## Equations of Motion

In keeping with our approach, we present the simplest analysis that treats the issues of salient interest to the vehicle designer. To this end, we consider the planar trajectory of a vehicle over a nonrotating spherical planet. The geometric situation is as shown in Fig. 5.13. The equations of motion are3

## Ryz

Fig. 7.22 Earth horizon scanner attitude determination concept. Fig. 7.22 Earth horizon scanner attitude determination concept. altitude by the onboard control logic. This is required because the reference Earth width as seen by the scanner depends upon this altitude. The use of sun sensors and Earth horizon scanners together can provide a very powerful attitude determination and control system for a LEO spacecraft. A system can be configured using these sensors in a scheme that uses a...

## Space Vehicle Disturbance Torques

As mentioned, operating spacecraft are subject to numerous disturbance forces which, if not acting through the center of mass, result in a net torque being imparted to the vehicle. Assessment of these influences in terms of both absolute and relative magnitude is an essential part of the ADCS designer's task. The role of the upper atmosphere in producing satellite drag was discussed in Chapter 4 in connection with orbit decay. The same drag force will, in general, produce a disturbance torque...

## Engine Cooling

A variety of cooling concepts have been proposed for use in rocket engines, many of which have seen operational use, often in combination. The most common approach for large engines with lengthy operating times is regenerative cooling, mentioned earlier, where one of the propellants is passed through cooling passages in the thrust chamber and nozzle wall before being injected into the combustion chamber. This very effective and efficient approach is usually supplemented by film or...

## Weightlessness and Microgravlty

It is common to assume that orbital flight provides a weightless environment for a spacecraft and its contents. To some level of approximation this is true, but as with most absolute statements, it is inexact. A variety of effects result in acceleration levels (i.e., weight per unit mass) between 10 3 and 10 ug, where 1 g is the acceleration due to gravity at the Earth's surface, 9.81 m s2. The acceleration experienced in a particular case will depend on the size of the spacecraft, its...

## Plane Changes

Most missions, in cases where any orbit adjustment at all is required, will require some adjustment of the orbital plane. This plane (see Fig. 4.13) is perpendicular to the angular momentum vector h, which for a Keplerian orbit is permanently fixed in space. A pure plane rotation alters h in direction but not in magnitude and thus requires an applied torque normal to h. This in turn requires the application of a force on the spacecraft, e.g., a thruster firing, parallel to h. For pure plane...

## Gravity Gradient Stabilization

From our previous discussion, it is clear that a spacecraft in a reasonably low orbit will tend to stabilize with its minimum-inertia axis in a vertical orientation. This property can obviously be used to advantage by the designer when a nadir or zenith orientation is desired for particular instruments. The principal design feature of such a satellite again involves the inertia ratio the vehicle must possess an axis such that lz < Ix, v. As noted previously, even when the spacecraft is...

## Inner Planetary Missions

The target bodies included in this category are those from Mercury to the inner reaches of the asteroid belt. The energy required to reach these extremes from Earth is roughly the same, a vis-viva energy of 30-40 km2 s2 (see Chapter 4). Even though the region encompasses a variation in solar radiative and gravitational intensity of about 60, it can be said to be dominated by the sun. Within this range, it is feasible to design solar-powered spacecraft and to use solar orientation as a factor in...

## Solar Radiation Pressure

Observed solar radiation intensity at the Earth's orbit about the sun is closely approximated (to within 0.3 ) by Is integrated intensity (in W m2) on the area normal to the sun D phase of year D 0 on July 4 (aphelion) given by Smith and Gottlieb.44 Note that 1358 W m2 is the mean intensity observed at a distance of 1 A.U. solar radiation intensity for planets at other distances from the sun is computed from the inverse square law. For practical purposes in spacecraft design, the observed...

## Spacecraft Thermal Environment

Comments on the space thermal environment were offered in Chapter 3 as part of our discussion of the overall space environment. However, it is useful to expand on our earlier discussion prior to considering the design features that are intended to deal with that environment. The spacecraft thermal environment can vary considerably, depending upon a variety of naturally occurring effects. Orbital characteristics are a major source of variation. For example, most spacecraft orbits will have an...

## Motion In Elliptic Orbits

Figure 4.3 defines the parameters of interest in elliptic orbit motion. The conic section results given earlier are sufficient to describe the size and shape of the orbit, but do not provide the position of body m as a function of time. Because it is awkward to attempt a direct solution of Eq. (4.21) to yield 6 (and hence r) as a function of time, the auxiliary variable E, the eccentric anomaly, is introduced. The transformation between true and eccentric anomaly is Figure 4.3 defines the...

## Spacecraft Configuration

Figure 8.1 shows the Voyager spacecraft. Two of these vehicles, built and tested by the Jet Propulsion Laboratory (JPL) for NASA, were THERMAL BLANKETS FOR CLARITY) Fig. 8.1 Voyager spacecraft. (Courtesy of Jet Propulsion Laboratory.) THERMAL BLANKETS FOR CLARITY) Fig. 8.1 Voyager spacecraft. (Courtesy of Jet Propulsion Laboratory.) launched in 1977 to explore the outer planets of the solar system. The baseline mission was to be a four-year trip involving a flyby of Jupiter and...

