The simplest method for determining the diameter of a planet is to measure its apparent angular diameter when the planet is at a

FIGURE 4.1. Determination of the diameter of

Mars.

FIGURE 4.1. Determination of the diameter of

Mars.

known distance from Earth. In figure 4.1, for example, in which the size of the planet is greatly exaggerated for clarity, the angular diameter is indicated by 0. If D is the distance from the planet to Earth and 6 is expressed in radians (1 radian=57.296 degrees of arc), then the linear diameter d of the planet is given by d = 0XD

where d and D are in the same units of length. The angle 6 is determined by visual observation in a telescope or by measurement of a photographic image, and the distance D is derived from the position of the planet in its orbit at the time the angle is measured.

Because the angular diameter varies with the distance of the planet, it is the usual practice to refer the value to a standard distance. This standard distance is the astronomical unit; that is, the average distance of Earth from the Sun. The reduced angular diameter referred to the astronomical unit is equal to 6XD/AU (or dfAU), where AU, the astronomical unit, is 149.6 million kilometers (92.95 million miles). The linear diameter of the planet, expressed in astronomical units, is then equal numerically to the reduced angular diameter in radians.

The procedure described above is simple in principle, but it has proved difficult to put into practice for Mars, for reasons which will be apparent shortly. Consequently, a more indirect method has been developed for this planet. It involves observations on the ap parent path, as a function of time, of a characteristic surface marking as the planet rotates.

Apart from systematic errors in measurement, arising from instrumental problems and from conditions in Earth's atmosphere, which apply to all planets, there is a peculiar difficulty in determining the angular diameter of Mars. In 1925, W. H. Wright, at the Lick Observatory in California, noted that the size of a photographic image of Mars varied with the color of the light in which the image was obtained.

A transparent blue filter allows blue light to pass through, but it will absorb, and hence stop, light of longer wavelength, such as green, yellow, orange, and red. A red filter, on the other hand, will absorb light of shorter wavelength, namely, blue, green, orange, and yellow, but will transmit red light. By the use of a series of colored filters, it is possible to obtain a photographic image of Mars in light of a particular color; that is, of a particular wavelength. Wright found that the angular diameter of Mars as determined from an image obtained in blue light was about 3 percent larger than in red light. This dependence of the apparent diameter of Mars on the wavelength of the light in which the planet is observed is called Wright's phenomenon or the Wright effect.

Although some astronomers have doubted the reality of the Wright effect, it seems that, in part at least, it is not merely caused by instrumental errors or by variations in the sensitivity of the photographic plates. A possible contributory factor is that in red light the brightness of the Martian surface appears to diminish outward from the center to the edge of the disk, a phenomenon called limb darkening (p. 44). On the other hand, in blue light, there is limb brightening, with the intensity increasing outward from the center of the disk. The limb brightening in blue light is probably caused by the scattering (or reflection in random directions) of sunlight by the Martian atmosphere (ch. V). As a result of the limb darkening in red light and the brightening in blue light, the photographic image would appear larger in the latter case. This, however, may not be sufficient to account completely for the Wright effect.

Wright and others have suggested that the phenomenon is caused by differences in scattering of light of different wavelengths at high altitudes, for example, about 100 kilometers (62 miles), in the Martian atmosphere. But the density of the gas molecules at such an altitude is now thought to be too small to account for the Wright effect. Another suggestion is that a layer of fine dust particles high in the atmosphere causes the preferential scattering of blue light. Such a layer, which does not app sar to be too probable, lias been postulated to explain the blue-haze phenomenon described more fully in chapter VII. Recently, the possibility has been mentioned that the Wright effect is not a Martian phenomenon, but arises in part from conditions in Earth's atmosphere (p. 150).

No matter whether the Wright effect is real or illusory, and regardless of its cause, the fact remains that photographic (and visual) images seem to give different diameters for Mars in light of different wavelengths. Which wavelength, if any, then gives the true diameter? It has been generally accepted that images of Mars in red light give the best approximation to the actual size of the planet. Even if this were so, however, there is still a subjective problem in interpreting the telescopic images; it is difficult to determine, either by the eye or even with instruments, the precise boundary of the small bright disk of Mars. Consequently, the estimation of the angular diameter is bound to involve an element of uncertainty.

