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1961Apj. . .133. .6575 PHYSICAL and ORBITAL BEHAVIOR OF .6575 PHYSICAL AND ORBITAL BEHAVIOR OF COMETS* .133. Roy E. Squires University of California, Davis, California, and the Aerojet-General Corporation, Sacramento, California 1961ApJ. AND DaVid B. Beard University of California, Davis, California Received August 11, I960; revised October 14, 1960 ABSTRACT As a comet approaches perihelion and the surface facing the sun is warmed, there is evaporation of the surface material in the solar direction The effect of this unidirectional rapid mass loss on the orbits of the parabolic comets has been investigated, and the resulting corrections to the observed aphelion distances, a, and eccentricities have been calculated. Comet surface temperature has been calculated as a function of solar distance, and estimates have been made of the masses and radii of cometary nuclei. It was found that parabolic comets with perihelia of 1 a u. may experience an apparent shift in 1/a of as much as +0.0005 au-1 and that the apparent shift in 1/a is positive for every comet, thereby increas- ing the apparent orbital eccentricity over the true eccentricity in all cases. For a comet with a perihelion of 1 a u , the correction to the observed eccentricity may be as much as —0.0005. The comet surface tem- perature at 1 a.u. is roughly 180° K, and representative values for the masses and radii of cometary nuclei are 6 X 10" gm and 300 meters. I. INTRODUCTION Since most comets are observed to move in nearly parabolic orbits, questions concern- ing their origin and whether or not they are permanent members of the solar system are not clearly resolved. Because of the motion of the solar system in our Galaxy, the ab- sence of any significantly hyperbolic orbit casts considerable doubt on an interstellar origin, but the existence of approximately parabolic and sometimes slightly hyperbolic orbits must be explained if comets are to be considered as long-time members of the solar system. Comets are visible only while near the sun (<8 a.u.), and their observed orbital eccentricities have been computed from only a small fraction of their orbits. The possibility exists that seemingly parabolic comets actually approach the sun along ellipti- cal trajectories and have their apparent eccentricity increased by perturbations as they pass close to the sun. Investigations (Sinding 1948; Van Woerkem 1948) of the effect of Jupiter’s perturba- tion on cometary orbits revealed that the resultant change in the reciprocal major axis or reciprocal of the aphelion,1 1/a, on an average passage is ±0.00053 and ±0.00071 a.u.“1 for comets with perihelia of 1 and 4.5 a.u., respectively. Van Woerkem has shown that the cumulative effect of this perturbation is such that after a million years the aphelia distribution of the comets should be constant for equal intervals of 1/a, and the aphelia will range from 25 to 100 a.u. for comets with perihelion distances between 4.5 and 1 a.u. This theory is not in agreement with observations which show nearly all comets to have negligible 1/a. Oort (1950a, b) has concluded that the great majority of the comets are observed on only their first or first few passages about the sun. He has given evidence of a cloud of hundreds of millions of comets surrounding the sun at a ra- dius of the order of 50000-150000 a.u. which become visible as their perihelia are lessened by occasional collisions with passing stars. * Part of this work was completed in partial fulfilment of the M.A. degree in physics for R. E. Squires 1 The major axis of a nearly parabolic comet and the aphelion differ by less than one part in 105 and are not distinguished in this communication. 657 © American Astronomical Society • Provided by the NASA Astrophysics Data System .6575 658 ROY E. SQUIRES AND DAVID B. BEARD .133. It is reasonable to suppose that virtually all the visible, nearly parabolic comets do indeed survive but a few passages about the sun because of their rapid evolution of mat- ter in the solar heat. It has been shown (Whipple 1950,1951) that even the old, extremely 1961ApJ. dim comets in Jupiter’s family, which have lost much of their volatile matter in repeated apparitions, probably lose up to 1 per cent or more of their mass in each apparition. Whipple has amassed considerable evidence in support of a comet model composed of a low-density, rather cobwebby conglomeration of ices of molecules such as water, am- monia, and possibly methane, carbon dioxide, and cyanogen, mixed with meteoric ma- terial, all initially at extremely low temperatures (T < 50° K). When the nucleus is warmed by solar heat, the comet is made visible by matter evaporating from the surface facing the sun, leaving an outer crust