orbital mechanics pdf
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. When solving these equations it is important to work in radians rather than degrees, where 2 radians equals 360 degrees. 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. Below we describe several types of orbits and the advantages of each: Geosynchronous orbits (GEO) are circular orbits around the Earth having a period of 24 hours. Authors: Richard Fitzpatrick. Earth orbiting satellites typically have very high drag coefficients in the range of about 2 to 4. Click here for example problem #4.21 Knowing the position of one node, the second node is simply This three-burn maneuver may save propellant, but the propellant savings comes at the expense of the total time required to complete the maneuver. Finally, when the satellite reaches perigee of the second transfer orbit, another coplanar maneuver places the satellite into the final orbit. Figure 4.5 shows a particle revolving around C along some arbitrary path. To achieve escape velocity we must give the spacecraft enough kinetic energy to overcome all of the negative gravitational potential energy. Mean anomaly is a function of eccentric anomaly by the formula. The discussion thus far has focused on the elliptical orbit, which will result whenever a spacecraft has insufficient velocity to escape the gravity of its primary. In other words, it has already slowed down to very nearly its hyperbolic excess velocity. We may allow low-altitude orbits to decay and reenter the atmosphere or use a velocity change to speed up the process. Indeed, Newton used Kepler's work as basic information in the formulation of his gravitational theory. 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. Newton's laws of motion describe the relationship between the motion of a particle and the forces acting on it. If we assume that m is negligible compared to M, then R is negligible compared to r. Thus, equation (4.7) then becomes, If we express the angular velocity in terms of the period of revolution, = 2/P, we obtain. That is, m2r must equal M2R. Consequently, in practice, geosynchronous transfer is done with a small plane change at perigee and most of the plane change at apogee. 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. Note that if v∞ = 0 (as it is on a parabolic trajectory), the burnout velocity, vbo, becomes simply the escape velocity. We may also boost satellites at all altitudes into benign orbits to reduce the probability of collision with active payloads, especially at synchronous altitudes. Precise orbit determination requires that the periodic variations be included as well. If the satellite crosses the plane going from south to north, the node is the ascending node; if moving from north to south, it is the descending node. For a spacecraft to achieve Earth orbit, it must be launched to an elevation above the Earth's atmosphere and accelerated to orbital velocity. It may be easier to first calculate e and a, and then calculate true anomaly using equation (4.43), rearranged as follows: Whenever is positive, should be taken as positive; whenever is negative, should be taken as negative. We thus have For Earth, a equals 6,378,137 meters, b equals 6,356,752 meters, and f equals 1/298.257. Thus, a particle undergoing uniform circular motion is under the influence of a force, called centripetal force, whose magnitude is given by. The interval of time in which a spacecraft must be launched in order to complete its mission is called a launch window. The first burn is a coplanar maneuver placing the satellite into a transfer orbit with an apogee much higher than the final orbit. Let's now consider this case. The most energy efficient orbit, that is one that requires the least amount of propellant, is a direct low inclination orbit. Latitude is the angular distance of a point on Earth's surface north or south of Earth's equator, positive north and negative south. An eccentricity of zero indicates a circle. Two particular cases of note are satellites with repeating ground tracks and geostationary satellites. The Hyperbolic Orbit For example, we may specify the size of the transfer orbit, choosing any semi-major axis that is greater than the semi-major axis of the Hohmann transfer ellipse. In a broad sense the V budget represents the cost for each mission orbit scenario.
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