Source: in part from Refs 4-2 and 4-3.

"Earth mass is 5.976 x 1024 kg.

"Earth mass is 5.976 x 1024 kg.

planets, and the moon. Launching from the earth's surface at escape velocity is not practical. As a vehicle ascends through the earth's atmosphere, it is subject to severe aerodynamic heating and dynamic pressures. A practical launch vehicle has to traverse the atmosphere at relatively low velocity and accelerate to the high velocities beyond the dense atmosphere. For example, during a portion of the Space Shuttle's ascent, its main engines are actually throttled to a lower thrust to avoid excessive pressure and heating. Alternatively, an escape vehicle can be launched from an orbiting space station or from an orbiting Space Shuttle.

A rocket spaceship can become a satellite of the earth and revolve around the earth in a fashion similar to that of the moon. Satellite orbits are usually elliptical and some are circular. Low earth orbits, typically below 500 km altitude, are designated by the letters LEO. Satellites are useful as communications relay stations for television or radio, weather observation, or reconnaissance observation. The altitude of the orbit is usually above the earth's atmosphere, because this minimizes the expending of energy to overcome the drag which pulls the vehicle closer to the earth. The effects of the radiation in the Van Allen belt on human beings and sensitive equipment sometimes necessitate the selection of an earth orbit at low altitude.

For a circular trajectory the velocity of a satellite must be sufficiently high so that its centrifugal force balances the earth's gravitational attraction.

For a circular orbit, the satellite velocity us is found by using Eq. 4—12, which is smaller than the escape velocity by a factor of -Jl. The period r in seconds of one revolution for a circular orbit relative to a stationary earth is

The energy E necessary to bring a unit of mass into a circular satellite orbit neglecting drag, consists of kinetic and potential energy, namely, mu2/R = mg us = R0Jg0/(R0 + h) = ,/ß/R

The escape velocity, satellite velocity, satellite period, and satellite orbital energy are shown as functions of altitude in Fig. 4—6.

A satellite circulating around the earth at an altitude of 300 miles or 482.8 km has a velocity of about 7375 m/sec or 24,200 ft/sec, circles a stationary earth in 1.63 hr, and ideally requires an energy of 3.35 x 107 J to place 1 kg of spaceship mass into its orbit. An equatorial satellite in a circular orbit at an altitude of 6.611 earth radii (about 26,200 miles, 42,200 km, or 22,700 nautical miles) has a period of revolution of 24 hr. It will appear stationary to an observer on earth. This is known as a synchronous satellite in geosynchronous earth orbit, usually abbreviated as GEO. It is used extensively for communications satellite applications. In Section 4.7 on launch vehicles we will describe how the payload of a given space vehicle diminishes as the orbit circular altitude is increased and as the inclination (angle between orbit plane and earth equatorial plane) is changed.

Elliptical Orbits

The circular orbit described above is a special case of the more general elliptic orbit shown in Fig. 4-7; here the earth (or any other heavenly body around which another body is moving) is located at one of the focal points of this ellipse. The equations of motion may be derived from Kepler's laws, and the elliptical orbit can be described as follows, when expressed in polar coordinates:






Perigee radius

FIGURE 4-7. Elliptical orbit; the attracting body is at one of the focal points of the ellipse.


Perigee radius

FIGURE 4-7. Elliptical orbit; the attracting body is at one of the focal points of the ellipse.

where u is the velocity of the body in the elliptical orbit, R is the instantaneous radius from the center of the attracting body (a vector quantity, which changes direction as well as magnitude), a is the major axis of the ellipse, and ¡1 is the earth's gravitational constant with a value of 3.986 x 1014 m3/sec2. The symbols are defined in Fig. 4-7. From this equation it can be seen that the velocity up is a maximum when the moving body comes closest to its focal point at the orbit's perigee and that its velocity ua is a minimum at its apogee. By substituting for R in Eq. 4-29, and by defining the ellipse's shape factor e as the eccentricity of the ellipse, e — y/a2 — b2/a, then the apogee and perigee velocities can be expressed as ua = ^TT^ (4—30)

Another property of an elliptical orbit is that the product of velocity and instantaneous radius remains constant for any location a or b on the ellipse, namely, uaRa = ubRb — uR. The exact path that a satellite takes depends on the velocity (magnitude and vector orientation) with which it is started or injected into its orbit.

