Orbital mechanics, also called flight mechanics, is the study of the motions of artificial satellites and space vehicles moving under the influence of forces such as gravity, atmospheric drag, thrust, etc. Orbital mechanics is a modern offshoot of celestial mechanics which is the study of the motions of natural celestial bodies such as the moon and planets. The root of orbital mechanics can be traced back to the 17th century when mathematician Isaac Newton (1642-1727) put forward his laws of motion and formulated his law of universal gravitation. The engineering applications of orbital mechanics include ascent trajectories, reentry and landing, rendezvous computations, and lunar and interplanetary trajectories.
| Conic Section | Eccentricity, e | Semi-major axis | Energy |
|---|---|---|---|
| Circle | 0 | = radius | < 0 |
| Ellipse | 0 < e < 1 | > 0 | < 0 |
| Parabola | 1 | infinity | 0 |
| Hyperbola | > 1 | < 0 | > 0 |
For additional useful constants please see the appendix Basic Constants.
静止遷移軌道、静止トランスファ軌道（せいしせんいきどう、せいしトランスファきどう、geostationary transfer orbit, GTO）は、人工衛星を静止軌道にのせる前に、一時的に投入される軌道で、よく利用されるのは、遠地点が静止軌道の高度、近地点が低高度の楕円軌道である. So, specifically, the GTO is the blue path from the yellow orbit to the red orbit. The ESA telescope Gaia orbits around an L-point. The point is exactly behind Earth, so at this point Gaia would be in Earth's shadow and unable to receive the sunlight needed to power its solar panels. 1970 pontiac gto judge wt1 - orbit orange original - $98,000 (hamilton) fully restored pristine condition this car is immaculate true original orbit orangematching number car -all numbers match - phs docs includedfactory original:original 400 4bbl - block code ws = matches vin#ram airm20 4 speed factory air. Orbit Orange GTO convertible Pontiac 400 V8 5-speed 12-bolt PS PB A/C Leather /// Like your muscle cars both head-turning AND fun to drive? This picturesque Poncho is just what you've been looking for! Fully sorted and ready to roll, this clean cruiser is the beneficiary of detailed restoration that mixes polished poise with proven athleticism.
For a refresher on SI versus U.S. units see the appendix Weights & Measures.
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Click here for example problem #4.3
| In celestial mechanics where we are dealing with planetary or stellar sized bodies, it is often the case that the mass of the secondary body is significant in relation to the mass of the primary, as with the Moon and Earth. In this case the size of the secondary cannot be ignored. The distance R is no longer negligible compared to r and, therefore, must be carried through the derivation. Equation (4.9) becomes More commonly the equation is written in the equivalent form where a is the semi-major axis. The semi-major axis used in astronomy is always the primary-to-secondary distance, or the geocentric semi-major axis. For example, the Moon's mean geocentric distance from Earth (a) is 384,403 kilometers. On the other hand, the Moon's distance from the barycenter (r) is 379,732 km, with Earth's counter-orbit (R) taking up the difference of 4,671 km. |
Kepler's second law of planetary motion must, of course, hold true for circular orbits. In such orbits both and r are constant so that equal areas are swept out in equal times by the line joining a planet and the sun. For elliptical orbits, however, both and r will vary with time. Let's now consider this case.
Figure 4.5 shows a particle revolving around C along some arbitrary path. The area swept out by the radius vector in a short time interval t is shown shaded. This area, neglecting the small triangular region at the end, is one-half the base times the height or approximately r(rt)/2. This expression becomes more exact as t approaches zero, i.e. the small triangle goes to zero more rapidly than the large one. The rate at which area is being swept out instantaneously is therefore
For any given body moving under the influence of a central force, the value r2 is constant.
