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Orbital Mechanics and Interplanetary Trajectories

written by: •edited by: RC Davison•updated: 3/19/2012

Navigating in space is not the same as moving about on Earth. You are dealing with the interaction of two bodies--the Earth and the spacecraft. That brings into play laws developed by Kepler and Newton that dictate how a spacecraft can move in orbit.

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    Vagaries of Space Flight

    When we fly in space, the rules of the road here on Earth don’t apply. We can’t catch up with another spacecraft by speeding up. We actually have to slow down. That’s because increasing speed at a certain point moves the craft into a higher orbit. In a higher orbit, the speed of the craft decreases. This is because the higher the orbit of a satellite, the slower it needs to move to balance the pull of gravity. The other craft will move further away.

    To catch another craft, the chasing spaceship must drop down to a lower orbit—where it would speed up and catch up with the target. Then it can move into the target’s orbit and rendezvous with it.

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    Fundamentals of Orbital Mechanics

    Such are the vagaries of orbital mechanics. It’s all because orbits are conic sections—ellipses and circles. Parabolas and hyperbolas are also conic sections, but they are open to infinity and so would take a spacecraft away from the planet. Johannes Kepler defined theConic sections  laws of bodies orbiting other bodies in the 17th century, and Sir Isaac Newton refined them later. Basically, Newton explained Kepler in this way:

    If two bodies interact gravitationally, they will orbit each other about the common mass of the two in an ellipse.

    His other two laws describe the speed at which the smaller body will travel around the larger body. We will come back to these laws later.

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    Falling Through Space

    Now, let’s talk about how spacecraft get into orbit and what happens then. Let’s do a mental experiment to see what is actually happening. Imagine a cannonball being fired from a very high mountain (we assume no atmospheric drag in this experiment). If it is a very strong cannon and you can put enough gunpowder in it, the cannonball will go fast enough that even though it begins to fall back to earth, as it falls it begins to speed up under the pull of gravity at 32 feet per second/per second. Half way around the Earth it reaches its lowest point, but it is now going as fast as it was when it left the cannon. It doesn’t hit the ground because it’s high enough to continue its flight. Now it begins to gain altitude again. As it does it slows, but still has sufficient speed to come back to the point from which it was fired, and continue its flight.

    In other words, it was ‘falling’ around the planet.

    That is what a spacecraft is effectively doing. A rocket boosts it to a speed and altitude at which it experiences little or no air resistance, and it is going fast enough to fall around the Earth without falling back to Earth. That’s why weightlessness is sometimes called ‘freefall.’ The speed required to achieve a low earth orbit (LEO) of about 186 miles is about 18,000 mph or 25,000 fps.

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    Maneuvering in Space

    The point at which the spacecraft goes into orbit is generally the high point of its orbit, called the apogee. It then ‘falls’ around the earth to a low point, called the perigee. Here is where Kepler’s laws come into play again. At apogee, the spaceship’s speed is at its slowest. At perigee it is at its fastest. Very often, it is necessary to raise the perigee, and sometimes to change the elliptical orbit into a circular orbit.Perigee and apogee 

    How is this done?

    The speed at apogee must be increased. Firing a rocket motor there will raise the perigee without affecting the apogee. By the same Orbit change token, firing a motor at perigee will raise the apogee without affecting the perigee. This is how a LEO orbit can be changed into a higher orbit. Usually it is desired to raise both the apogee and perigee, and circularize the orbit. This requires two burns—one at each of the critical points in the orbit.Perigee burn 

    These burns are called delta v, or changing velocity. In those we’ve discussed, the burns are done in the direction of travel and are Circularizing called posigrade burns. It is also possible to move a spacecraft from a high orbit to a lower orbit with retrograde burns. In this case the burns are done opposite to the direction of travel. This of course is how the shuttle ends its flights and reenters the atmosphere.

    It is not just altitude and shape that can be changed by engine burns. A spacecraft enters orbit at a particular orbital inclination—that is, an angle to the equator. Inclination can also be changed with delta v, incline but it must be applied perpendicular to the direction of travel and in the direction of the desired inclination change. In low orbits, only very small inclination changes can be made. At very high orbits, inclinations can be changed as much as 30 degrees. This is possible because the orbital velocity is low, as low as around 6000 mph in some, and the pull of gravity is much less than in LEOs. Therefore, less energy is required to change the inclination.