For interplanetary transfers the ideal mission can be achieved with minimum energy in a simple transfer ellipse, as suggested originally by Hohmann (see Ref. 4-6). Assuming the planetary orbits about the sun to be circular and coplanar, it can be demonstrated that the path of minimum energy is an ellipse tangent to the planetary orbits as shown in Fig. 4-8. This operation requires a velocity increment (relatively high thrust) at the initiation and another at ter-

Planet B at t2

Planet B at t2

FIGURE 4-8. Schematic diagram of interplanetary transfer paths. These same transfer maneuvers apply when going from a low-altitude earth satellite orbit to a higher orbit.

mination; both increments are the velocity differences between the respective circular planetary velocities and the perigee and apogee velocity which define the transfer ellipse. The thrust levels at the beginning and end maneuvers of the Hohmann ellipse must be high enough to give a short operating time and the acceleration of at least 0.01 g0, but preferably more. With electrical propulsion these accelerations would be about 10~5 g0, the operating time would be weeks or months, and the best transfer trajectories would be very different from a Hohmann ellipse; they are described in Chapter 19.

The departure date or the relative positions of the launch planet and the target planet for a planetary transfer mission is critical, because the spacecraft has to meet with the target planet when it arrives at the target orbit. The Hohmann transfer time (t2 — 11) starting on earth is about 116 hours to go to the moon and about 259 days to Mars. If a faster orbit (shorter transfer time) is desired (see dashed lines in Fig. 4-8), it requires more energy than a Hohmann transfer ellipse. This means a larger vehicle with a larger propulsion system that has more total impulse. There also is a time window for a launch of a spacecraft that will make a successful rendezvous. For a Mars mission an earth-launched spacecraft may have a launch time window of more than two months. A Hohmann transfer ellipse or a faster transfer path apply not only to planetary flight but also to earth satellites, when an earth satellite goes from one circular orbit to another (but within the same plane). Also, if one spacecraft goes to a rendezvous with another spacecraft in a different orbit, the two spacecraft have to be in the proper predetermined positions prior to the launch for simultaneously reaching their rendezvous. When the launch orbit (or launch planet) is not in the same plane as the target orbit, then additional energy will be needed by applying thrust in a direction normal to the launch orbit plane.

Example 4-2. A satellite is launched from a circular equatorial parking orbit at an altitude of 160 km into a coplanar circular synchronous orbit by using a Hohmann transfer ellipse. Assume a homogeneous spherical earth with a radius of 6374 km. Determine the velocity increments for entering the transfer ellipse and for achieving the synchronous orbit at 42,200 km altitude. See Fig. 4-8 for the terminology of the orbits.

SOLUTION. The orbits are RA = 6.531 x 106 m; RB = 48.571 x 106 m. The major axis a of the transfer ellipse a,e = \{Ra + Rb) = 27.551 x 106 m/ sec

The orbit velocities of the two satellites are uA = y/ß/RA = [3.986005 x 1014/6.571 x 106]i = 7788 m/sec uB = y/ß/RB = 2864.7 m/ sec

The velocities needed to enter and exit the transfer ellipse are

(;ute)A = JjH(2/RA) - (l/a)]i = 10, 337 m/ sec (u,e)B = Vm[(2/Rb) - (l/a)]l/2 = 1394 m/sec

The changes in velocity going from parking orbit to ellipse and from ellipse to final orbit are:

AuA = |(ute)A - uA| = 2549 m/sec AuB = |uB - (ute)B = 1471 m/sec

The total velocity change for the transfer maneuvers is:

Figure 4—9 shows the elliptical transfer trajectory of a ballistic missile or a satellite ascent vehicle. During the initial powered flight the trajectory angle is adjusted by the guidance system to an angle that will allows the vehicle to reach the apogee of its elliptical path exactly at the desired orbit altitude. For the ideal satellite orbit injection the simplified theory assumes an essentially instantaneous application of the total impulse as the ballistic trajectory reaches its apogee or zenith. In reality the rocket propulsion system operates over a finite time, during which gravity losses and changes in altitude occur.

