An analytical procedure is developed to solve the magnetohydrodynamic equations for the stellar wind problem in the strong-magnetic field, optically thick limit for hot stars. The slow-mode, Alfven, and fast-mode critical points are modified by the radiation terms in the force equation but in a manner that can be treated relatively easily. Once the velocities at the critical points and the distances to the points are known, the streamline constants are determined in a straight-forward manner. This allows the structure of the wind to be elucidated without recourse to complicated computational schemes

Nerney, S.

English

NASA Technical Reports Server

N86-24539
1985
NASA/ASEE SUMMER FACULTY RESEARCH FELLOWSHIP PROGRAM
MARSHALL SPACE FLIGHT CENTER
THE UNIVERSITY OF ALABAMA
RADIATIVELY DRIVEN WINDS FROM MAGNETIC,
FAST-ROTATING STARS
Prepared By:
Academic Rank:
University and Department;
NASA/MSFC:
Division:
Branch:
MSFC Counterpart:
Date:
Contract No.:
Steven Nerney, Ph.D.
Associate Professor
Naval Postgraduate School
Physics Department
Space Science Laboratory
Solar-Terrestrial Physics
Solar Physics
S. T. Suess
August 28, 1985
NGT 01-008-021
The University of
Alabama in Huntsville
XXXII-i
https://ntrs.nasa.gov/search.jsp?R=19860015068 2020-03-20T15:39:55+00:00Z
' ' *' "\ - "^ <;*„"* ~" Jf
» -• \; "^  .^ "*" O *> 'f
RADIATIVELY DRIVEN WINDS FROM MAGNETIC,
FAST-ROTATING STARS
BY
Steven Nerney
ABSTRACT
An analytical procedure is developed to solve the
magnetohydrodynamic equations for the stellar wind
problem in the strong-magnetic field, optically thick
limit for hot stars. The slow-mode, Alfven, and fast-
mode critical points are modified by the radiation terms
in the force equation but in a manner that can be treated
relatively easily. Once the velocities at the critical
points and the distances to the points are known, the
streamline constants are determined in a straight-forward
manner. This allows the structure of the wind to be
elucidated without recourse to complicated computational
schemes.
XXXIl-ii
I. INTRODUCTION
Since the study of Weber and Davis (1967, WD) showed
that the '"sun could be spun down by the solar wind over
its main sequence lifetime, work has continued on angular
momentum loss in the solar wind and in stellar winds
(Pizzo, et al., 1983; Nerney and Suess, 1975; Nerney,
1980; Goldreich and Julian, 1970; and Friend and
MacGregor, 1984).
Early-type stars present additional problems in
understanding angular momentum loss as stellar winds from
0 and B stars are primarily driven by their intense flux
of ultraviolet radiation. The radiation has an
associated outward force due to Thomson scattering by
photospheric electrons as well as a line radiation force
that acts on metals in the wind. Unfortunately,
analytical solutions are not possible as the basic
stream-line constants are not known until a full
numerical solution is generated. The complexity arises
because the force due to line radiation depends
nonlinearly on the acceleration in the wind. The
critical point analysis shows that there is a focus of
singular points which the numerical solution must be
tangent to. This is accomplished through an iteration
scheme that produces an acceptable physical solution.
The purpose of this study is to attempt to
analytically model radiatively driven stellar winds with
strong magnetic fields from fast-rotating stars in the
optically-thick limit. This will allow a simplification
of the problem so that it becomes readily apparent how
the streamline constants determine the solution. Also,
the modified magnetohydrodynamic critical points are
treated and a procedure is developed for constructing
wind models which does not require a complicated
interaction scheme.
XXXII-1
II. EQUATIONS OF MOTION
The steady, spherically-symmetric', dissipationless,
magnetohydrodynamic (MHD) equations, together with terms
that give rise to radiative bulk forces may be written as
follows:
A
(1) 4irr pv = M, which is the continuity equation
where M is £he mass-loss rate in kg/sec, r is radial
distance from the center of the star in the usual r,8,<f>
spherical polar coordinate system, p is the mass density
in kg/m , and vr is the radial velocity of the wind.
r
which is the radial momentum equation where P is the
pressure, G is the gravitational constant, M is the mass
of the star, r is a constant (see below) related to the
radiative force due to Thomson scattering of continuum
photons by electrons, v is the rotational (azumuthal)
velocity of the wind, B^ is the azimuthal component of
the magnetic field, and f.. is the force/kg due to line
radiation. The term in B is the radial component of
the 5 x ^ force. ^
(3) r = . p -- where L is the stellar luminosity, c is
the speed of light, and a is the electron scattering
opacity per unit mass.
which is an approximate law derived in the original
theory of Castor, Abbott, and Klein (1975, CAK) where k
is a constant which is a measure of the mixture of the
number of strong radiation lines in the wind, t is an
optical depth parameter, and a is a constant which
depends on the importance of optically thick and thin
radiation lines and is model-dependent.
XXXII-2
ay'1
(5) t = a p v , (-j—) and vfcn is the thermal velocity
of the ions which absorb and scatter radiation. CAR used
values of k = 1/30 and a = .7, but I will use a = 1 and
choose various values for the constant that k appears
