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198Lapj. . .251 . .2 66P the Astrophysical Journal, 251:266-270 66P .2 . .251 The Astrophysical Journal, 251:266-270, 1981 December 1 . © 1981. The American Astronomical Society. All rights reserved. Printed in U.S.A. 198lApJ. PHOTOSPHERIC FLOW AND STELLAR WINDS E. N. Parker Department of Physics and Department of Astronomy and Astrophysics, University of Chicago Received 1981 February 9; accepted 1981 June 15 ABSTRACT The dwarf main-sequence stars, such as the Sun, have strongly bound coronas whose extended temperatures cause them to expand and to form tenuous stellar winds. There is a slight mean upward motion of the gas in the photosphere of the star to replenish the mass lost through the coronal expansion. Formal solution of the time-dependent hydrodynamic equations provides an illustration of the effect of the photospheric velocity on the corona and on the wind at large distances. Using the Sun as an example, the effect of blocking the upward flow in the photosphere is calculated. The solar wind velocity would be reduced by a few cm s “1, and the solar wind density would decline about 1 % over the next three centuries. It appears that the photospheric flow has no directly sensible effects on the wind. Subject headings: hydrodynamics — stars: atmospheres — stars: winds I. INTRODUCTION thermal energy is of the order of 400 eV per atom (at 6 The theoretical hydrodynamic expansion of stellar 2 X 10 K). The pressure scale height A is a corre- spondingly small fraction of the solar radius, with coronas to form stellar winds is generally illustrated 5 through stationary (8/dt = 0) solutions of the radial A ~ 1.2 x 10 km Rö. The gas is near hydrostatic hydrodynamic equations (see Parker 1958, 1963, 1964, equilibrium and the density declines rapidly with height. 1965; Hundhausen 1972; Hartmann and MacGregor Coronal temperatures extend far out from the Sun into 1980; Summers 1980a, b, and references therein). The space (Kohl et al. 1980) evidently as a consequence of expansion velocity v(r) is uniquely determined by the extended coronal heating and, in any case, as a con- temperature T(r) (if the temperature is given); more sequence of the extraordinarily high thermometric con- generally, both p(r) and T(r) are determined by ductivity of the hot, tenuous coronal gas (Chapman 1957). The characteristic conduction scale height of the the coronal heating mechanism. The dynamical expan- 2/7 sion has been shown to be stable (Carovillano and King temperature field (T oc l/r in a static tenuous corona) 1966; lockers 1968). exceeds that of the gravitational potential (O oc 1/r), so It is shown below that the calculations are readily that the strong gravitational binding near the Sun gives extended to time-dependent solutions in which the ex- way to free thermal expansion beyond several solar radii. pansion velocity in the photosphere is a free parameter. In The rapid escape of gas from the outer corona, beyond particular, we examine the consequences of blocking the 5 Rq, relieves slightly the overburden on the lower flow at the photosphere while maintaining coronal tem- corona, so that a tiny upward expansion of the dense peratures T(r) above. The solution for vanishing flow in lower corona (and chromosphere and photosphere) supplies the loss of gas from the tenuous outer corona. the photosphere is of particular interest, in view of the 12 x assertion by Cannon and Thomas (1977) that the velocity The observed mass loss (10 g s" ) from the solar corona 1 has a significant cooling effect on the corona, consuming in the photosphere is the prime mover of the stellar wind. 27 The formal solution shows that the consequences of some 2 x 10 ergs s" ^ Radiation losses are less, at least blocking the photospheric flow are so small as to be in the quiet corona and coronal holes from which the undetectable. principal expansion occurs; downward thermal conduc- This paper examines the effect of the photospheric tion from the corona into the chromosphere may be somewhat larger. In any case, the corona consumes outflow on the expansion of the strongly bound coronas 27 -1 of dwarf main-sequence stars, such as the Sun, wherein energy at a rate of several times 10 ergs s . It was the general physical properties are fairly well known from pointed out many years ago that the corona must be direct observation. The distinctive features of the corona heated in some way by the mechanical work performed of the dwarf main-sequence star are (1) the strong gravi- by the convection below the photosphere, although the tational binding and (2) the direct deposition of energy to specific mechanisms are still far from clear. The degree to produce the high temperature. The gravitational binding which gas circulates back and forth between the corona energy is 5 or more times the thermal energy. In the low and the photosphere is not known precisely (but see the corona of the Sun, for instance, the binding energy is study by Foukal 1978). It probably exceeds the mean out- approximately 2000 eV per hydrogen atom, while the ward flow through the photosphere into the corona (the outflow being the main point of interest in the present 1 See also the remarks by Andriesse (1979). writing). 