Let's now consider two points P1 and P2 in an orbit with radii r1 and r2, and velocities v1 and v2. Since the velocity is always tangent to the path, it can be seen that if is the angle between r and v, then
where vsin is the transverse component of v. Multiplying through by r, we have
or, for two points P1 and P2 on the orbital path
Note that at periapsis and apoapsis, = 90 degrees. Thus, letting P1 and P2 be these two points we get
Let's now look at the energy of the above particle at points P1 and P2. Conservation of energy states that the sum of the kinetic energy and the potential energy of a particle remains constant. The kinetic energy T of a particle is given by mv2/2 while the potential energy of gravity V is calculated by the equation -GMm/r. Applying conservation of energy we have
From equations (4.14) and (4.15) we obtain
Rearranging terms we get
Gto Orbit Parameters
Click here for example problem #4.5
The eccentricity e of an orbit is given by
If the semi-major axis a and the eccentricity e of an orbit are known, then the periapsis and apoapsis distances can be calculated by
Equation (4.26) gives the values of Rp and Ra from which the eccentricity of the orbit can be calculated, however, it may be simpler to calculate the eccentricity e directly from the equation
To pin down a satellite's orbit in space, we need to know the angle , the true anomaly, from the periapsis point to the launch point. This angle is given by
In most calculations, the complement of the zenith angle is used, denoted by . This angle is called the flight-path angle, and is positive when the velocity vector is directed away from the primary as shown in Figure 4.8. When flight-path angle is used, equations (4.26) through (4.28) are rewritten as follows:
The semi-major axis is, of course, equal to (Rp+Ra)/2, though it may be easier to calculate it directly as follows:
If e is solved for directly using equation (4.27) or (4.30), and a is solved for using equation (4.32), Rp and Ra can be solved for simply using equations (4.21) and (4.22).
Orbit Tilt, Rotation and Orientation
Above we determined the size and shape of the orbit, but to determine the orientation of the orbit in space, we must know the latitude and longitude and the heading of the space vehicle at burnout.
Figure 4.9 above illustrates the location of a space vehicle at engine burnout, or orbit insertion. is the azimuth heading measured in degrees clockwise from north, is the geocentric latitude (or declination) of the burnout point, is the angular distance between the ascending node and the burnout point measured in the equatorial plane, and is the angular distance between the ascending node and the burnout point measured in the orbital plane. 1 and 2 are the geographical longitudes of the ascending node and the burnout point at the instant of engine burnout. Figure 4.10 pictures the orbital elements, where i is the inclination, is the longitude at the ascending node, is the argument of periapsis, and is the true anomaly.
If , , and 2 are given, the other values can be calculated from the following relationships:
In equation (4.36), the value of is found using equation (4.28) or (4.31). If is positive, periapsis is west of the burnout point (as shown in Figure 4.10); if is negative, periapsis is east of the burnout point.
The longitude of the ascending node, , is measured in celestial longitude, while 1 is geographical longitude. The celestial longitude of the ascending node is equal to the local apparent sidereal time, in degrees, at longitude 1 at the time of engine burnout. Sidereal time is defined as the hour angle of the vernal equinox at a specific locality and time; it has the same value as the right ascension of any celestial body that is crossing the local meridian at that same instant. At the moment when the vernal equinox crosses the local meridian, the local apparent sidereal time is 00:00. See this sidereal time calculator.
| Geodetic Latitude, Geocentric Latitude, and Declination Latitude is the angular distance of a point on Earth's surface north or south of Earth's equator, positive north and negative south. The geodetic latitude (or geographical latitude), , is the angle defined by the intersection of the reference ellipsoid normal through the point of interest and the true equatorial plane. The geocentric latitude, ', is the angle between the true equatorial plane and the radius vector to the point of intersection of the reference ellipsoid and the reference ellipsoid normal passing through the point of interest. Declination, , is the angular distance of a celestial object north or south of Earth's equator. It is the angle between the geocentric radius vector to the object of interest and the true equatorial plane. R is the magnitude of the reference ellipsoid's geocentric radius vector to the point of interest on its surface, r is the magnitude of the geocentric radius vector to the celestial object of interest, and the altitude h is the perpendicular distance from the reference ellipsoid to the celestial object of interest. The value of R at the equator is a, and the value of R at the poles is b. The ellipsoid's flattening, f, is the ratio of the equatorial-polar length difference to the equatorial length. For Earth, a equals 6,378,137 meters, b equals 6,356,752 meters, and f equals 1/298.257. When solving problems in orbital mechanics, the measurements of greatest usefulness are the magnitude of the radius vector, r, and declination, , of the object of interest. However, we