Deep Space

Lunar and interplanetary missions include circumnavigation, landing, and return flights to the moon, Venus, Mars, and other planets. The energy necessary to escape from earth can be calculated as |mv2e from Eq. 4-25. It is 6.26 x 107J/kg, which is more than that required for a satellite. The gravitational attraction of various heavenly bodies and their respective escape velocities depends on their masses and diameters; approximate values are listed in Table 4—1. An idealized diagram of an interplanetary landing mission is shown in Fig. 4-10.

The escape from the solar system requires approximately 5.03 x 10s J/kg. This is eight times as much energy as is required for escape from the earth. There is technology to send small, unmanned probes away from the sun to outer space; as yet there needs to be an invention and demonstrated proof of a long duration, novel, rocket propulsion system before a mission to the nearest star can be achieved. The trajectory for a spacecraft to escape from the sun is either a parabola (minimum energy) or a hyperbola.

Local Elliptical ballistic

Local Elliptical ballistic

FIGURE 4-9. Long-range ballistic missiles follow an elliptical free-flight trajectory (in a drag-free flight) with the earth's center as one of the focal points. The surface launch is usually vertically up (not shown here), but the trajectory is quickly tilted during early powered flight to enter into the ellipse trajectory. The ballistic range is the arc distance on the earth's surface. For satellites, another powered flight period occurs (called orbit injection) just as the vehicle is at its elliptical apogee (as indicated by the velocity arrow), causing the vehicle to enter an orbit.

FIGURE 4-9. Long-range ballistic missiles follow an elliptical free-flight trajectory (in a drag-free flight) with the earth's center as one of the focal points. The surface launch is usually vertically up (not shown here), but the trajectory is quickly tilted during early powered flight to enter into the ellipse trajectory. The ballistic range is the arc distance on the earth's surface. For satellites, another powered flight period occurs (called orbit injection) just as the vehicle is at its elliptical apogee (as indicated by the velocity arrow), causing the vehicle to enter an orbit.


This section gives a brief discussion of the disturbing torques and forces which cause perturbations or deviations from any space flight path or satellite's flight trajectory. For a more detailed treatment of flight paths and their perturbations, see Refs. 4-2 and 4-3. A system is needed to measure the satellite's position and deviation from the intended flight path, to determine the needed periodic correction maneuver and then to counteract, control, and correct them. Typically, the corrections are performed by a set of small reaction control thrusters which provide predetermined total impulses into the desired directions. These corrections are needed throughout the life of the spacecraft (for 1 to 20 years) to overcome the effects of the disturbances and maintain the intended flight regime.

Operation of retro rocket to stow vehicle down to satellite velocity (1-5%)

Operation of retro rocket to stow vehicle down to satellite velocity (1-5%)

Interplanetary orbit (1-10%)

Coast in orbit

FIGURE 4-10. Schematic diagram of typical powered flight maneuvers during a hypothetical interplanetary mission with a landing. The numbers indicate typical thrust magnitudes of the maneuvers in percent of launch takeoff thrust. This is not drawn to scale. Heavy lines show powered flight segments.

Acceleration maneuver to attain

Interplanetary orbit (1-10%)

Coast in orbit

FIGURE 4-10. Schematic diagram of typical powered flight maneuvers during a hypothetical interplanetary mission with a landing. The numbers indicate typical thrust magnitudes of the maneuvers in percent of launch takeoff thrust. This is not drawn to scale. Heavy lines show powered flight segments.

Perturbations can be cateogirzed as short-term and long-term. The daily or orbital period oscillating forces are called diurnal and those with long periods are called secular.

High-altitude each satellites (36,000 km and higher) experience perturbing forces primarily as gravitational pull from the sun and the moon, with the forces acting in different directions as the satellite flies around the earth. This third-body effect can increase or decrease the velocity magnitude and change its direction. In extreme cases the satellite can come very close to the third body, such as the moon, and undergo what is called a hyperbolic maneuver that will radically change the trajectory. This encounter can be used to increase or decrease the energy of the satellite and intentionally change the velocity and the shape of the orbit.

