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Both transit and occultation must occur during the orbit if a<pp and P>DA cosi (condition 3). If none of the above three sets of conditions are fulfilled, then whether transit or occultation occurs depends on both the eccentricity of the orbit and the orientation of the observer relative to the line of apsides.

Eclipse Conditions. To determine the conditions under which eclipses occur, we first determine the length, C, and angular radius, pc, of the shadow cone for any of the planets or natural satellites. (See Fig. 3-25.) Let S be the distance from the planet to the Sun, Rp be the radius of the planet, Rs be the radius of the photosphere (i.e., the visible surface) of the Sun, and C be measured from the center of the planet to the apex of the shadow cone. Then,

Fig. 3-25. Variables for Eclipse Geometry

fined

Fig. 3-25. Variables for Eclipse Geometry

For the Earth, the size of the shadow cone for its mean distance from the Sun is C—1.385 X106 km and pr = 0.264°. For the Moon, the mean size is C-3.75X105 km and pc = 0.266°. The length of the shadow cone for the Moon is just less than the semimajor axis of the Moon's orbit of 3.84 X 10s km. Therefore, eclipses of the Sun seen on the Earth are frequently annular eclipses, and when they are total eclipses they are seen over a very narrow band on the Earth because the maximum radius of the Moon's shadow cone at the distance of the Earth's surface is 135 km.

The presence of an atmosphere on some planets and the non-neglible radius of the natural satellites may be taken into account by adjusting the radius of the planet, as will be discussed later. Initially, we will assume that there are no atmospheric effects and that we are concerned with eclipses seen by objects of neglible size, such as spacecraft. The conditions for the satellite to see a total eclipse of the Sun are exactly those for a transit of the satellite as viewed from the apex of the shadow cone. Similarly, the conditions for spacecraft to see a partial eclipse are nearly the same as those for occultation of the spacecraft viewed from a point in the direction of the Sun equidistant from the planet as the apex of the shadow cone.

To develop specific eclipse conditions, let Ds be the vector from the spacecraft to the Sun and let Hp be the vector from the spacecraft to the center of the planet. Three quantities of interest are the angular radius of the Sun, ps, the angular radius of the planet, pp, and the angular separation, 6, between the Sun and planet as viewed by the spacecraft, as shown in Fig. 3-22(a). These are given by:

The necessary and sufficient eclipse conditions are

1. Partial Eclipse:

2. Total Eclipse:

3. Annular Eclipse:

These eclipses are illustrated in Fig. 3-22.

The surface brightness of the Sun is nearly uniform over the surface of the disk. Therefore, the intensity, I, of the illumination on the spacecraft during a partial or an annular eclipse is directly proportional to the area of the solar disk which can be seen by the spacecraft. These relations may be obtained directly from Appendix A as:

1. Partial Eclipse

sinp,sin0 j

smpssmpp )J

2. Annular Eclipse

where /0 is the fully illuminated intensity, and the inverse trigonometric functions in Eq. (3-59) are expressed in radians.

The effect of a planetary atmosphere is difficult to compute analytically because the atmosphere absorbs light, scatters it in all directions, and refracts it into the shadow cone. Close to the surface of the Earth, only a small fraction of the incident light is transmitted entirely through the atmosphere. Thus, the major effects are an increase in the size of the shadow and a general lightening of the entire umbra due to scattering. The scattering becomes very apparent in some eclipses of the Moon, as seen from Earth when the Moon takes on a dull copper color due to refracted and scattered light. The darkness of individual lunar eclipses is noticeably affected by cloud patterns and weather conditions along the boundary of the Earth where light is being scattered into the umbra. The atmosphere of the Earth increases the size of the Earth's shadow by about 2% at the distance of the Moon over the size the shadow would be expected to have from purely geometrical considerations. (See Supplement to the Astronautical Ephemeris and the American Ephemeris and Nautical Almanac [1961].) Some ambiguity exists in such measurements because the boundary of the umbra is diffuse rather than sharp. If the entire 2% at the Moon's distance is attributed to an increase in the effective linear radius of the Earth, this increase corresponds to about 90 km.

In considering the general appearance of the solar system as seen by a spacecraft, we may be interested in eclipses of the* natural satellites as well as eclipses of spacecraft. In the case of natural satellites, the large diameter of the satellite will have a considerable effect on the occurrence of eclipses. This may be taken into account easily by changing the effective linear diameter of the planet. Let Rp be the radius of the planet, Rm be the radius of the natural satellite, and define the effective planetary radii Re, = Rp+ Rm and Re2=R^—,Rm. Then, when the center of the satellite is within the shadow formed by an object of radius Rel, at least part of the real satellite is within the real shadow cone. Similarly, when the center of the satellite is within the shadow cpne defined by an object of radius Rel, then all of the real satellite is within the real shadow cone; this is referred to as a total eclipse of the satellite when seen from another location. This procedure of using effective radii ignores a correction term comparable to the angular radius of the satellite at the distance of the Sun.