are often given, or required to report, data in other forms. For instance, at the time of some specific event, such as 'orbit insertion', we may be given the spacecraft's altitude along with the geodetic latitude and longitude of the sub-vehicle point. In such cases, it may be necessary to convert the given data to a form more suitable for our calculations. The relationship between geodetic and geocentric latitude is, The radius of the reference ellipsoid is given by, The length r can be solved from h, or h from r, using one of the following, And declination is calculated using, For spacecraft in low earth orbit, the difference between and ' is very small, typically not more than about 0.00001 degree. Even at the distance of the Moon, the difference is not more than about 0.01 degree. Unless very high accuracy is needed, for operations near Earth we can assume ≈ ' and r ≈ R + h. It is important to note that the value of h is not always measured as described and illustrated above. In some applications it is customary to express h as the perpendicular distance from a reference sphere, rather than the reference ellipsoid. In this case, R is considered constant and is often assigned the value of Earth's equatorial radius, hence h = r – a. This is the method typically used when a spacecraft's orbit is expressed in a form such as '180 km × 220 km'. The example problems presented in this web site also assume this method of measurement. |
where Mo is the mean anomaly at time to and n is the mean motion, or the average angular velocity, determined from the semi-major axis of the orbit as follows:
This solution will give the average position and velocity, but satellite orbits are elliptical with a radius constantly varying in orbit. Because the satellite's velocity depends on this varying radius, it changes as well. To resolve this problem we can define an intermediate variable E, called the eccentric anomaly, for elliptical orbits, which is given by
where is the true anomaly. Mean anomaly is a function of eccentric anomaly by the formula
For small eccentricities a good approximation of true anomaly can be obtained by the following formula (the error is of the order e3):
The preceding five equations can be used to (1) find the time it takes to go from one position in an orbit to another, or (2) find the position in an orbit after a specific period of time. When solving these equations it is important to work in radians rather than degrees, where 2 radians equals 360 degrees.
Click here for example problem #4.14
At any time in its orbit, the magnitude of a spacecraft's position vector, i.e. its distance from the primary body, and its flight-path angle can be calculated from the following equations:
And the spacecraft's velocity is given by,
Perturbations due to Non-spherical Earth
When developing the two-body equations of motion, we assumed the Earth was a spherically symmetrical, homogeneous mass. In fact, the Earth is neither homogeneous nor spherical. The most dominant features are a bulge at the equator, a slight pear shape, and flattening at the poles. For a potential function of the Earth, we can find a satellite's acceleration by taking the gradient of the potential function. The most widely used form of the geopotential function depends on latitude and geopotential coefficients, Jn, called the zonal coefficients.
The potential generated by the non-spherical Earth causes periodic variations in all the orbital elements. The dominant effects, however, are secular variations in longitude of the ascending node and argument of perigee because of the Earth's oblateness, represented by the J2 term in the geopotential expansion. The rates of change of and due to J2 are
where n is the mean motion in degrees/day, J2 has the value 0.00108263, RE is the Earth's equatorial radius, a is the semi-major axis in kilometers, i is the inclination, e is the eccentricity, and and are in degrees/day. For satellites in GEO and below, the J2 perturbations dominate; for satellites above GEO the Sun and Moon perturbations dominate.
Molniya orbits are designed so that the perturbations in argument of perigee are zero. This conditions occurs when the term 4-5sin2i is equal to zero or, that is, when the inclination is either 63.4 or 116.6 degrees.
where CD is the drag coefficient, is the air density, v is the body's velocity, and A is the area of the body normal to the flow. The drag coefficient is dependent on the geometric form of the body and is generally determined by experiment. Earth orbiting satellites typically have very high drag coefficients in the range of about 2 to 4. Air density is given by the appendix Atmosphere Properties.
The region above 90 km is the Earth's thermosphere where the absorption of extreme ultraviolet radiation from the Sun results in a very rapid increase in temperature with altitude. At approximately 200-250 km this temperature approaches a limiting value, the average value of which ranges between about 700 and 1,400 K over a typical solar cycle. Solar activity also has a significant affect on atmospheric density, with high solar activity resulting in high density. Below about 150 km the density is not strongly affected by solar activity; however, at satellite altitudes in the range of 500 to 800 km, the density variations between solar maximum and solar minimum are approximately two orders of magnitude. The large variations imply that satellites will decay more rapidly during periods of solar maxima and much more slowly during solar minima.
