A special class of magnetostatic equilibria is described, which are mathematically simple and yet sufficiently versatile so as to fit any arbitrary normal magnetic flux prescribed at the photosphere. With these solutions, the corona can be modeled with precisely the same mathematically simple procedure as has previously been done with potential fields. The magnetostatic model predicts, in addition to the coronal magnetic field, the three dimensional coronal density which can be compared with coronagraph observations

Bogdan, T. J.

Low, B. C.

English

NASA Technical Reports Server

N87-20914
THREE-DIMENSIONAL MAGNETOSTATIC MODELS OF THE
LARGE-SCALE CORONA
T. J. Bogdan
B. C. Low
High Altitude Observatory
National Center for Atmospheric Research 1
INTRODUCTION
There is a relatively stable component of the large-scale corona that persists for periods of the
order of a solar rotation. This component is captured in the synoptic charts constructed from the
daily coronagraph observation of the density structures visible in Thomson-scattered light from
the photosphere (e.g., Fisher and Sime 1984). The Thomson-scattered light is optically thin and
the coronagraph images correspond to electron densities integrated along the line of sight. A long
standing need therefore exists for the development of theoretical models of the corona capable of
relating the observed integrated density to the actual three-dimensional coronal structures. It is
the magnetic field and electric current in the highly conducting medium of the corona that order
the density into various structures. Unfortunately, the coronal magnetic field cannot be measured
to any useful spatial resolution. Only the longitudinal component of the magnetic field in the
photosphere can be measured routinely with a magnetograph. An obvious approach for a theoret-
ical model to adopt is to use the measured photospheric magnetic field as an input and extrapo-
late for both the magnetic field and density in the corona. Until recently, this approach has been
feasible only with the potential field model (e.g., Newkirk and Altschuler 1970). The disadvan-
tage, as is well known, is that a potential magnetic field does not interact with the plasma and
the model predicts no density structure. The idea to use linear force-free magnetic fields with a
constant a, proposed by Nakagawa (1973), has difficulties of nonuniqueness and infinite total
magnetic energy when applied to coronal models in spherical geometry (Seehafer 1978). Non-
constant a force-free magnetic fields and general non-potential magnetic fields in static equili-
brium have so far proved to be mathematically intractable in three-dimensional geometry. In this
paper, we describe a special class of magnetostatic equilibria, which are mathematically simple
and yet sufficiently versatile so as to fit any arbitrary normal magnetic flux prescribed at the pho-
tosphere. With these solutions, the corona can be modeled with precisely the same mathematical
procedure as has previously been done with potential fields. The magnetostatic model predicts, in
addition to the coronal magnetic field, the three-dimensional coronal density which can be com-
pared with coronagraph observations. This is an ongoing project being conducted at the High
Altitude Observatory (HAO), using data from the Kitt Peak Magnetograph and several corona-
graphs operated by HAO.
THREE-DIMENSIONAL MAGNETOSTATIC MODELS
The corona is in a state of expansion (Parker 1963, Hundhausen 1972). The expansion is believed
to be negligible in the low corona, and, as a first approximation, the low corona is taken to be in
static equilibrium and described by
47r
pGMe
r 2
= 0, [1]
1The National Center for Atmospheric Research is sponsored by the National Science Foundation.
275 i_,_CED|NG p,_L_E _ NOT
https://ntrs.nasa.gov/search.jsp?R=19870011481 2020-03-20T12:03:44+00:00Z
= o, [2]
where/_, p and p are, respectively, the magnetic field, pressure and density, G is Newton's grav-
itational constant, Mo the solar mass and r the heliocentric distance. To close the equations,
we may relate the temperature to p and p through the ideal gas law and specify an equation for
steady energy balance. The balance of energy in the corona plays an important role in determin-
ing coronal structures. The detailed processes are complicated and our current knowledge about
them is very incomplete. In any case, to account for these processes would render the magntos-
tatic problem quite intractable. It therefore seems reasonable, as a first step, to ignore the energy
equation and seek solutions to equations (1) and (2) with no further constraints. Even within such
an artificial consideration, the magnetostatic problem is not readily tractable (see the review by
Low (1985a)). Equations (1) and (2) have not been treated in fully three-dimensional geometry
except for the cases of the potential magnetic field and the constant-a force-free magnetic field.
In a recent development, Low (19855) showed that equations (1) and (2) can be reduced to a
simpler problem without assuming any special geometry if the electric current density J is
assumed to flow everywhere perpendicular to the solar gravity. In this case, employing the usual
spherical coordinates, we can express the magnetic field and electric current density as
!8¢# I a¢
/_ _---_r + r O0 + [3]rsinO _¢'