Medium- and low-altitude satellites (500 to 35,000 km) experience perturbations because of the earth's oblateness. The earth bulges in the vicinity of the equator and a cross section through the poles is not entirely circular. Depending on the inclination of the orbital plane to the earth equator and the altitude of the satellite orbit, two perturbations result: (1) the regression of the nodes, and (2) shifting of the apsides line (major axis). Regression of the nodes is shown in Fig. 4-11 as a rotation of the plane of the orbit in space, and it can be as high as 9° per day at relatively low altitudes. Theoretically, regression does not occur in equatorial orbits.

Figure 4-12 shows an exaggerated shift of the apsidal line, with the center of the earth remaining as a focus point. This perturbation may be visualized as the movement of the prescribed elliptical orbit in a fixed plane. Obviously, both the apogee and perigee points change in position, the rate of change being a func-

FIGURE 4-11. The regression of nodes is shown as a rotation of the plane of the orbit. The direction of the movement will be opposite to the east-west components of the earth's satellite motion.

tion of the satellite altitude and plane inclination angle. At an apogee altitude of 1000 nautical miles (n.m.) and a perigee of 100 n.m. in an equatorial orbit, the apsidal drift is approximately 10° per day.

Satellites of modern design, with irregular shapes due to protruding antennas, solar arrays, or other asymmetrical appendages, experience torques and forces that tend to perturb the satellite's position and orbit throughout its orbital life. The principal torques and forces result from the following factors:

FIGURE 4-12. Shifting of the apsidal line of an elliptic orbit from position 1 to 2 because of the oblateness of the earth.

1. Aerodynamic drag. This factor is significant at orbital altitudes below 500 km and is usually assumed to cease at 800 km above the earth. Reference 4-7 gives a detailed discussion of aerodynamic drag which, in addition to affecting the attitude of unsymmetrical vehicles, causes a change in elliptical orbits known as apsidal drift, a decrease in the major axis, and a decrease in eccentricity of orbits about the earth.

2. Solar radiation. This factor dominates at high altitudes (above 800 km) and is due to impingement of solar photons upon satellite surfaces. The solar radiation pressure p (N/m2) on a given surface of the satellite in the vicinity of the earth exposed to the sun can be determined as p = 4.5 x. I0"6cos6»[(l -ks) cos 6 + 0.67^] (4-32)

where 6 is the angle (degrees) between the incident radiation vector and the normal to the surface, and ks and k¿ are the specular and diffuse coefficients of reflectivity. Typical values are 0.9 and 0.5, respectively, for ks and kd on the body and antenna, and 0.25 and 0.01 respectively, for ks and kj with solar array surfaces. The radiation intensity varies as the square of the distance from the sun (see Ref. 4-8). The torque T on the vehicle is given by T = pAl, where A is the projected area and I is the offset distance between the spacecraft's center of gravity and the center of solar pressure.

3. Gravity gradients. Gravitational torque in spacecraft results from a variation in the gravitational force on the distributed mass of a spacecraft. Determination of this torque requires knowledge of the gravitational field and the distribution of spacecraft mass. This torque decreases as a function of the orbit radius and increases with the offset distances of masses within the spacecraft (including booms and appendages), it is most significant in large spacecraft or space stations operating in relatively low orbits (see Ref. 4-9).

4. Magnetic field. The earth's magnetic field and any magnetic moment within the satellite interact to produce torque. The earth's magnetic field precesses about the earth's axis but is very weak (0.63 and 0.31 gauss at poles and equator, respectively). This field is continually fluctuating in direction and intensity because of magnetic storms and other influences. Since the field strength decreases with l /i?3 with the orbital altitude, magnetic field forces are often neglected in the preliminary design of satellites (see Ref. 4-10).

5. Internal accelerations. Deployment of solar array panels, the shifting of propellant, movement of astronauts or other mass within the satellite, or the "unloading" of reaction wheels produce torques and forces.