For circular orbits we can approximate the changes in semi-major axis, period, and velocity per revolution using the following equations:
where a is the semi-major axis, P is the orbit period, and V, A and m are the satellite's velocity, area, and mass respectively. The term m/(CDA), called the ballistic coefficient, is given as a constant for most satellites. Drag effects are strongest for satellites with low ballistic coefficients, this is, light vehicles with large frontal areas.
A rough estimate of a satellite's lifetime, L, due to drag can be computed from
where H is the atmospheric density scale height. A substantially more accurate estimate (although still very approximate) can be obtained by integrating equation (4.53), taking into account the changes in atmospheric density with both altitude and solar activity.
Perturbations from Solar Radiation
Solar radiation pressure causes periodic variations in all of the orbital elements. The magnitude of the acceleration in m/s2 arising from solar radiation pressure is
where A is the cross-sectional area of the satellite exposed to the Sun and m is the mass of the satellite in kilograms. For satellites below 800 km altitude, acceleration from atmospheric drag is greater than that from solar radiation pressure; above 800 km, acceleration from solar radiation pressure is greater.
Another option for changing the size of an orbit is to use electric propulsion to produce a constant low-thrust burn, which results in a spiral transfer. We can approximate the velocity change for this type of orbit transfer by
where the velocities are the circular velocities of the two orbits.
From equation (4.73) we see that if the angular change is equal to 60 degrees, the required change in velocity is equal to the current velocity. Plane changes are very expensive in terms of the required change in velocity and resulting propellant consumption. To minimize this, we should change the plane at a point where the velocity of the satellite is a minimum: at apogee for an elliptical orbit. In some cases, it may even be cheaper to boost the satellite into a higher orbit, change the orbit plane at apogee, and return the satellite to its original orbit.
Typically, orbital transfers require changes in both the size and the plane of the orbit, such as transferring from an inclined parking orbit at low altitude to a zero-inclination orbit at geosynchronous altitude. We can do this transfer in two steps: a Hohmann transfer to change the size of the orbit and a simple plane change to make the orbit equatorial. A more efficient method (less total change in velocity) would be to combine the plane change with the tangential burn at apogee of the transfer orbit. As we must change both the magnitude and direction of the velocity vector, we can find the required change in velocity using the law of cosines,
where Vi is the initial velocity, Vf is the final velocity, and is the angle change required. Note that equation (4.74) is in the same form as equation (4.69).
As can be seen from equation (4.74), a small plane change can be combined with an altitude change for almost no cost in V or propellant. Consequently, in practice, geosynchronous transfer is done with a small plane change at perigee and most of the plane change at apogee.
Another option is to complete the maneuver using three burns. The first burn is a coplanar maneuver placing the satellite into a transfer orbit with an apogee much higher than the final orbit. When the satellite reaches apogee of the transfer orbit, a combined plane change maneuver is done. This places the satellite in a second transfer orbit that is coplanar with the final orbit and has a perigee altitude equal to the altitude of the final orbit. Finally, when the satellite reaches perigee of the second transfer orbit, another coplanar maneuver places the satellite into the final orbit. This three-burn maneuver may save propellant, but the propellant savings comes at the expense of the total time required to complete the maneuver.
When a plane change is used to modify inclination only, the magnitude of the angle change is simply the difference between the initial and final inclinations. In this case, the initial and final orbits share the same ascending and descending nodes. The plane change maneuver takes places when the space vehicle passes through one of these two nodes.
In some instances, however, a plane change is used to alter an orbit's longitude of ascending node in addition to the inclination. An example might be a maneuver to correct out-of-plane errors to make the orbits of two space vehicles coplanar in preparation for a rendezvous. If the orbital elements of the initial and final orbits are known, the plane change angle is determined by the vector dot product. If ii and i are the inclination and longitude of ascending node of the initial orbit, and if and f are the inclination and longitude of ascending node of the final orbit, then the angle between the orbital planes, , is given by
The plane change maneuver takes place at one of two nodes where the initial and final orbits intersect. The latitude and longitude of these nodes are determined by the vector cross product. The position of one of the two nodes is given by
Knowing the position of one node, the second node is simply
Orbit Rendezvous
Orbital transfer becomes more complicated when the object is to rendezvous with or intercept another object in space: both the interceptor and the target must arrive at the rendezvous point at the same time. This precision demands a phasing orbit to accomplish the maneuver. A phasing orbit is any orbit that results in the interceptor achieving the desired geometry relative to the target to initiate a Hohmann transfer. If the initial and final orbits are circular, coplanar, and of different sizes, then the phasing orbit is simply the initial interceptor orbit. The interceptor remains in the initial orbit until the relative motion between the interceptor and target results in the desired geometry. At that point, we would inject the interceptor into a Hohmann transfer orbit.