J' = 41r r sinO 0¢ _l---_-r 47r r O0 Or j '
O_
where _P is an arbitrary function of _ and r. The magnetostatic problem reduces to solving
D(b
for • as a boundary value problem given 02(-_-_-,r ). The freedom to choose the form of 02 arises
from our not imposing an energy equation. From the solution _, the magnetic field is given by
equation (3) and the equilibrium pressure and density are given by
1 _(0qb ,r 1 orp = po(r)- _ _ ) + -_n f dv@(v,r), [61
P -_ po(r ) +
r 2 [ 1 1 04p 002 1 0_ 002 1 1 0 , 0_,2 1 0 . 0ep ,2)
GM® ( 47r r2sin20 0¢ 0¢ + 4rcr 2 00 00 8rr r2sin20 _rr(--_ --) 8_rr 20r(--_-) , [7]J
where p0 and P0 are the pressure and density of an arbitrary spherically symmetric hydrostatic
0_
atmosphere. The familiar case of a potential magnetic field is recovered if we set o2 -- Or ' and
then only the hydrostatic components in equations (6) and (7) remain. In general, equation (5) is
276
nonlinearandsuitablenumericalmethodsneedto bedeveloped to solve it. If we set
----"(r)Or ' [8]
where r/is given, equation (7) is linear in (I) and standard numerical methods for elliptic equations
can be employed to solve the boundary value problem. Bogdan and Low (1986) considered
several cases of given t? for which the boundary value problem is soluble analytically in terms of
classical functions. In particular if
,(, ) = (1+ [9]
where a is a constant, equation (7) admits the series solution
(I)(,,0,¢) = _n°°=l_mn =_ n (Am,(r +a)-"-'+Bm,(r +a)"}Y_"(O,f_), [101
where Amn and Bran are constant coefficients and Ynm are the usual spherical harmonics. As to be
expected, the classical spherical-harmonic expansion for the potential field is obtained if a ---_ 0.
In fact, equation (9) shows that @ -* (I) in the limit of r --+ co so that in that limit, the magnetic
field is potential. In other words, the electric current is confined to the low corona within a
heliocentric distance of the order of a . For a given non-zero a, equation (10) can he fitted with a
prescribed radial magnetic field at an inner spherical surface identified with the solar surface,
using B r _ _. An outer boundary condition is needed to determine the coefficients Am_ and
Bm_ uniquely. This outer condition may take the form of requiring the magnetic field to vanish
as r --* oo or that the magnetic field becomes purely radial at some outer spherical surface (e.g.,
Newkirk and Altschuler 1970). The procedure follows essentially the same steps as that of the
classical potential field model. However, the magnetic field is not potential for a non-zero with
the equilibrium pressure and density is given by equations (6) and (7).
DISCUSSION
We plan to generate three-dimensional magnetospheric models of the corona based on equations
(9) and (10), with an input of the photospheric normal magnetic flux from the Kitt Peak Magne-
tograph. This is a first opportunity for constructing a three-dimensional magnetostatic corona
which is based on photospheric magnetograms and can be tested against coronagraph observa-
tions. By varying the free parameter a, it will be interesting to see if we can achieve a reason-
able agreement between the density predicted by the model and the observed line-of-sight
integrated density. The model is based on several artificial assumptions and we do not have
apriori reasons to expect excellent agreement. The assumption of the electric current flowing
perpendicular to gravity is not as severe as it may appear. Such a distribution of the electric
current may result from stretching a magnetic field radially outward, starting from a potential
state, as demonstrated by the solutions described by equation (10) for a _ 0 (Bogdan and Low
1986). The outward stretch of the magnetic field may result from the buoyancy of magnetic
structures. The assumption of equation (9) is more restrictive. It requires that the presence of
electric current can be characterized by a single parameter, namely, a. The illustrative examples
in Bogdan and Low (1986) look promising, but it remains to be seen from actual modelling with
data if this is a workable hypothesis.
REFERENCES
Bogdan, T. J., and Low, B. C. 1986, "Three Dimensional Structures of Magnetostatic
277
Atmospheres. II. Modelling the Large-Scale Corona" , Ap.J. , in press.
Fisher, R., and Sime, D. G. 1984, "Rotational Characteristics of the White-Light Corona: 1965 -
1983" , Ap.J. 287, 959.
Hundhausen, A. J. 1972, "Coronal Expansion and the Solar Wind" (Springer-Verlag).
Low, B. C. 1985a, "Some Recent Developments in the Theoretical Dynamics of Magnetic Fields" ,
SolarPhys. 100, 309.
Low, B. C. 1985b, "Three-Dimensional Structures of Magnetostatic Atmospheres. I. Theory" ,
Ap.J. 293, 31.
Nakagawa, Y. 1973, "A Practical Representation of the Solar Magnetic Field" , Astron.Astrophys
27, 95.
Newkirk, Jr., G., and Altsehuler, M. D. 1970, "Magnetic Fields and the Solar Corona. III. The
Observed Connection between Magnetic Fields and the Density Structures of the Corona" , Solar-
Phys. 13, 131.
Parker, E. N. 1963, "Interplanetary Dynamical Processes" (Interscience).
Seehafer, N. 1978, "Determination of Constant a Force-Free Solar Magnetic Fields from Magneto-
graph Data" , SolarPhys. 58,215.
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Three-dimensional magnetostatic models of the large-scale corona

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