Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton’s laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

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### Kepler’s laws of planetary motion:

**Orbits are elliptical, with the heavier body at one focus of the ellipse.**A special case of this is a circular orbit (a circle is a special case of ellipse) with the planet at the center.

An ellipse is defined to be a curve with the following property: for each point on an ellipse, the sum of its distances from two fixed points, called foci, is constant. The longest and shortest lines that can be drawn through the center of an ellipse are called the major axis and minor axis, respectively. The *semi-major axis* is one-half of the major axis and represents a satellite’s mean distance from its primary. *Eccentricity* is the distance between the foci divided by the length of the major axis and is a number between zero and one. An eccentricity of zero indicates a circle.

**A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.****The square of a satellite’s orbital period is proportional to the cube of its average distance from the planet.**Kepler’s third law of orbital motion gives us a precise relationship between the speed of the satellite and its distance from the earth. An Earth-orbiting satellite’s motion is mostly controlled by Earth’s gravity. As satellites get closer to Earth, the pull of gravity gets stronger, and the satellite moves more quickly.

a is the semi-major axis, which describes the size of the orbit and is called the gravitational constant G is the universal gravitational constant and M is the mass of the central body. This unsurprisingly means that the larger the orbit, the longer it takes for the object to traverse it For Earth, this gravitational constant is approximately 398,600.5 km^{3}/s^{2}.

Without applying force (such as firing a rocket engine), the period and shape of the satellite’s orbit won’t change. If thrust is applied at only one point in the satellite’s orbit, it will return to that same point on each subsequent orbit, though the rest of its path will change. Thus one cannot move from one circular orbit to another with only one brief application of thrust.

From a circular orbit, thrust applied in a direction opposite to the satellite’s motion changes orbit to elliptical; the satellite will descend and reach the lowest orbital point (the periapse) at 180 degrees away from the firing point; then it will ascend back. Thrust applied in the direction of the satellite’s motion creates an elliptical orbit with its highest point (apoapse) 180 degrees away from the firing point.

### Orbital Elements

To mathematically describe an orbit one must define six quantities, called *orbital elements*. They are

**Epoch:**A set of orbital elements is a snapshot, at a particular time, of the orbit of a satellite. Epoch is simply a number that specifies the time at which the snapshot was taken.The orbit ellipse lies in a plane known as the orbital plane. The orbital plane always goes through the center of the earth, but may be tilted any angle relative to the equator. Inclination is the angle between the orbital plane and the equatorial plane. By convention, inclination is a number between 0 and 180 degrees.Orbits with inclination near 0 degrees are called equatorial orbits (because the satellite stays nearly over the equator). Orbits with inclination near 90 degrees are called polar (because the satellite crosses over the north and south poles). The intersection of the equatorial plane and the orbital plane is a line which is called the line of nodes.**Orbital Inclination:****Right Ascension of Ascending Node (R.A.A.N.):**Two numbers orient the orbital plane in space. The first number was Inclination. “right ascension of ascending node” is an angle, measured at the center of the earth, from the vernal equinox to the ascending node. The intersection of the equatorial plane and the orbital plane is a line which is called the line of nodes. One of two nodes is called the ascending node (where the satellite crosses the equator going from south to north). The other is called the descending node (where the satellite crosses the equator going from north to south). By convention, we specify the location of the ascending node. By convention, RAAN is a number in the range 0 to 360 degrees. Vernal equinox is nothing more than the ascending node of the Sun’s orbit. The Sun’s orbit has RAAN = 0 simply because we’ve defined the Sun’s ascending node as the place from which all ascending nodes are measured. The RAAN of your satellite’s orbit is just the angle (measured at the center of the earth) between the place the Sun’s orbit pops up past the equator, and the place your satellite’s orbit pops up past the equator.**Argument of Perigee:**Now that we’ve oriented the orbital plane in space, we need to orient the orbit ellipse in the orbital plane. We do this by specifying a single angle known as argument of perigee. The line-of-apsides passes through the center of the earth. We’ve already identified another line passing through the center of the earth: the line of nodes. The angle between these two lines is called the argument of perigee. Where any two lines intersect, they form two supplementary angles, so to be specific, we say that argument of perigee is the angle (measured at the center of the earth) from the ascending node to perigee.**Eccentricity:**In the Keplerian orbit model, the satellite orbit is an ellipse. Eccentricity tells us the “shape” of the ellipse. When e=0, the ellipse is a circle. When e is very near 1, the ellipse is very long and skinny.**Mean Motion:**Kepler’s third law of orbital motion gives us a precise relationship between the speed of the satellite and its distance from the earth. Satellites that are close to the earth orbit very quickly. Satellites far away orbit slowly. This means that we could accomplish the same thing by specifying either the speed at which the satellite is moving or its distance from the earth. Satellites in circular orbits travel at a constant speed. We just specify that speed, and we’re done. Satellites in non-circular (i.e., eccentricity > 0) orbits move faster when they are closer to the earth, and slower when they are farther away. The common practice is to average the speed. You could call this number “average speed”, but astronomers call it the “Mean Motion”. Mean Motion is usually given in units of revolutions per day. Sometimes “orbit period” is specified as an orbital element instead of Mean Motion. The period is simply the reciprocal of Mean Motion. A satellite with a Mean Motion of 2 revs per day, for example, has a period of 12 hours.**Mean Anomaly:**Now that we have the size, shape, and orientation of the orbit firmly established, the only thing left to do is specify where exactly the satellite is on this orbit ellipse at some particular time. Our very first orbital element (Epoch) specified a particular time, so all we need to do now is specify where, on the ellipse, our satellite was exactly at the Epoch time. Anomaly is yet another astronomer word for angle. Mean anomaly is simply an angle that marches uniformly in time from 0 to 360 degrees during one revolution. It is defined to be 0 degrees at perigee, and therefore is 180 degrees at apogee.Drag caused by the earth’s atmosphere causes satellites to spiral downward. As they spiral downward, they speed up. The Drag orbital element simply tells us the rate at which Mean Motion is changing due to drag or other related effects. Precisely, Drag is one half the first time derivative of Mean Motion. Its units are revolutions per day per day. It is typically a very small number. Common values for low-earth-orbiting satellites are on the order of 10^-4. Common values for high-orbiting satellites are on the order of 10^-7 or smaller.**Drag (optional):**