Launch Windows
Similar to the rendezvous problem is the launch-window problem, or determining the appropriate time to launch from the surface of the Earth into the desired orbital plane. Because the orbital plane is fixed in inertial space, the launch window is the time when the launch site on the surface of the Earth rotates through the orbital plane. The time of the launch depends on the launch site's latitude and longitude and the satellite orbit's inclination and longitude of ascending node.
Orbit Maintenance
Once in their mission orbits, many satellites need no additional orbit adjustment. On the other hand, mission requirements may demand that we maneuver the satellite to correct the orbital elements when perturbing forces have changed them. Two particular cases of note are satellites with repeating ground tracks and geostationary satellites.
After the mission of a satellite is complete, several options exist, depending on the orbit. We may allow low-altitude orbits to decay and reenter the atmosphere or use a velocity change to speed up the process. We may also boost satellites at all altitudes into benign orbits to reduce the probability of collision with active payloads, especially at synchronous altitudes.
V Budget
To an orbit designer, a space mission is a series of different orbits. For example, a satellite might be released in a low-Earth parking orbit, transferred to some mission orbit, go through a series of resphasings or alternate mission orbits, and then move to some final orbit at the end of its useful life. Each of these orbit changes requires energy. The V budget is traditionally used to account for this energy. It sums all the velocity changes required throughout the space mission life. In a broad sense the V budget represents the cost for each mission orbit scenario.
A space vehicle that has exceeded the escape velocity of a planet will travel a hyperbolic path relative to the planet. The hyperbola is an unusual and interesting conic section because it has two branches. The arms of a hyperbola are asymptotic to two intersecting straight line (the asymptotes). If we consider the left-hand focus, f, as the prime focus (where the center of our gravitating body is located), then only the left branch of the hyperbola represents the possible orbit. If, instead, we assume a force of repulsion between our satellite and the body located at f (such as the force between two like-charged electric particles), then the right-hand branch represents the orbit. The parameters a, b and c are labeled in Figure 4.14. We can see that c2 = a2+ b2 for the hyperbola. The eccentricity is,
Super Gto Orbit
At any known true anomaly, the magnitude of a spacecraft's radius vector, its flight-path angle, and its velocity can be calculated using equations (4.43), (4.44) and (4.45).
Early we introduced the variable eccentric anomaly and its use in deriving the time of flight in an elliptical orbit. In a similar manner, the analytical derivation of the hyperbolic time of flight, using the hyperbolic eccentric anomaly, F, can be derived as follows:
where,
Whenever is positive, F should be taken as positive; whenever is negative, F should be taken as negative.
It is, of course, absurd to talk about a space vehicle 'reaching infinity' and in this sense it is meaningless to talk about escaping a gravitational field completely. It is a fact, however, that once a space vehicle is a great distance from Earth, for all practical purposes it has escaped. In other words, it has already slowed down to very nearly its hyperbolic excess velocity. It is convenient to define a sphere around every gravitational body and say that when a probe crosses the edge of this sphere of influence it has escaped. Although it is difficult to get agreement on exactly where the sphere of influence should be drawn, the concept is convenient and is widely used, especially in lunar and interplanetary trajectories. For most purposes, the radius of the sphere of influence for a planet can be calculated as follows:
where Dsp is the distance between the Sun and the planet, Mp is the mass of the planet, and Ms is the mass of the Sun. Equation (4.89) is also valid for calculating a moon's sphere of influence, where the moon is substituted for the planet and the planet for the Sun.
Compiled, edited and written in part by Robert A. Braeunig, 1997, 2005, 2007, 2008, 2011, 2012, 2013.
Bibliography
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