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Passive attitude control refers to the use of mechanisms which stabilize the satellite without putting a drain on the satellite s energy supplies; at most, infrequent use is made of these supplies, for example, when thruster jets are impulsed to provide corrective torque Examples of passive attitude control are spin stabilization and gravity gradient stabilization The latter depends on the interaction of the satellite with the gravitational field of the central body and has been used, for example, with the Radio Astronomy Explorer-2 satellite, which was placed in orbit around the moon (Wertz, 1984) For communications satellites, spin stabilization is often used, and this is described in detail in Sec 731 The other form of attitude control is active control With active attitude control, there is no overall stabilizing torque present to resist the disturbance torques Instead, corrective torques are applied as required in response to disturbance torques.

AutoCAD allows you to read, add, or modify a linetype s description using the Description property of the Linetype object. This property is a string that typically describes the linetype, often with a simplified symbolized version of the actual linetype. The following code snippet shows the Description property being set and retrieved: Dim objLineType As AcadLineType Dim strLineTypeDescription As String objLineType.Description = "Linetype Description: -.-.-." strLineTypeDescription = objLineType.Description The following example changes a Linetype description based on user input: Public Sub DescribeLinetype() On Error Resume Next Dim strLinetypeName As String Dim strLinetypeDescription As String Dim objLinetype As AcadLineType

Methods used to generate active control torques include momentum wheels, electromagnetic coils, and mass expulsion devices, such as gas jets and ion thrusters The electromagnetic coil works on the principle that the earth s magnetic field exerts a torque on a current-carrying coil and that this torque can be controlled through control of the current However, the method is of use only for satellites relatively close to the earth The use of momentum wheels is described in more detail in Sec 732..

Figure 7.4 (a) Roll, pitch, and yaw axes. The yaw axis is directed toward the earth s center, the pitch axis is normal to the orbital plane, and the roll axis is perpendicular to the other two. (b) RPY axes for the geostationary orbit. Here, the roll axis is tangential to the orbit and lies along the satellite velocity vector.

The three axes which define a satellite s attitude are its roll, pitch, and yaw (RPY) axes. These are shown relative to the earth in Fig. 7.4. All three axes pass through the center of gravity of the satellite. For an equatorial orbit, movement of the satellite about the roll axis moves the antenna footprint north and south; movement about the pitch axis moves the footprint east and west; and movement about the yaw axis rotates the antenna footprint.

Spin stabilization may be achieved with cylindrical satellites. The satellite is constructed so that it is mechanically balanced about one particular axis and is then set spinning around this axis. For geostationary satellites, the spin axis is adjusted to be parallel to the N-S axis of the earth, as illustrated in Fig. 7.5. Spin rate is typically in the range of 50 to 100 rev/min. Spin is initiated during the launch phase by means of small gas jets. In the absence of disturbance torques, the spinning satellite would maintain its correct attitude relative to the earth. Disturbance torques are generated in a number of ways, both external and internal to the satellite. Solar radiation, gravitational gradients, and meteorite impacts are all examples of external forces which can give rise to disturbance torques. Motor-bearing friction and the movement of satellite elements such as the antennas also can give rise to disturbance torques. The

ceives from consuming it. The more units of a commodity the individual consumes per unit of time, the greater is the total utility he receives. Although total utility increases, the extra, or marginal, utility received from consuming each additional unit of the commodity decreases. This is referred to as the law of diminishing marginal utility. Example 11.1 For purposes of illustration, we assume in Table 11.1 that satisfaction can actually be measured in terms of units of utility called utils. The rst two columns of Table 11.1 give an individual s hypothetical total utility (TU) schedule from consuming various quantities of commodity X (say oranges) per unit of time. Table 11.1