    Changing orbital inclination 
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    This second page of Orbital Mechanics and Maneuvering in Space discusses the different types of orbits that are used when putting spacecraft and satellites in orbit by everyone from NASA to ESA to the Russian space agency, Roscosmos. Detailing the geosynchronous and geostationary orbits, Molniya and polar orbits, also discussing the Holman transfer orbits.
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    Types of Orbits

    Spacecraft are inserted into several types of orbits:

    • equatorial
    • polar
    • geosynchronous or geostationary
    • precessional
    • sun synchronous
    • Molniya
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    Equatorial and Polar Orbits

    Equatorial orbits are, obviously, orbits that track directly over the equator. Some of the EU space ventures are launched from near the equator and are inserted into such orbits. This also gives them the extra 1000 mph of the Earth’s rotation on their way to space. U.S. rockets launched from Cape Canaveral get an extra 914 mph from the rotation.

    Polar orbiting satellites are craft that have an inclination to the equator of 90 degrees and fly over the poles. These are desirable for mapping and spy-in-the-sky satellites because as the Earth rotates under them, they cover every part of the globe in 24 hours. U.S. polar orbiting craft are launched from Vandenberg Air Force Base in California as it provides a flight path for polar launch, plus the security of a military installation. Polar orbits require more power than equatorial launches because there is no help from Mother Earth.

    Vandenberg Space Launch Complex-6 
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    Geosynchronous, Precessional, and Sun Synchronous Orbits

    Geosynchronous orbits are orbits in which the satellite has the same orbital period as the Earth’s rotation—one orbit every 24 hours. If it is circular and its inclination is in the plane of the equator it remains in one spot over the planet at all times--it is stationary and is a geostationary orbit.This type of orbit is used for communications and some GPS satellites. To be in a geosynchronous orbit a satellite must be about 22,000 miles up. At that altitude its' velocity is around 6800 mph and it will make one orbit every 24 hours. But it obviously takes a great deal of energy to get to that altitude. We will discuss how we get to that rarefied altitude below.

    Precessional orbits and sun synchronous orbits: these are orbits that take advantage of the fact that the Earth is not a perfect sphere. It is bloated, and so there are minute differences in the gravitational influence on a satellite as it orbits the planet. These minute differences are sufficient to cause the orbit to precess, or gradually change its orbital inclination. This phenomenon can be used in sun synchronous orbits to cause a satellite to reach perigee at the same local time each orbit. This is useful for mapping and spy satellites that require a specific sun angle.

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    Molniya Orbits

    Molniya orbits are used to keep a satellite over a specific section of the planet for nearly 12 hours. These are highly eccentric orbits with a period of about 12 hours, or two revolutions per day. The inclination is chosen so perigee precession is zero, to keep the satellite over a given part of the Earth. For U.S. satellites, the orbital inclinations chosen are 63.4 and 116.6 degrees. This keeps the satellite over the northern hemisphere for about 11 hours at apogee.

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    Hohmann Transfer Orbits

    The Hohmann orbit is not one a spacecraft is put into from launch. It is actually a maneuver used to move from one orbit to another, or even into interplanetary space (which we will discuss in the second installment). It was promulgated by the German scientist Walter Hohmann in 1925 as the lowest energy method of interplanetary travel. Hohmann transfer orbits require two rocket burns—one at the perigee of the original orbit, and one at the apogee of the new orbit—called an apogee kick by NASA—to circularize the new orbit. Hohmann transfers are used to put geosynchronous satellites into their final orbits.Hohmann transfer maneuver They are also used for rendezvous. Remember we said you can’t just speed up to catch another spacecraft. The Shuttle uses a Hohmann transfer to rendezvous with the International Space Station (ISS), but uses just one burn—to change from its LEO to the ISS’s higher orbit. It catches up to the ISS and passes it. But as it moves into the ISS orbit it slows down, and the ISS catches up to it. Once the two get within sight of each other, the vagaries of orbital mechanics no longer apply and shuttle pilots can fly their craft directly to the docking port just as if they were refueling in the atmosphere.Hohmann rendezvous The Hohmann transfer is a time consuming maneuver. It is used because it uses less fuel than a fast transfer maneuver. A fast transfer would require a posigrade burn and a perpendicular burn, requiring a great deal more propellant.Fast transfer maneuver This is particularly important for lunar and interplanetary flights, as we will see in part 2 of this series.

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    Conic sections: Rocket and Space Technology

    All Orbital illustrations: University of Washington

    Vandenberg SLC-6: USAF

Orbital Mechanics and Interplanetary Trajectories

Flying is space requires a completely different way of thinking. You don't simply move in a straight line from one point to another. You are always moving in an ellipse as a result of gravitational forces from the planet or the sun in the case of interplanetary flight. That is different.
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