of thermally insulating meteoric material behind in the case of the periodic comets. Because of its loose structure, the cometary nucleus is subject to slow rotations only, resulting in the evolution of material predominantly in the direction of the sun. The far brighter, nearly parabolic comets lacking a thick outer crust of insulating material must lose much more of their mass through evaporation and may well become unobservable after a few passages, depending on their original size and constitution. It is the purpose of this communication to describe an investigation into the effect of rapid mass loss on the trajectories of the parabolic comets, as has been in- dicated by Hamid and Whipple (1953). II. SURFACE TEMPERATURE AS A FUNCTION OF SOLAR DISTANCE In order to obtain the differential equation for the perturbed orbit, it is necessary to know the behavior of the reaction force due to vapor pressure, FeVj as a function of solar distance, r. This force is given by the integral of the vapor pressure, ÿ, of the ices at the nuclear surface of the comet over the hemisphere exposed to the sun. Since ÿ is a func- tion of the surface temperature, T, one must first determine the dependence of T on r. This can be done by considering the conservation of energy at the surface. Because of the structure of cometary nuclei, it was assumed that the thermal con- ductivity from the surface to the interior was effectively zero. In this case the solar energy incident on the nucleus is equal to the sum of the energy reradiated and the energy evolved through the evaporation process. If Es is the total solar radiation, the total energy incident per second per unit surface area normal to the radius vector from the sun is Ei = eiE*/47rr2, where €i is the emissivity of the surface at the solar temperature. 4 The energy reradiated is €2öT , where €2 is the emissivity of the comet surface at the comet temperature and <r is the Stefan-Boltzmann constant. The energy of evaporation is given by = nmU, where n is the number of molecules evaporated per second per unit area, m is the mass of these molecules, and R is their average heat of sublimation per gram. Thus the expression for the conservation of energy at any unit area normal to the solar direction is given by -^ = e rr4 + imtf. (D 47Tf2 2( The total incident energy depends on the cross-sectional area of the nucleus, whereas evaporation and reradiation of energy occur over the entire hemispherical surface facing the sun. However, since the parts of the surface not normal to the incident radiation re- ceive less energy by a cosine factor and are at a lower temperature, the energy lost from these regions by evaporation and radiation is less. Therefore, in view of the great un- certainties in the physical properties of comet material, the comet was treated as a two- dimensional disk facing the sun. If b is the nuclear radius, this area is given by tt^2, and the force in the solar direction due to the evaporation of material from the surface of the 2 nucleus becomes simply Eev = 7r& ^. © American Astronomical Society • Provided by the NASA Astrophysics Data System .6575 659 .133. BEHAVIOR OF COMETS . To express the evaporation energy in terms of temperature, the vapor pressure is written in the form p = nmvnj where vn is the normal component of the root-mean-square 1961ApJ. thermal velocity, z), of the molecules at the surface temperature. If v is assumed to be random in direction, then vn = The mass loss per unit area per second, then, is nm — 2p/vy and the energy of evaporation becomes Eev — 2pH/v. Although the comet nucleus is a heterogeneous mixture of ices, it was assumed that water, being the least volatile constituent, would control the evaporation rate. As the comet approaches the sun, the more volatile substances on the surface evaporate away, leaving a layer of water ice, which prevents the other substances beneath the surface from being lost until more water ice is evaporated. It was assurned that for every water molecule there are, depend- ing on the relative concentrations, 7 molecules of other ices leaving the surface. Thus Eev can be rewritten The subscript w corresponds to water and x to the other molecules, and v and p are the r.m.s. thermal velocity and the vapor pressure of water. In terms of temperature, v = V(3RT/M)y where R is the universal gas constant and M is the molecular weight (water in this case). The empirical formula relating vapor pressure and temperature is of the form p = B X \0~A/T so that equation (1) becomes jH m \ tiZ* - T* 'JBXIO-VT x x 4^2 62<ri ^ViSRT/M) (3) mw J Because of the structure of the nucleus, the emissivity of the surface must be low, and hence €1 and €2 were both taken to be 0.10.
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