## A5 Power Spectral Density

In the Fourier transform domain, these convolution integrals yield the much simpler algebraic relationships where Sxx((t)), Sxy(o)), and Syr(< ) are the Fourier transforms of Rxx(t), RxAt) and Ryy(t), respectively, defined via Eq. (A.23). Once Eqs. (A.28)-and (A.29) have been used to obtain Syy( D) and SxA > )< Ryy(t) and Rxy(t) can be obtained using the inverse Fourier transform, Eq. (A.24). From Eq. (A. 16), we note that E y 2(r) Ryy(0) then gives the variance of the output random...

## Planetary Envlronmehts

Interplanetary spacecraft designers face environmental problems that may be unique even in what is, after all, a rather specialized field. Flyby spacecraft, such as Pioneers 10 and 11 and Voyagers 1 and 2, may encounter radiation environments greatly exceeding those in near-Earth space. The Mariner 10 mission to Mercury required the capability to cope with a factor of 10 increase in solar heating compared to Earth orbit, whereas Voyager 2 at Neptune received only about 0.25 of the illumination...

## Passive Thermal Control

The techniques applied for passive thermal control include the use of geometry, coatings, insulation blankets, sun shields, radiating fins, and heat pipes. By geometry we imply the process of configuring the spacecraft to provide the required thermal radiating area, placing low-temperature objects in shadow, and exposing high-temperature objects to the sun or burying them deeply within the structure, and other similar manipulation of the spacecraft configuration to optimize thermal control....

## In Orbital Operation I He Spacecraft Is3 Axis Stabilized With The Bodyfixed Antenna Pointing Constantly At The Earth

Fig. 8.5 FLTSATCOM spacecraft (Courtesy of TRW.) boresight and fold between segments to form a hexagonal cylinder around the spacecraft. This configuration is maintained until after orbit insertion so that the deployed arrays do not have to withstand the insertion g loads. Only a portion of the arrays is illuminated in this configuration, but power requirements are so low in cruise mode compared to the operational relay mode that ample power is available. The advantage of the two-bus...

## Coordinate Frames

Within the fixed orbit plane of two-body Keplerian motion, the coordinate system of choice is the polar coordinate system depicted in Fig. 4.2. The position of the object is given by the coordinates (r, ff), with the true anomaly 6 measured from periapsis. This system of perifocal or orbit plane coordinates is both natural and sufficient as long as the orientation of the orbit in space need not be considered. However, we have seen in the preceding discussion that a particular orbit is defined...

## Viscous Interaction Parameter

Fig. 6.4 Lift modulation for STS-2 reentry. centrifugal forces. This is essentially the first-order ballistic entry model with lift added. With proper selection of parameters, the so-called skip or skip-glide entry may be obtained. Consider the high-speed entry of a lifting vehicle at an initially negative flight-path angle. As always, the vehicle and atmosphere parameters are considered constant. With lift dominant over gravity, the flight path will be turned upward (d-y df > 0) so that the...

## Active Thermal Control

Active thermal control of spacecraft may require devices such as heaters and coolers, shutters or louvers, or cryogenic materials. Thermal transport may be actively implemented by pumped circulation loops. Heaters usually are wire-wound resistance heaters, or possibly deposited resistance strip heaters. Control may be by means of ground command, or automatically with onboard thermostats, or both. For very small heaters where on off control is not required, radioisotope heaters are sometimes...

## A2 Concept of a Random Process

If a random variable X is a function of time, i.e., X X(t), then X(t) is said to be a random process or stochastic process. Unlike simple random variables, random processes are characterized both by their properties at a given time and by their behavior as it evolves across time. The value of X(t) at any particular time, for example X(t0) x0, is a random variable characterized by a probability density function (.t, t0) and having a mean, variance, etc., just as for any random variable. For...

## Coplanar Transfers

We now consider maneuvers that leave the orientation of the orbit unchanged but that may alter the elements a, e, and cj and the period r. Because the direction of h is to remain fixed, all maneuvers must produce torques parallel to h and are thus confined to the orbit plane. 4.4.2.1 Single-impulse transfer. The geometry of a general singleimpulse orbit transfer is shown in Fig. 4.16. An impulsive maneuver is executed at some position (n, 0i) in the orbit plane, with velocity and flight-path...

## Info

When the solar array is perpendicular to the velocity vector, the vector to the aerodynamic center of pressure was rcp 22 m k. Assume the station to be in a 400-km circular orbit, with standard atmospheric density at this altitude of p 2.8 x 10-12 kg m3. The projected area of the station with its array normal to the velocity vector is 600 m2, with a drag coefficient of CD 2 assumed. (a) What is the nominal orientation of the station in the absence of aerodynamic or solar radiation pressure...