The method of determining the diameter of Mars based on measurement of the displacement of a characteristic feature during the rotation of the planet would seem to offer the prospect of improved accuracy, at least in principle. This turns out not to be the case in practice, however. There are very few, if any, sharply distinguishable markings on the surface the positions of which can be observed precisely enough to make possible an accurate calculation of the radius of Mars.

Like Earth, the planet Mars is not exactly spherical, but is somewhat flattened at the poles. In other words, the polar diameter is less than the equatorial diameter. This is shown in the section in figure 4.2 which is not drawn to scale; rp is the polar radius, that is, half the polar diameter, and re is the equatorial radius, or half the equatorial diameter. Provided its interior is not completely rigid, any body, like Earth or Mars, which rotates at a relatively high speed is expected to have a larger diameter in the equatorial direction because of the centrifugal effect of the rotation. The force in the outward direction caused by this effect is a maximum at the equator of a rotating mass. Because the equatorial and polar diameters of Mars are different, they must, of course, be measured

separately. The magnitude and significance of the difference will be discussed later in the chapter, but the results of the measurements will be given here.

In 1964, G. de Vaucouleurs in the United States published an extensive critical review of the results of determinations of the equatorial and polar diameters of Mars dating back to 1879. He concluded that the equatorial angular diameter of Mars, adjusted to an Earth-Mars distance of 1 AU, was between 9.285 and 9.415 seconds of arc. Because 1 degree of arc contains 36C0 seconds, 1 radian is equivalent to 57.296 X 3600 = 206 265 seconds. The linear equatorial diameter of Mars in astronomical units is then obtained upon dividing the angular diameter in seconds by 206 265. To derive the diameter in kilometers, the result is multiplied by 149.6 million.

The equatorial diameter of Mars is thus found to lie between 6723 and 6828 kilometers. By taking the various factors into consideration, de Vaucouleurs suggested a rounded figure of 6750 kilometers (or 4190 miles) for the equatorial diameter. In the same way, a value of 6700 kilometers (4160 miles) was suggested for the polar diameter. The mean diameter of the planet is thus approximately 6720 kilometers (4170 miles) and the average radius is about 3360 kilometers.

The trajectory of the Mariner IV spacecraft in the vicinity of Mars provided another means for obtaining an indication of the radius of the planet. By timing the radio signal from Mariner IV just before the spacecraft went behind the planet and again when it reappeared on the other side (fig. 5.6), it was possible to calculate the distance of the closest approach of the radio beam to the center of gravity (or center of mass) of the planet. The values found were 3384 kilometers for the distance when the radio signal was cut off (at a latitude of 60° N) and 3379 kilometers when it reappeared again (at a latitude of 50° S). The corresponding diameter, which should be less than the equatorial diameter, would then be about 6760 kilometers.

It appears, therefore, that the values of the equatorial and polar diameters suggested by de Vaucouleurs may be too low, and that a more reliable value for the average radius of Mars is approximately 3380 kilometers (2100 miles). For purposes of comparison, it may be noted that the average radius of Earth is 6371 kilometers (3959 miles). Consequently, the radius of Mars is a little more than half that of Earth. The volume of Mars, which depends on the cube of its radius, is thus approximately one-seventh the volume of Earth.

The flattening, also sometimes referred to as the ellipticity or oblateness, of a planet is defined by where re and rp are the equatorial and polar radii, respectively. Although the radii of Mars determined by individual observers have varied significantly, as seen above, the Battenings derived from the measurements have been in good agreement. The average value obtained from studies of photographic and visual telescopic images of the planet is given as approximately 0.011. This is called the optical flattening because it is determined from optical measurements. A flattening of this magnitude would mean that the equatorial radius of Mars is 36 kilometers (22 miles) greater than the polar radius.

There is another method for evaluating the flattening of a planet, depending on the motion of a satellite (or moon) in an orbit that is not too far distant from the parent planet. For Mars, the required conditions are satisfied by its two satellites. The orbits of these satellites are not quite in the same plane as the Martian equator, but each orbit crosses the equatorial plane at two points, called the nodes of the orbit. The line, lying in the equatorial plane, that joins the two nodes is referred to as the line of the nodes.

