# Keplerian elements from position and velocity relationship

### Basics of Space Flight: Orbital Mechanics

Compute orbital position and velocity at time t given the orbital elements. where is The equation above will automatically yield the correct value of.) Compute. Orbital elements are the parameters required to uniquely identify a specific orbit. In celestial True anomaly (ν, θ, or f) at epoch (M0) defines the position of the orbiting position (x, y, z in a Cartesian coordinate system), plus the velocity in each of .. Rocket equation · Rendezvous · Transposition, docking, and extraction. Conic Sections; Orbital Elements; Types of Orbits; Newton's Laws of Motion and The table below shows the relationships between eccentricity, semi-major .. A space vehicle's orbit may be determined from the position and the velocity of the .

A spacecraft in a geostationary orbit appears to hang motionless above one position on the Earth's equator. For this reason, they are ideal for some types of communication and meteorological satellites.

### Elliptic orbit - Wikipedia

A spacecraft in an inclined geosynchronous orbit will appear to follow a regular figure-8 pattern in the sky once every orbit. To attain geosynchronous orbit, a spacecraft is first launched into an elliptical orbit with an apogee of 35, km 22, miles called a geosynchronous transfer orbit GTO.

The orbit is then circularized by firing the spacecraft's engine at apogee. Polar orbits PO are orbits with an inclination of 90 degrees. An orbiting satellite is subjected to a great many gravitational influences.

First, planets are not perfectly spherical and they have slightly uneven mass distribution. These fluctuations have an effect on a spacecraft's trajectory. Also, the sun, moon, and planets contribute a gravitational influence on an orbiting satellite. With proper planning it is possible to design an orbit which takes advantage of these influences to induce a precession in the satellite's orbital plane.

The resulting orbit is called a walking orbit, or precessing orbit. Sun synchronous orbits SSO are walking orbits whose orbital plane precesses with the same period as the planet's solar orbit period.

In such an orbit, a satellite crosses periapsis at about the same local time every orbit. This is useful if a satellite is carrying instruments which depend on a certain angle of solar illumination on the planet's surface. In order to maintain an exact synchronous timing, it may be necessary to conduct occasional propulsive maneuvers to adjust the orbit. 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 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.

## Elliptic 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. To send a spacecraft to an inner planet, such as Venus, the spacecraft is launched and accelerated in the direction opposite of Earth's revolution around the sun i.

It should be noted that the spacecraft continues to move in the same direction as Earth, only more slowly. To reach a planet requires that the spacecraft be inserted into an interplanetary trajectory at the correct time so that the spacecraft arrives at the planet's orbit when the planet will be at the point where the spacecraft will intercept it.

This task is comparable to a quarterback "leading" his receiver so that the football and receiver arrive at the same point at the same time. The interval of time in which a spacecraft must be launched in order to complete its mission is called a launch window. Newton's Laws of Motion and Universal Gravitation Newton's laws of motion describe the relationship between the motion of a particle and the forces acting on it.

The first law states that if no forces are acting, a body at rest will remain at rest, and a body in motion will remain in motion in a straight line. Thus, if no forces are acting, the velocity both magnitude and direction will remain constant. The second law tells us that if a force is applied there will be a change in velocity, i. This law may be summarized by the equation where F is the force, m is the mass of the particle, and a is the acceleration.

The third law states that if body 1 exerts a force on body 2, then body 2 will exert a force of equal strength, but opposite in direction, on body 1.

This law is commonly stated, "for every action there is an equal and opposite reaction". In his law of universal gravitation, Newton states that two particles having masses m1 and m2 and separated by a distance r are attracted to each other with equal and opposite forces directed along the line joining the particles.

The common magnitude F of the two forces is where G is an universal constant, called the constant of gravitation, and has the value 6. Let's now look at the force that the Earth exerts on an object. If we drop the object, the Earth's gravity will cause it to accelerate toward the center of the Earth. Many of the upcoming computations will be somewhat simplified if we express the product GM as a constant, which for Earth has the value 3.

The product GM is often represented by the Greek letter. For additional useful constants please see the appendix Basic Constants. For a refresher on SI versus U. Uniform Circular Motion In the simple case of free fall, a particle accelerates toward the center of the Earth while moving in a straight line.

The two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with fewer degrees of freedom are the circular and parabolic orbit. Because at least six variables are absolutely required to completely represent an elliptic orbit with this set of parameters, then six variables are required to represent an orbit with any set of parameters.

Another set of six parameters that are commonly used are the orbital elements. Solar System[ edit ] In the Solar Systemplanetsasteroidsmost comets and some pieces of space debris have approximately elliptical orbits around the Sun. Strictly speaking, both bodies revolve around the same focus of the ellipse, the one closer to the more massive body, but when one body is significantly more massive, such as the sun in relation to the earth, the focus may be contained within the larger massing body, and thus the smaller is said to revolve around it.

The following chart of the perihelion and aphelion of the planetsdwarf planets and Halley's Comet demonstrates the variation of the eccentricity of their elliptical orbits. For similar distances from the sun, wider bars denote greater eccentricity. The sixth parameter is still necessary to define the orbit; it is merely numerically set to zero by convention or "moved" into the definition of the epoch with respect to real-world clock time.

Alternative parametrizations[ edit ] Keplerian elements can be obtained from orbital state vectors a three-dimensional vector for the position and another for the velocity by manual transformations or with computer software. When orbiting the Earth, the last two terms are known as the apogee and perigee. It is common to specify the period instead of the semi-major axis in Keplerian element sets, as each can be computed from the other provided the standard gravitational parameterGM, is given for the central body.

Using, for example, the "mean anomaly" instead of "mean anomaly at epoch" means that time t must be specified as a seventh orbital element. Sometimes it is assumed that mean anomaly is zero at the epoch by choosing the appropriate definition of the epochleaving only the five other orbital elements to be specified. Different sets of elements are used for various astronomical bodies. The eccentricity, e, and either the semi-major axis, a, or the distance of periapsis, q, are used to specify the shape and size of an orbit.

## Orbital elements

Either the longitude at epoch, L0, the mean anomaly at epoch, M0, or the time of perihelion passage, T0, are used to specify a known point in the orbit. The choices made depend whether the vernal equinox or the node are used as the primary reference. The semi-major axis is known if the mean motion and the gravitational mass are known. This method of expression will consolidate the mean motion n into the polynomial as one of the coefficients. The appearance will be that L or M are expressed in a more complicated manner, but we will appear to need one fewer orbital element.

Mean motion can also be obscured behind citations of the orbital period P. Sets of orbital elements.