## Sphere of Influence

Of the various possible perturbations to basic two-body motion, the most obvious are those due to the presence of additional bodies. Such bodies are always present and cannot be easily included in an analysis, particularly at elementary levels. It is then necessary to determine criteria for the validity of Keplerian approximations to real orbits when more than two bodies are present. If we consider a spacecraft in transit between two planets, it is clear that when close to the departure planet,...

## Problems

10.1 Size a spacecraft power system consisting of a solar array and Ni-Cd batteries to supply 7.5 kW of prime power using 12 efficient 2 x 4 cm silicon cells with sun-tracking flat panels. The orbit is 500-km circular at 28.5 deg, i.e., approximately that of the Hubble Space Telescope. Assume a five-year life, with a minimum of 28 V required during eclipse. For the battery, assume an average discharge voltage of 1.1 V cell and a minimum allowed discharge voltage of 1.0 V cell. Use good design...

## Solar Arrays

Regardless of the size of the total array, each array is made up of a very large number of individual cells arranged on a substrate of some type. Although each cell puts out a relatively small current and voltage, proper series and parallel connection can provide any desired current and voltage within reasonable physical limitations. Individual cells are made in a variety of shapes and sizes. Probably the most common as this is written is the rectangular cell with dimensions on the order of 2 x...

## Deployable Structures

The requirement for deployable structures arises from the limitations on dimensions and geometry of the launch vehicle payload volume as compared with the need for large antennas and solar arrays, the requirements for instrument field of view and isolation, and the requirement to isolate radiation producing objects such as RTGs and reactors. A variety of concepts have been developed, of which a representative few will be discussed here. 8.3.3.1 Solar arrays. Deployable solar arrays have for the...

## Cross Range Maneuvers

Thus far we have assumed that the entry trajectory lies in the plane of the initial orbit. However, if a lifting entry vehicle is banked (lift vector rotated out of the'vertical plane defined by r, V), then a force normal to the original orbit plane is generated and the vehicle flies a three-dimensional trajectory. This may be done with both gliding and skip entry profiles as discussed earlier. The dynamics of three-dimensional flight within the atmosphere are beyond the intended scope of this...

## Earth Environment

Throughout its tenure on Earth, the spacecraft and its components are subjected to a variety of potentially degrading environments. The atmosphere itself is a primary source of problems. Containing both water and oxygen, the Earth's atmosphere is quite corrosive to a variety of materials, including many of those used in spacecraft, such as lightweight structural alloys. Corrosion of structural materials can cause stress concentration or embrittlement, possibly leading to failure during launch....

## Free Molecular Heating

Thus far we have discussed only continuum flow results stagnation heating in rarefied flow may be important when considering satellites that orbit, or at least have periapsis, at very low altitudes. Free molecular heating is also relevant during launch vehicle ascent flight indeed, it is usually this constraint that determines the lowest altitude, typically around 100 km, at which the payload shroud can be jettisoned. The free molecular heating rate will be of the form where a is an unknown...

## Aerodynamic and Solar Pressure Stabilization

As with gravity gradient, the existence of aerodynamic and solar radiation pressure torques implies the possibility of their use in spacecraft control. This has in fact been accomplished, although the flight history is considerably reduced compared to the gravity-gradient case. The most prominent example of aerodynamic stabilization occurred with MAGSAT, a low-altitude spacecraft intended to map the Earth's magnetic field.10 This vehicle used an aerodynamic trim boom to assist in orienting the...

## Power Conditioning and Control

The power conditioning or processing portion of the space power system carries the responsibility for many of the functions listed earlier in this chapter. Power conditioning is necessary because the voltage from the power source may vary substantially, especially with solar arrays, which are of course the most common source of primary spacecraft power, for a variety of reasons including load variability, array temperature, and other external environmental factors. Broadly considered, the power...

## Method of Patched Conics

As the name implies, this approach uses a series of Keplerian orbits to define the trajectory. Each separate conic section is assumed to be solely due to the influence of the dominant body for that portion of the mission. The different segments are patched at the sphere of influence boundaries between different bodies given by Eq. (4.93). Thus, a spacecraft trajectory between Earth and Mars will be modeled for departures as a geocentric escape hyperbola, which at great distances from Earth...

## Newtons Law of Cooling

For forced convection of a single-phase fluid over a surface at a moderate temperature difference, it was discovered by Newton that the heat transfer is proportional to both the surface area and the temperature difference. The convective heat flux into the wall may then be written according to Newton's law of cooling as Q hcAAT hcA(Tf Tw) (9.17) where Q is the power, h the convection or film coefficient, A the area, and AT the driving temperature differential from 7 and Tw, the fluid and wall...

## Attitude Jitter

Spacecraft attitude jitter is almost universally discussed in statistical terms, a view consistent with the fact that the jitter is, by definition, not subject to ADCS influence, and is therefore random in that sense. Continuing in this vein, we note that by subtracting the average value Qa from the data, we produce by definition a zero-mean history such as shown in Fig. 7.4. The smooth central curve results from filtering the data to remove the jitter, Le., the components above the spacecraft...