266 © American Astronomical Society • Provided by the NASA Astrophysics Data System 66P .2 . .251 PHOTOSPHERIC FLOW AND STELLAR WINDS 267 . The mean outflow of gas, to replenish the mass lost in corona, and finally the stellar wind beyond the corona. the continual expansion of the corona, is readily They draw an analogy between the outflow from a star _ 1 198lApJ. computed. A typical, quiet-day solar wind of 300 km s and a supersonic wind tunnel (wherein gas is pumped at and 5 hydrogen atoms cm“3 (1.5 x Ifr8 atoms cm“2 s) high pressure through a Laval nozzle, expanding at observed at the orbit of Earth implies a loss to the Sun of supersonic speed into a large evacuated chamber). Our 0.7 x 1012 g s“ \ if we make the simple assumption that difficulty in undertaking their analogy stems from (1) the the wind has about the same strength everywhere around Mach number of 3 x 10“6 in the chromosphere, and the Sun. Conservation of matter implies, then, that the 10“ 2 in the low corona, which both seem a trifle low to mean outward flow velocity at a radial distance r from the produce shocks, and (2) our dogmatic acceptance of the center of the Sun, where the number density of the gas is first law of thermodynamics. We have difficulty in under- N(r), is standing the “self-heating” of a flow of gas. Whatever 12 2 validity their ideas may have elsewhere, they do not 7 x 10 /Ko\ v(r)* cm s (i) appear to have anything to do with the Sun or similar W) W stars. To investigate the analogy proposed by Cannon and 8 In the low corona, where r = RQ and N ^ 10 atoms Thomas (1977), note that the supersonic wind tunnel cm“3, the mean outward flow velocity is approximately involves a flow of gas under nearly adiabatic conditions. 0.7 km s“1, with a Mach number of about 3 x 10“3. In The energy input to the wind tunnel is just the enthalpy in the chromosphere, where N ~ 1012 atoms cm“3, the the gas pumped in at the high-pressure end. The kinetic mean velocity is 7 cm s“1 with a Mach number of about energy per molecule of the supersonic gas at the output 3 x 10“ 6. In the photosphere where iV ^2 x 1017 atoms does not exceed the enthalpy per molecule at the input. cm”3, the mean outward velocity is 3 x 10”5 cm s“1 The solar wind is a different situation altogether. The with a Mach number of 3 x 10“11. input enthalpy at the photosphere is less than 1 eV per In view of the small mean outward velocity in the lower atom, while the kinetic energy plus potential energy at the atmosphere, it is of interest to calculate the mean travel output end in interplanetary space is approximately time from the photosphere to the outer corona and the 400 eV + 2000 eV = 2400 eV per atom. The traditional solar wind. The travel time is most easily calculated from concept of conservation of energy would seem to require conservation of mass, noting that the mass above the some other source beyond the input enthalpy of 1 eV per photospheric level where iV = 2 x 1017 atoms cm“3 and atom. Indeed, in keeping with conventional terminology, T & 5600 K is approximately 10 g cm”2 or 6 x 1023 g it would not be unfair to say that the other source—which total. This mass supplies the mass loss in the solar wind is conventionally called the “coronal heating mechan- for a period of about 1012 s or 3 x 104 years. The trip ism ” and supplies about 2400 eV per atom—is the basic from the photosphere to the chromosphere is a leisurely cause, or prime mover, of the solar wind. Without direct one, requiring some 30 millenia, with Mach numbers of energy input into the corona, by some potent, inde- 10“ 6 and less. The mass above the chromospheric level, pendent heating mechanism, there would be no solar where N = 1012 atoms cm“3 and T & 104 K, is 10“4 g wind. Coronal heating to 1-2 x 106 K appears to be both cm“2, or 6 x 1018 g overall, which is replenished every a necessary and sufficient condition for the existence of 107 s.
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