If the planet were a uniform sphere, the line of the nodes of a satellite's orbit would remain stationary in space. But if the planet is flattened, the orbit of the satellite is continuously disturbed in such a way that the line of the nodes rotates in the equatorial plane. From the rate of rotation, the flattening of the planet can be calculated. This is called the dynamical flattening.

From a study of the orbits of the Martian satellites, the dynamical flattening of the planet has been calculated to be 0.00525, which is about half the optical flattening. There has been much discussion, but no conclusion, concerning this discrepancy between the dynamical and optical flattening of Mars. Because the telescopic image may depend on the possible effects arising from the scattering of light in the Martian atmosphere, it was thought that the results of the optical method might be incorrect.

In answer to this, however, reference has been made to the value reported in 1927 by R. J. Trumpler, based on observations at the Lick Observatory during the 1924 apparition. By tracking a conspicuous feature on the surface of Mars as the planet rotated, Trumpler found the flattening to be 0.011. This result, which is essentially independent of scattering in the Martian atmosphere, is the same as the optical flattening obtained by direct measurement of the equatorial and polar diameters (or radii). But the agreement is not altogether convincing because the indirect method used by Trumpler is not considered to be very accurate.

There is a possibility, however, that both the optical and dynamical flattening values may be correct. The optical flattening is determined from the actual (linear) dimensions of the planet, but the dynamical flattening depends on the mass distribution. If the mass is not distributed uniformly throughout the interior of the planet, then the optical and dynamical flattenings may not be the same. It is of interest to note in this connection that the flattening of Earth as determined from direct measurements on the surface is essentially identical with the dynamic flattening calculated from the orbital motions of artificial satellites.

The American scientist H. C. Urey suggested in 1950 that the difference between the optical and dynamic flattening of Mars could be explained if there were a belt of mountains around the equator, accompanied by a compensation of subsurface material of lower density than average. This compensation effect, known as isostasy, leading to a uniform pressure below a certain depth in the interior, is believed to occur on Earth. In the situation postulated by Urey, the mass distribution, as it affected the motion of a satellite, would not correspond with the actual linear dimensions of the planet. The optical flattening would then be larger than the dynamical value.

In order to account for the actual difference between the two values of the flattening, the elevation of the high land around the equator would have to be about 15 kilometers (49 000 feet). Consequently, in 1952, Urey withdrew his original suggestion because, as he said, "such high plateaus or mountains . . . even in tropical regions should be covered by snow as on Earth, and . . . [this is] not obs.erved."

It will be seen in chapter VI, however, that elevation differences on the order of 10

to 15 kilometers on Mars are not improbable. And, furthermore, the ground temperature variations with altitude, which lead to the permanent snow cover on high terrestrial mountains, may be quite different on Mars. Thus, in 1966, C. Sagan and J. B. Pollack of the Smithsonian Institution Astrophysical Observatory, in Cambridge, Mass., stated that in their view: "While the entire effect [i.e., the difference between dynamic and optical flattening] may not be due to equatorial elevations ... we believe that Urey's explanation . . . now has some measure of observational support."

A theory, similar in principle but different in detail to that of Urey, was proposed in the United States in 1962 by D. L. Lamar. He suggested that the elevation of the Martian surface increases gradually from the poles to the equator, and that this increase is compensated isostatically by adjustment in the thickness of the planet's crust; that is, in its outermost layer (p. 67). In Lamar's opinion there is an equatorial bulge, but not necessarily high mountains or plateaus in the equatorial region. Because of the isostatic compensation, the dynamic ellipticity would be less than the optical value. But no explanation has been offered for the existence of such an isostatically compensated equatorial bulge on Mars.

Apart from the suggestion, which cannot be ruled out, that the flattening discrepancy is caused by observational errors in the optical data, an elevated equatorial belt of some kind seems to be the only satisfactory way out of the dilemma. It would appear, however, that a complete solution of the problem must await the time when more reliable information concerning both the linear dimensions and mass distribution of Mars can be obtained from spacecraft passing close by the planet and from artificial satellites that orbit around it.