The real world is slightly more complex than the Keplerian model, and tracking programs compensate for this by introducing minor corrections to the Keplerian model. These corrections are known as perturbations.

### Types of orbits

There are essentially three types of Earth orbits: high Earth orbit, medium Earth orbit, and low Earth orbit. Many weather and some communications satellites tend to have a high Earth orbit, farthest away from the surface. Satellites that orbit in a medium (mid) Earth orbit include navigation and specialty satellites, designed to monitor a particular region. Most scientific satellites, including NASA’s Earth Observing System fleet, have a low Earth orbit.

#### Geosynchronous Orbit (GSO) & Geostationary Orbit (GEO)

Objects in GSO have an orbital speed that matches the Earth’s rotation, yielding a consistent position over a single longitude. GEO is a kind of GSO. It matches the planet’s rotation, but GEO objects only orbit Earth’s equator, and from the ground perspective, they appear in a fixed position in the sky. GSO and GEO are used for telecommunications and Earth observation. They cannot cover the high latitudes.

#### Polar Orbit

Within 30 degrees of the Earth’s poles, the polar orbit is used for satellites providing reconnaissance, weather tracking, measuring atmospheric conditions, and long-term Earth observation. They fly several hundred km over the earth’s surface with a rotation period of about 110 minutes. They can cover most of the earth’s surface except for regions immediately adjacent to the poles. They have an inclination that measures the angle at which they cross the equator and which determines how close they get to passing directly over the poles.

#### Sun-Synchronous Orbit (SSO)

A type of polar orbit, SSO objects are synchronous with the sun, such that they pass over an Earth region at the same local time every day

#### Highly Elliptical Orbit (HEO)

An HEO is oblong, with one end nearer the Earth and other more distant. Satellites in HEO are suited for communications, satellite radio, remote sensing and other applications.

* Molniya orbits* are highly eccentric Earth orbits with periods of approximately 12 hours (2 revolutions per day). The orbital inclination is chosen so the rate of change of perigee is zero, thus both apogee and perigee can be maintained over fixed latitudes. This condition occurs at inclinations of 63.4 degrees and 116.6 degrees. For these orbits the argument of perigee is typically placed in the southern hemisphere, so the satellite remains above the northern hemisphere near apogee for approximately 11 hours per orbit. This orientation can provide good ground coverage at high northern latitudes.

* Hohmann transfer orbits* are interplanetary trajectories whose advantage is that they consume the least possible amount of propellant. A Hohmann transfer orbit to an outer planet, such as Mars, is achieved by launching a spacecraft and accelerating it in the direction of Earth’s revolution around the sun until it breaks free of the Earth’s gravity and reaches a velocity which places it in a sun orbit with an aphelion equal to the orbit of the outer planet. Upon reaching its destination, the spacecraft must decelerate so that the planet’s gravity can capture it into a planetary orbit.

### References and resources also include:

https://www.amsat.org/keplerian-elements-tutorial/