We can categorize satellite propulsion needs according to function as listed in Table 4-2, which shows the total impulse "budget" applicable to a typical

TABLE 4.2. Propulsion Functions and Total Impulse Needs of a 2000-lbm Geosynchronous Satellite with a 7-Year Life

Total Impulse

Function (N-sec)

Acquisition of orbit 20,000

Attitude control (rotation) 4,000

Station keeping, E-W 13,000

Station keeping, N-S 270,000

Repositioning (Am, 200 ft/sec) 53,000

Control apsidal drift (third body attraction) 445,000

Deorbit 12,700

Total 817,700

high altitude, elliptic orbit satellite. The control system designer often distinguishes two different kinds of stationary-keeping orbit corrections needed to keep the satellite in a synchronous position. The east-west correction refers to a correction that moves the point at which a satellite orbit intersects the earth's equatorial plane in an east or west direction; it usually corrects forces caused largely by the oblateness of the earth. The north-south correction counteracts forces usually connected with the third-body effects of the sun and the moon.

In many satellite missions the gradual changes in orbit caused by perturbation forces are not of concern. However, in certain missions it is necessary to compensate for these perturbing forces and maintain the satellite in a specific orbit and in a particular position in that orbit. For example, a synchronous communications satellite in a GEO needs to maintain its position and its orbit, so it will be able to (1) keep covering a specific area of the earth or communicate with the same stations on earth within its line of sight, and (2) not become a hazard to other satellites in this densely occupied synchronous equatorial orbit. Another example is a LEO communications satellite system with several coordinated satellites; here at least one satellite has to be in a position to receive and transmit RF signals to specific points on earth. Their orbits, and the positions of these several satellites with respect to each other, need to be controlled and maintained (see Refs. 4-11 to 4-13).

Orbit maintenance means applying small correcting forces and torques periodically; for GEO it is typically every few months. Typical velocity increments for the orbit maintenance of synchronous satellites require a Au between 10 and 50 m/sec per year. For a satellite mass of about 2000 kg a 50 m/sec correction for a 10-year orbit life would need a total impulse of about 100,000 N-sec, which corresponds to a propellant mass of 400 to 500 kg (about a quarter of the satellite mass) if done by a small monopropellant or bipropellant thrust. It would require much less propellant if electrical propulsion were used, but in some spacecraft the inert mass of the power supply would increase.

Mission Velocity

A convenient way to describe the magnitude of the energy requirement of a space mission is to use the concept of the mission velocity. It is the sum of all the flight velocity increments needed to attain the mission objective. In the simplified sketch of a planetary landing mission of Fig. 4-10, it is the sum of all the Am velocity increments shown by the heavy lines (rocket-powered flight segments) of the trajectories. Even though some of the velocity increments were achieved by retro-action (a negative propulsion force to decelerate the flight velocity), these maneuvers required energy and their absolute magnitude is counted in the mission velocity. The initial velocity from the earth's rotation (464 m/sec at the equator and 408 m/sec at a launch station at 28.5° latitude) does not have to be provided by the vehicle's propulsion systems. For example, the required mission velocity for launching at Cape Kennedy, bringing the space vehicle into an orbit at 110 km, staying in orbit for a while, and then entering a de-orbit maneuver has the Au components shown in Table 4-3.

The required mission velocity is the sum of the absolute values of all translation velocity increments that have forces going through the center of gravity of the vehicle (including turning maneuvers) during the flight of the mission. It is the theoretical hypothetical velocity that can be attained by the vehicle in a gravity-free vacuum, if all the propulsive energy of the momentum-adding thrust chambers in all stages were to be applied in the same direction. It is useful for comparing one flight vehicle design with another and as an indicator of the mission energy.

The required mission velocity has to be equal to the "supplied" mission velocity, that is, the sum of all the velocity increments provided by the propulsion systems of each of the various vehicle stages. The total velocity increment to be "supplied" by the shuttle's propulsion systems for the shuttle mission described below (solid rocket motor strap-on boosters, main engines and, for orbit injection, also the increment from the orbital maneuvering system—all shown in Fig. 1-13) has to equal or exceed 9621 m/sec. With chemical propulsion systems and a single stage, we can achieve a space mission velocity of 4000

TABLE 4-3. Space Shuttle Incremental Flight Velocity Breakdown

Ideal satellite velocity

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