Two characteristics of a planet, namely, its average density and the force of gravity at the surface, are related to the mass and the radius. Because the mass of Mars is about one-tenth and the volume one-seventh of the corresponding properties of Earth, it is evident that the average density (mass per unit volume) of Mars must be less than Earth's density. The actual average density of Mars, expressed in grams per cubic centimeter (g/cm3), can be calculated from the mass and radius given above. The volume is equal to %77-r3, where r is the average radius of the planet in centimeters (3380 X105). The volume of Mars is found to be 1.62 XI026 cubic centimeters. If the mass is taken to be 6.42 XI028 grams, then

Because neither the mass nor the radius of Mars is known with any degree of accuracy, it is evident that the calculated density is equally uncertain. Values ranging from 3.85 to 4.25 g/cm3 have, in fact, been reported. All that can be said at present, therefore, is that the average density of Mars is in the vicinity of 4 g/cm3. For comparison it may be noted that the average density of Earth is 5.52 g/cm3.

The gravitational acceleration at a specified point on the surface of a planet is given by the expression

Accel eration=^4^ r2

where G is the universal constant of gravitation (6.67 XI0"8 in the centimeter-gram-second system of units), M is the mass of the planet, and r is the radius of the planet at the point under consideration. The radii of flattened bodies, like Mars and Earth, vary with the latitude, and so, consequently, also do the gravitational accelerations. For the present purpose, however, it is sufficient to determine an average value for Mars. Thus, by using the mass (in grams) and average radius (in centimeters) given above, it is found that

Gravitational acceleration

Like the density, the gravitational acceleration at the surface of Mars cannot be determined precisely because of the uncertainties in the mass and radius involved in the calculation. Published values range from about 360 to 390, and an approximate mean is 375 cm/sec/sec for the average acceleration due to gravity at the Martian surface. The acceleration is somewhat greater at higher latitudes, that is, nearer to the poles, and smaller at lower latitudes, toward the equator. The average value of the gravitational acceleration at Earth's surface is 983 cm/sec/sec. Hence, the gravitational force at the surface of Mars is 0.38 (almost two-fifths) of the force on the same mass on Earth.

The length of a day depends on how the day is defined; for practical purposes, the sidereal day and the solar day may be distinguished. The sidereal day of a planet is the exact time required for the planet to make a complete rotation about its axis; it is also referred to as the period of rotation of the planet. For Mars, this period is determined by timing the passage of conspicuous markings on the planet past a distant fixed star. The accepted value of the period of rotation (or sidereal day) for Mars is 24 hours 37 minutes 22.67 seconds. The hours, minutes, and seconds refer to time intervals on Earth, which are defined in the manner explained below.

Timekeeping on Earth is based on the solar day, rather than on the sidereal day. Because Earth cannot be continuously observed from a great distance, as Mars is, the period of rotation must be obtained in a different manner. It is equal to the time interval between two consecutive transits (or crossings) by a distant fixed star of the meridian at any given point on Earth. The meridian is an imaginary circle in the sky that passes directly overhead and extends in a north-south direction.

The solar day, on the other hand, is the interval between two consecutive meridian transits of the Sun; that is to say, it is the interval between two consecutive noons. The solar day is longer than the sidereal day because the planet moves in an orbit around the Sun. The difference may be explained with the aid of figure 4.3. An observer at position a in figure 4.3, I-A, sees a distant fixed star

cross the meridian, and exactly one sidereal day later Earth has rotated to position I-B (and has moved in its orbit) when the fixed star is again seen to cross the meridian. In figure 4.3, II-A, the Sun makes a transit of the meridian at a, and the next transit, a solar day later, is indicated by b in figure 4.3, II-B. In order to reach the point b where the transit of the Sun occurs, Earth will have rotated the additional amount ab. Consequently, the solar day on Earth is longer than the sidereal day. The same is true for Mars, although the difference in length is not the same.

It can be seen from figure 4.3 that the reason the sidereal and solar days for a given planet are different is that the planet has moved in its orbit around the Sun. If the planet simply rotated in a stationary position, the two kinds of day would be the same. The difference in the lengths depends on the distance the planet travels in its orbit between two successive noons. Since the average orbital velocity of Mars (24.1 km/sec) is less than that of Earth (29.8 km/sec), but the days are approximately the same length, Mars moves a smaller distance in its orbit than does Earth in this period. The difference between the sidereal and solar days on Mars, therefore, is somewhat less than on Earth.