We may use Eqs. (3-49) through (3-52) to determine the conditions on a satellite orbit such that eclipses will always occur or never occur. Let Dp be the perifocal distance, DA be the apofocal distance, / be the angle between the vector to the Sun and the satellite orbit plane, and C and pc be defined by Eqs. (3-53) and (3-54). We define y and S by:

An eclipse will not occur in any orbit for which y>pc. An eclipse will always occur in an orbit for which 8<pc.

Planetary and Satellite Magnitudes. The magnitude, m, of an object is a logarithmic measure of its brightness or flux density, F, defined by m = m0—2.5 log

F, where m^ is a scale constant. Two objects of magnitude difference Am differ in intensity by a factor of (VlOO >^«2.51*"1 with smaller numbers corresponding to brighter objects; e.g., a star of magnitude - 1 is 100 times brighter than a star of magnitude +4. As discussed in detail in Section 5.6, the magnitude of an object depends on the spectral region over which the intensity is measured. In this section, we are concerned only with the visual magnitude, V, which has its peak sensitivity at about 0.55 pm.

Let S be the distance of an object from the Sun in Astronomical Units (AU), r be the distance of the object from the observer in AU, £ be the phase angle at the object between the Sun and the observer, and P (g) be the ratio of the brightness of the object at phase £ to its brightness at zero phase (i.e., fully illuminated). Because the brightness falls off as S~2 and r-2, the visual magnitude as a function of £ and r times S is given by:

where K(1,0) is the visual magnitude at opposition relative to the observer* (i.e., |=0) and at a distance such that rS= 1. Note that />(£) is independent of distance only as long as the observer is sufficiently far from the object that he is seeing nearly half of the object at any one time; for example, for a low-Earth satellite, the illuminated fraction of the area seen by the satellite depends both on the phase and the satellite altitude.

If the mean visual magnitude, Vq, at opposition to the Earth is the known quantity, then where D is the mean distance of the object from the Sun in AU. Values of V0 and |^(1,0) for the Moon and planets are tabulated in Table L-3.

For the planets, or other objects for which V0 or K(1,0) is known, the major difficulty is in determining the phase law, />(£). Unfortunately, there is no theoretical model which is thought to predict P(Q accurately for the various phases of the planets. Thus, the best phase law information is empirically determined and, for the superior planets, only a limited range of phases around ¿ = 0 are observed from the Earth. Although no method is completely satisfactory, the three most convenient methods for predicting the phase law for an object are: (1) assume that the intensity is proportional to the observed illuminated area, that is, P(Q=0.5(1 + cos£); (2) for objects similar in structure to the Moon, assume that the Moon's phase law, which is tabulated numerically in Table L-9, holds; or (3) for the planets, assume that the phase dependence of the magnitude for small £ is of the form V= VQ+aii, where the empirical coefficients al are given in Table L-3. For Saturn, the magnitude depends strongly on the orientation of the observer relative

V( rS, £) = K( 1,0) + 5 log(rS ) — 2.5 log P(£)

* Equation (3-63) holds only for objects which shine by reflected sunlight. Additional terms are needed if lighting is generated internally or by planetary reflections. See Section S.6 for a discussion of stellar magnitudes.

to the ring system. Because the ring system is inclined to the ecliptic, the orientation of the rings relative to the Earth changes cyclically with a period equal to the period of revolution of Saturn, or about 30 years (Allen [I973D- Additional information on planetary photometry and eclipses is given by Kuiper and Middle-hurst [1961] and Link [1969].

For objects for which no a priori magnitude is known, but which shine by reflected sunlight, we may estimate K(1,0) from the relation:

where Vq is the visual magnitude of the Sun at 1 AU, Rp is the radius of the object in AU, and g is the geometric albedo or the ratio of the brightness of the object to that of a perfectly diffusing disk of the same apparent size at £=0. For the planets, g ranges from 0.10 for Mercury to 0.57 for Uranus; it is about 037 for the Earth, although it is a function of both weather and season. Table L-3 lists the Bond albedo, A, of the planets, which is the ratio of total light reflected from an object to the total light incident on it. The Bond and geometric albedos are related by

Jo where P($) is the phase law. The quantity q represents the reflection of the object at different phase angles and has the following values for .simple objects: q= 1.00 for a perfectly diffusing disk; ^=1.50 for a perfectly diffusing (Lambert) sphere; q=2.00 for an object for which the magnitude is proportional to the illuminated area; and ^=4.00 for a metallic reflecting sphere. For the planets, q ranges from 0.58 for Mercury to about 1.6 for Jupiter, Saturn, Uranus, and Neptune.

