from the center of the Earth. The Y axis is perpendicular to the plane created by the other two axes.  The X,Y, and Z coordinates always remain the same in relation to space while the Earth rotates within them.  The orbital tilt angle describes the inclination of the orbit relative to the Earth's equator. An orbit that lies in the XY axis has a tilt of 0 and is called an equatorial orbit. 

Tilting the orbit towards the North or South Poles inclines the orbit towards a tilt value of 90. 
A satellite in this orbit will fly over the poles. This is called a polar orbit.

Satellites can orbit in either direction through the Earth's orbit. Posigrade satellites move from west to east and retrograde satellites move from east to west (clockwise or counterclockwise as seen from above the North Pole). To describe a satellite's tilt and direction engineers use the term inclination or the symbol i.


Hubble Space Telescope

An inclination value of between 0 and 90 indicate a posigrade orbit, while a value between 90 and 180 indicates a retrograde orbit. To describe these different inclinations, you can use the chart below. 

Inclination

Direction and Location

0 Posigrade (west to east ) equatorial orbit
0-45 Posigrade (west to east) orbit tilted close to the equator
45-90 Posigrade (west to east) orbit tilted close to the poles
90 Polar orbit, northerly or southerly
90-135 Retrograde (east to west) orbit tilted close to the poles
135-180 Retrograde (east to west) orbit tilted close to the equator
180 Retrograde (east to west) equatorial orbit

Ground Tracks

A ground track map provides engineers with information about what points on the Earth a satellite is flying over.  This was especially important in the early days of spaceflight, when only ground stations were used to communicate with spacecraft and satellites. 


Ground track map in the Mission Control Center

We now use three TDRSS (tracking data and relay satellite system) satellites located in space so that there are no periods when the Mission Control Center is out of contact with a spacecraft. A ground track map uses the common longitude and latitude system to determine the coordinates on Earth that the satellite is flying over.

Latitude measures how far north or south of the equator an object is, and longitude measures how far east or west of the Prime Meridian (a line intersecting Greenwich, England) it is. Latitudinal values range from 0 to 90 north or south, and longitudal values range from 0 to 180 west of the Prime Meridian and 0 to 180 east of the Prime Meridian. (Some people use 360 to measure longitude with the numbers from 180 to 360 describe locations in the Western Hemisphere.)

Most ground tack maps appear on Mercator projections of the Earth. These are flat representations of the globe. When drawn on a globe, an orbit looks like a closed loop around the Earth. 

When drawn on a Mercator map, it looks like an S-shaped curve with half of the S being above and half of the S being below the equator. Consider the equator itself, which on a globe is a circular loop around the center of the Earth but which, on a Mercator map, is a straight line. The Shuttle's launch site is Cape Canaveral, Florida, which is at a latitude of 28.5 deg. north. The initial launch point is a point on the ground track.



As you see, the ground track of a spacecraft in low-Earth orbit resembles a sine wave varying between the launch latitude and the negative of the launch latitude. A vehicle launched in an orbit inclined 28.5 to the equator can never fly over latitudes above 28.5 N or below 28.5 S.

To see where the shuttle and a variety of other spacecraft in orbit around the Earth are right now, click here.

To see the orbit of the space station (or shuttle if it is flight) or any of over 500 satellites in 3-D click here!  (When the image loads use your mouse to click and drag the image.)

 

Orbital Maneuvers

When we talk about moving a spacecraft in orbit we use the following terms:

  • Posigrade horizontal - pointing forward along the orbital path

  • Retrograde horizontal - pointing backwards along the orbital path

  • Outward radial - pointing directly away from the center of the Earth

  • Inward radial - pointing directly towards the center of the Earth

At perigee and apogee, spacecraft only move in the horizontal direction since the radial component doesn't exist at these two points.  We use the term 'V (delta-V) to describe a change (delta = Greek for change) in velocity of a spacecraft. 

An orbital thrust maneuver or burn is used to change a spacecraft's 'V.  To change your orbit, you must burn your rocket engine a certain amount of time to produce the 'V necessary (change in velocity). 

On-Orbit Operations

A horizontal burn will increase or decrease your forward velocity. Your engine should fire in the opposite direction of your desired velocity change (or direction of motion).

When you fire your thrusters, fire in a reverse direction. You will increase your forward velocity. When you fire your engines in the forward direction, you will slow down. This is what the shuttle does when it makes its deorbit burn to come home. On a posigrade orbit, these maneuvers are called posigrade and retrograde burns.

A posigrade horizontal burn on a circular orbit will increase the eccentricity of the orbit.  The new orbit will intersect the old orbit at the burn point, which is now the perigee of the ellipse. The orbit also becomes larger.  The change of shape of the orbit will depend on where the burn occurs.  A posigrade horizontal burn, done at the apogee of the new orbit will cause it to become less eccentric and more circular in shape as noted in the A, B, and C orbits in the diagram. 

This maneuver is called a Hohmann transfer (named after the German engineer Dr. Walter Hohmann). It allows a spacecraft to attain a new, higher orbit using two consecutive burns at each intersecting point.

A posigrade burn done on an elliptical orbit works the same way. It increases the eccentricity of the orbit and increases the size of the orbit. A stronger burn will produce more 'V than a weaker burn, but it also requires more fuel.

 

Orbital Change Maneuvers

A retrograde horizontal burn will lower the altitude of your orbit on every point except for the burn point.  Therefore, the orbit becomes smaller.  If a retrograde burn is done at the apogee or an ellipse or on a circular orbit, the orbit becomes more eccentric.  If a retrograde burn is done at the perigee of an ellipse, the burn will become more circularized.  This would be used to change your orbit from C to B to A in the diagram above, or to lower your orbit. The new orbit that results from an orbital change maneuver must intersect or touch the old orbit at the location in which the maneuver occurred.

Posigrade Burns


Retrograde burns


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