The length of a sidereal day is essentially constant, but the solar day is not. It varies with the position of the planet in its orbit. The average of the lengths of the solar day throughout the year is called the mean solar day, and the mean solar day on Earth is defined as exactly 24 hours, with each hour having 60 minutes and each minute 60 seconds. The mean solar day is 3 minutes 55.91 seconds longer than the sidereal day on Earth; the latter is 23 hours 56 minutes 4.09 seconds.

On Mars, the solar day, the period between two successive transits of the meridian by the Sun, is 2 minutes 12.56 seconds longer than the sidereal day of 24 hours 37 minutes 22.67 seconds. The length of the solar day on Mars is consequently 24 hours 39 minutes 35.23 seconds. The average conventional day on Mars, the time interval between one noon and the next, is thus only about 40 minutes longer than the average terrestrial (solar) day.

From the known approximate equatorial radius of Mars and the period of rotation, the speed of rotation at the equator can be calculated. It is equal to 2ir times the radius, that is, the circumference of the planet at the equator, divided by the period of rotation. The value obtained in this manner is 0.26 km/sec (0.16 mps), compared with 0.47 km/sec (0.29 mps) for Earth.

An attempt to measure the magnetic field of Mars was made by means of instruments on Mariner IV. But even at its closest approach of 9846 kilometers (6118 miles) to the planet, the magnetometer on the spacecraft detected no increase in the magnetic field strength from the value in interplanetary space. From this observation it has been concluded that the strength of the Martian magnetic field cannot be more than a very small fraction (less than 0.03 percent) of the magnetic field of Earth.

Confirmation of the extremely small magnetic field of Mars was obtained from the failure of instruments on Mariner IV to detect trapped charged particles in the vicinity of the planet. Earth has an extensive radiation (Van Allen) belt in which electrically charged particles, namely positively charged hydrogen ions (protons) and negatively charged electrons, originating in the Sun, are trapped and confined by the magnetic field. The same charged particles, although in somewhat smaller numbers, are undoubtedly available near Mars, and the absence of a radiation belt can be ascribed only to the very small (or zero) magnetic field.

According to current views, there are two requirements for a planet to have a significant magnetic field. One is that the planet should rotate fairly rapidly, and the other is that it should have a central core of a liquid that is able to conduct electricity. Earth's core is believed to consist mainly of the heavy metals, iron and nickel. Mars satisfies one of the requirements, because it rotates quite rapidly. The absence of an appreciable magnetic field would imply that it either does not have a liquid-metal core or that such a core, if it is present, must be small in proportion to the size of the planet. The problem of the Martian core will be taken up shortly in the section dealing with the internal structure of the planet.

The albedo, from the Latin albus meaning white, is a measure of the ability of a body or material to reflect light. There are various ways of defining the albedo of a planet, and the one in common use is that suggested by W. G. Bond in the United States in 1861 and adopted by H. N. Russell in 1916. It is known as the Bond (or Russell-Bond) albedo, or as the spherical albedo. It is defined as the fraction of the solar radiation (of a ghcn wavelength) falling on a planet and being reflected in all directions. The Bond albedo A can be divided into two factors, the gm-metric albedo, p, and the phase integral, q, so that

A=pq

The geometric albedo is a measure of the sunlight (of a particular wavelength) that is reflected by the planet in the direction of an observer on Earth. It is determined from the measured brightness (or luminance) of the planet at full phase, referred to standard distances from the Sun and Earth. The phase integral can be calculated, in principle, from the observed variation of the brightness with the phase angle (p. 43). For Mars, this variation is relatively small and so the phase integral is calculated on the basis of certain assumptions.

The dependence of the Bond albedo, obtained in this manner, on the wavelength (or color) of the light is shown in figure 4.4. The albedo of Mars is seen to increase from quite small values (0.05 or less) in the ultraviolet and blue regions of the spectrum to more than 0.3 in the red and infrared. It is this increase which accounts for the reddish appearance of the planet. The Bond albedo of Mars in visible light is taken to be about 0.17. The foregoing values are averages which apply to the planet as a whole. The albedos of the bright areas are higher and those of the dark areas are lower than average (ch.

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