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35 Viewing and Lighting Conditions

The previous sections have been concerned with orbit kinematics and dynamics and with the general problem of determining the relative position of the spacecraft and other celestial objects. In this section, we assume that these quantities are known and consider the viewing and lighting conditions for planets and natural and artificial satellites. We also discuss the apparent brightness of objects as observed from space.

The geometry of viewing and lighting conditions for either natural or artificial satellites is shown in Fig. 3-21. Transit and occultation refer to the relative orientations of a planet, a satellite, and an observer. Transit occurs when a satellite

observer

Fig. 3-21. Definition of Viewing Conditions for Satellite. Sun, planet, observer, and orbit are all in the plane of the paper.

observer

Fig. 3-21. Definition of Viewing Conditions for Satellite. Sun, planet, observer, and orbit are all in the plane of the paper.

passes in front of the disk of a planet as seen by the observer. Occultation occurs when the satellite passes behind the disk of the planet and is occulted or blocked from the observer's view.

Eclipses are the phenomena of transits and occultations relative to the Sun. An eclipse of the Sun is a transit of an object in front of the Sun, blocking all or a significant part of the Sun's radiation from the observer. An eclipse of any other object (e.g., an eclipse of the Moon) is an occultation of that object by another object relative to the Sun. Because the Sun is the largest object in the solar system, the shadows of all the planets and natural satellites are shaped as shown in Fig. 3-21. The umbra, or shadow cone, is the conical region opposite the direction of the Sun in which the disk of the Sun is completely blocked from view by the disk of the planet Outside the umbra is the penumbra, where a portion of the disk of the Sun is blocked from view and, therefore, where the illumination on objects is reduced.

Unfortunately, the terminology of eclipses depends on whether the observer is thought of as being on the object which is entering the shadow or viewing the event from elsewhere. If the observer enters the shadow, the Sun is partially or wholly blocked from view and the event is referred to as an eclipse of the Sun or solar eclipse (Fig. 3-22). A total solar eclipse, which is frequently shortened to just eclipse

Fig. 3-22. Solar Eclipse Geometry

in spaceflight applications, occurs when the observer enters the umbra. If the observer is farther from the planet than the length of the shadow cone and enters the cone formed by the extension of the shadow cone through its apex, the observer will see an annular eclipse in which an annulus or ring of the bright solar disk is visible surrounding the disk of the planet. If the observer is within the penumbra, but outside the umbra, the observer will see a portion of the Sun's disk blocked by the planet and a partial eclipse of the Sun occurs.

If the observer is viewing the event from somewhere other than on the satellite being eclipsed, the event is called an eclipse of the satellite. A total eclipse of the satellite occurs when the entire satellite enters the umbra and a partial eclipse occurs when part of an extended satellite (such as the Moon) enters the umbra. If the satellite enters the penumbra only, the event is referred to as a penumbral edipse. Thus, a total eclipse of the Moon to an observer on the Earth is a total solar eclipse to a lunar observer and a penumbral eclipse of a satellite to an observer on the Earth is a partial eclipse of the Sun to an observer on the satellite.

Conditions for Transit and Occultation. Let X be a vector from the observer to the satellite, P be a vector from the observer to the center of the planet, and R

be the radius of the planet. The angular radius, pp, of the planet as seen by the observer is

The satellite will be in transit, that is, in front of the disk of the planet, whenever: p/,>arccos(P-X)

and »Transit Conditions (3-47)

Thé satellite will be in occultation, that is, behind the disk of the planet, whenever -, p/,>arccos(P-X)

Occultation Conditions (3-48)

To apply Eqs. (3-47) and (3-48) to the entire orbit of a satellite, we must determine the satellite position vector and test it against the transit and occultation equations at many places around the orbit. Therefore, it is convenient to determine from general orbital parameters those orbits for which transit and occultation' necessarily occur and those for which they cannot occur. We assume that the position of the observer remains fixed. If either transit or occultation occurs, it will be in progress when the angular separation between the spacecraft and the planet is a minimum. (See Fig. 3-23.) Therefore, the minimum angular separation between the planet and the spacecraft determines whether transit or occultation will occur.

In general, it is possible for either transit or occultation to occur in an orbit without the other. However, if the orbit is circular and transit occurs, then occultation must also. This is clear from Fig. 3-23, which shows the general appearance of a circular orbit viewed from nearby and out of the orbit plane. Point A is the closest point on the orbit to the observer and point B is the farthest point from the observer. If point A is in front of the disk, then point B is necessarily behind the disk.

For a noncircular orbit, the smallest minimum angular separation between the spacecraft and the planet will occur when perifocus is at B. As shown in Fig. 3-24, let Dp be the perifocal distance and p be the angular separation between the spacecraft and the planet when perifocus is at B. Then:

where / is the angle between the orbit plane and P, the vector from the observer to the center of the planet*, / may be determined from cosi = |PxN| (3-50)

where N is the unit vector normal to the orbit plane. Neither transit nor occultation will occur if /? > pP (condition 1).

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