As an example of the computation of magnitudes, we calculate the visual magnitude as seen from Earth of the S-IVB (the third stage of the Saturn V rocket) during the first manned flight to the Moon, Apollo 8. The S-IVB which orbited the Moon with the Command and Service modules and several miscellaneous panels, was a white-painted cylinder approximately 7 m in diamter and 18 m long. We assume that the overall Bond albedo was 0.8 because it was nearly all white paint, that q= 1.5 corresponding to a diffuse sphere, and that R/)='6 m=4x I0~n AU, corresponding to the radius of a sphere of the same cross section as the S-IVB viewed from the side. Therefore, the geometric albedo is 0.8/1.5ss0.5. From Eq. (3-65), we calculate ^(1,0) as K(1,0)= -26.7 + 52.0 + 0.7= +26.0.

During the time of the Apollo 8 flight, the angle at the EaqJi between the spacecraft and the Sun was about 60 deg; therefore, ¿»120 deg. Ifcwe assume that the intensity is proportional to the illuminated area, then />=0.5(1 +cos 120") = 0.25. Setting S= 1 AU and r= 100,000 km = 6.7x 10~4 AU for observations made en route, we find from Eq. (3-63) that the visual magnitude will be approximately V= +26.0-15.9+1.5= »+12. Thus, the S-IVB should be about magnitude + 12 at 100,000 km, dropping to magnitude +14.5 at the distance of the Moon.

The observed magnitudes are in general agreement with this [Liemohn, 1969], although in practice the actual brightness fluctuates by several magnitudes because of the changing cross section seen by the observers, bright specular reflections from windows or other shiny surfaces, and light scattered by exhaust gases during orbit maneuvers.

The visibility of both natural and artificial satellites is a function of both the magnitude of the object itself and its contrast with its surroundings. As illustrated in Fig. 3-26, spacecraft which are orbiting planets are most easily seen when the subsatellite region is in darkness but the spacecraft itself is still in sunlight. Thus, Earth satellites are best seen just after sunset or just before sunrise. Spacecraft orbiting the Moon have the greatest opportunity of being seen when they are not over the disk of the Moon or when they are near the terminator (the boundary between the illuminated and unilluminated portions) above the dark surface of the Moon as seen by the observer.

Fig. 3-26. Best viewing conditions for satellite (either from planet's surface or space) occur when the satellite is in sunlight over unilluminated surface.

References

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2. Baker, Robert M. L., Jr., Astrodynamics, Applications and Advanced Topics. New York: Academic Press, 1967.

3. - and Maud W. Makemson, An Introduction to Astroefynamics. New

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4. Battin, Richard H., Astronautical Guidance. New York: McGraw-Hill, Inc., 1964.

5. Dugas, René, A History of Mechanics, translated into English by J. R. Maddox. Neuchatel, Switzerland: Éditions du Griffon, 1955.

6. Escobal, Pedro Ramon, Methods of Orbit Determination. New York: John Wiley and Sons, Inc., 1976.

7. Girvin, Harvey F., A Historical Appraisal of Mechanics. Scranton. PA.: International Textbook Co., 1948.

8. Glasstone, Samuel, Sourcebook on the Space Sciences. Princeton, N.J.: D. Van Nostrand Company, Inc., 1965.

9. Herrick, Samuel, Astrodynamics (2 volumes). London: Van Nostrand Reinhold Company, 1971.

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11. Kaplan,.Marshall H., Modern Spacecraft Dynamics and Control. New York: John Wiley and Sons, Inc., 1976.

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13. Kuiper, G. P. and B. M. Middlehurst, editors, Planets and Satellites, Vol. Ill of The Solar System. Chicago: University of Chicago Press, 1961.

14. Liemohn, Harold B., "Optical Observations of Apollo 8," Sky and Telescope. Vol. 37, p. 156-160. 1969.

15. Link, F., Eclipse Phenomena. New York: Springer-Verlag, 1969.

16. McCuskey, S. W., Introduction to Celestial Mechanics. Reading, MA: Addi-son-Wesley Publishing Company, Inc., 1963.

17. O'Neill, Gerard K., testimony in Hearings Before the Subcommittee on Space Science and Applications of the Committee on Science and Technology, U.S. House of Representatives, U.S.G.P.O., Washington, D.C., 1975.

18. Roy, Archie E., The Foundations of Astrodynamics. New York: The Macmillan Company, 1965.

19. Ruppe, Harry O., Introduction to Astronautics (2 volumes). New York: Academic Press, 1966.

20. Thomson, William Tyrrell, Introduction to Space Dynamics. New York: John Wiley and Sons, Inc., 1963.

21. Watson, G. N., A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press, 1958.

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