The kinetic energy of the spacecraft, when it is launched, is mv2/2. It sums all the velocity changes required throughout the space mission life. Launch Windows
The table below shows the relationships between eccentricity, semi-major axis, and energy and the type of conic section. In order to maintain an exact synchronous timing, it may be necessary to conduct occasional propulsive maneuvers to adjust the orbit. 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. The direction of F at any instant must be in the direction of a at the same instant, that is radially inward.
In a similar manner, the analytical derivation of the hyperbolic time of flight, using the hyperbolic eccentric anomaly, F, can be derived as follows:
Thus, if m is the mass of the spacecraft, M is the mass of the planet, and r is the radial distance between the spacecraft and planet, the potential energy is -GmM /r. - Spacecraft Systems
These laws can be deduced from Newton's laws of motion and law of universal gravitation. If we let point P2 represent the perigee, then equation (4.13) becomes. Let's examine the case of two bodies of masses M and m moving in circular orbits under the influence of each other's gravitational attraction. At the United States' launch site in Cape Canaveral (28.5 degrees north latitude) a due east launch results in a "free ride" of 1,471 km/h (914 mph). This maneuver requires a component of V to be perpendicular to the orbital plane and, therefore, perpendicular to the initial velocity vector. 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. 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. tangent to the circle of position at this point.
compute the azimuth of the star using the estimated position and the data's
where the velocities are the circular velocities of the two orbits. In general these are ellipses with the center star in one of the two foci. Bibliography
As Kepler pointed out, all planets move in elliptical orbits, however, we can learn much about planetary motion by considering the special case of circular orbits. The orbit is then circularized by firing the spacecraft's engine at apogee. Orbit Maintenance
1 and 2 are the geographical longitudes of the ascending node and the burnout point at the instant of engine burnout.
the true anomaly at infinity, we have
Figure 4.11 represents a Hohmann transfer orbit. Sun synchronous orbits (SSO) are walking orbits whose orbital plane precesses with the same period as the planet's solar orbit period. 2.  A line joining any planet to the sun sweeps out equal areas in equal times. For this to happen, the gravitational force acting on each body must provide the necessary centripetal acceleration. Once in their mission orbits, many satellites need no additional orbit adjustment. For satellites in GEO and below, the J2 perturbations dominate; for satellites above GEO the Sun and Moon perturbations dominate. We can find the required change in velocity by using the law of cosines. Click here for example problem #4.21
R j (q)j2dq= 1. p is a geometrical constant of the conic called the parameter or semi-latus rectum, and is equal to
(4.20) The only equation still to be derived is that for the mean anomaly of an epoch. When the satellite reaches apogee of the transfer orbit, a combined plane change maneuver is done. Thus, we may choose the transfer orbit by specifying the size of the transfer orbit, the angular change of the transfer, or the time required to complete the transfer. Click here for example problem #4.27
A geosynchronous orbit with an inclination of zero degrees is called a geostationary orbit. 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. 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. This three-burn maneuver may save propellant, but the propellant savings comes at the expense of the total time required to complete the maneuver. The other root corresponds to the apoapsis radius, Ra. The stable orbits around a star are given by the Kepler's laws oft planetary motion.
Ordinarily we want to transfer a space vehicle using the smallest amount of energy, which usually leads to using a Hohmann transfer orbit.
The interceptor remains in the initial orbit until the relative motion between the interceptor and target results in the desired geometry. 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. 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 time of the launch depends on the launch site's latitude and longitude and the satellite orbit's inclination and longitude of ascending node. For example, we may need to transfer from an initial parking orbit to the final mission orbit, rendezvous with or intercept another spacecraft, or correct the orbital elements to adjust for the perturbations discussed in the previous section.
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.
If a space vehicle comes within 120 to 160 km of the Earth's surface, atmospheric drag will bring it down in a few days, with final disintegration occurring at an altitude of about 80 km. Don't confuse the intercept ITC with ITP -
We may allow low-altitude orbits to decay and reenter the atmosphere or use a velocity change to speed up the process. Once we know the semi-major axis of the ellipse, atx, we can calculate the eccentricity, angular distance traveled in the transfer, the velocity change required for the transfer, and the time required to complete the transfer. 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. 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. We can calculate this velocity from the energy equation written for two points on the hyperbolic escape trajectory – a point near Earth called the burnout point and a point an infinite distance from Earth where the velocity will be the hyperbolic excess velocity, v∞. Most frequently, we must change the orbit altitude, plane, or both. Orbit Maintenance
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. 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. The celestial navigation software ASNAv is now free to download.
We do this using equations (4.59) through (4.63) and (4.65) above, and the following equations: 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. The plane change maneuver takes places when the space vehicle passes through one of these two nodes. True anomaly, , is the angular distance of a point in an orbit past the point of periapsis, measured in degrees. In other words, it has already slowed down to very nearly its hyperbolic excess velocity.
where A is the cross-sectional area of the satellite exposed to the Sun and m is the mass of the satellite in kilograms. If we know the radius, r, velocity, v, and flight path angle, , of a point on the orbit (see Figure 4.15), we can calculate the eccentricity and semi-major axis using equations (4.30) and (4.32) as previously presented.
We can define all conic sections in terms of the eccentricity. where G is an universal constant, called the constant of gravitation, and has the value 6.67259x10-11 N-m2/kg2 (3.4389x10-8 lb-ft2/slug2). Orbit Altitude Changes
If, on the other hand, we give our vehicle more than escape velocity at a point near Earth, we would expect the velocity at a great distance from Earth to be approaching some finite constant value. Finally, when the satellite reaches perigee of the second transfer orbit, another coplanar maneuver places the satellite into the final orbit.
In this case, the transfer orbit's ellipse is tangent to both the initial and final orbits at the transfer orbit's perigee and apogee respectively. 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. Mean anomaly is a function of eccentric anomaly by the formula.
Click here for example problem #4.30, If you give a space vehicle exactly escape velocity, it will just barely escape the gravitational field, which means that its velocity will be approaching zero as its distance from the force center approaches infinity. 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. Two particular cases of note are satellites with repeating ground tracks and geostationary satellites. In this case, the transfer orbit's ellipse is tangent to both the initial and final orbits at the transfer orbit's perigee and apogee respectively. where Vi is the velocity before and after the burn, and is the angle change required.
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