in. This Sobolev high-velocity gradient approximation to
the line radiation force is discussed in Mihalas, 1978,
p.561.
The azimuthal equation of motion and the induction
equation (a form of Faraday's law) are the same as in
Weber and Davis (1967, WD) and Friend and MacGregor
(1984), namely:
(7) (V x E)^ = 0 - p [r
as V • § = 0 gives
2(8) B = fb/r for spherical symmetry and as B /pv is a
constant, (b) integrates to
Br(9) rv, - -: - rB. = L where L is the stream constant
<f> 4-irpv^ <J>
total angular momentum per unit mass which is the sum of
the angular momentum carried in particles, rv , as well
as that in the electromagnetic field. Faraday's law
integrates to
2(10) r(v B, - v.B ) = constant = - n ffe and (9) and (10)
together can be solved for v and B as in Weber and
Davis (1967). 9 *
r A l - M A r
XXXIl-3
r 2 - r2
(12) BA = - B_ ^  ~
where Q is the rotation rate of the region where the
field lines are rooted (usually taken to be the
photospheric value), f. is the conserved radial magnetic
flux, v
 A is the radial velocity at the Alfven
radius (r.) where vr equals the speed of an Alfven wave
in the meaium, and NL is the radial Alfven mach number.
AIT
(13) M 2 = 4*p -£=. = p /pA r l a
Closure is obtained by using the polytropic
approximation.
(14) P = P (p/p )Y where P and p refer to values at
the base of°the w?nd (whether the base is in a corona or
photosphere), and y is the polytropic index.
This approximation is often used when the heating
sources in the wind are not well-known. Generally,
modellers replace this with an isothermal approximation
for hot stars and use the stellar effective temperature.
This is formally incorrect as it leads to inappropriate
asymptotic states of the wind, Hundhausen (1972, p. 9).
However, the use of effective temperatures of even 50,000
K as for 0 stars does not lead to significant problems as
it would for stars with coronae. Most early-type stars
show x-ray emission. Therefore, I will examine a hot-
wind solution and the polytropic approximation is
appropriate.
The radial momentum equation, (2), can be simplified
using a = YP/P (a is the sound speed) together with
-
 Using a
 "
 lf
reduces to P
XXXII-4
•} O O O VO
dv (v - A ) (2a + v/ 1- (1-r)) + 2 v v A A,i, , . r r_ _ __ r r ^ 9 ± r 9 r 9lib; , - ^ s ^ = - —s
Vr dr (vr2- Ar2) [vr2 (1-A) - a2] - v/ A^2
2 2 2 2
where A = B /4irp, A. = B, /4 i rp , , the radial and
r r 9 9
^ O * /"'M
azimuthal Alfven speeds, v = —^— , the escape velocity
at r, and r
(17) A = (t ) F GMk or, for a = 1,
M a v. ,
e th
(18) A = kL
CM vfch
A is the essential radiation parameter in this wind
model and can be estimated for a particular star. For
instance, Friend and MacGregor (1984) model the 06ef
star X Cephei with a = .7, M * 5.2 x 10~6Mo/year,
T = 6.5xl04K, L = 6.76xl05L0, and k was specified by
using the CAK value of 1/30. These numbers give A = .66
in the current model. It is difficult to estimate A
because k is model dependent, vfcn depends on both the
base temperature as well as the mass of the ions that
absorb and scatter radiation, and mass-loss rates are not
well-known for many stars.
Therefore A will be specified for a range of values
and a variety of models will be calculated.
The general results of this study are based on the
strong magnetic field limit of the MHD equations. This
allows the explicit calculation of the position of the
critical points based upon a similar prescription for
non-radiative strong-field winds in the study of Hartmann
and MacGregor (1982, HM).
XXXH-5
III. THE CRITICAL POINTS
A. Critical Velocities
The WD theory gives rise to three critical points in
the radial momentum equation. These points correspond to
distances from the star where the speed of the stellar
wind equals the speed of the characteristic MHD
disturbances that occur in the fluid approximation;
namely, the slow-mode wave, Alfven wave, and fast-mode
wave. Setting A = r = 0 for the moment, the denominator
of equation (16) goes to zero for three different wind
speeds and, hence, the numerator must be zero at these
critical points. These boundary conditions partially
specify the physical characteristics of the wind. In the
current study, radiation changes the nature of the
critical points in a manner that can be explicitly shown.
2 2The strong-field limit requires that v « A at
the modified slow-mode point, rg. This is equivalent
to mA r « 1 at rs.
This will be true for sufficiently strong magnetic
fields and is justified a posteriori. Setting the
denominator of (16) equal to zero:
(1 - A) - - v YAr2} = 0
Faraday's law can be written as
and as v, is equal to the corotation value at rg to
9 2 2 20(M z A r ) we can neglect v A /A in the strong-field
limit . Therefore,
(19) v
rs
XXXII -6
which reduces to a well-known result when
A =
 °'
 vrs = as'
At the Alfven point, the radial velocity is assumed to be
much greater than its value at r so that
S
vrA
2
 (1 - A) » a2A
Setting the denominator of (16) equal to zero.
2 2 d>(20) v = A + j-?-r at r at which reduces to the well-
known r r i-A a
result v = A at ra when A = 0.IT A
The modified^fast-mode velocity is found by
assumming v > A (M > 1) at r^. In a similar manner,
B. Optically-Thick Slow-Mode Point
Expanding Equation (11) for v.
(22) v = nr(l - M + — MAr + ...)
rA
For clarity, let v = nr(l - 5) where 6 < 1.
Setting the numerator equal to zero at rco
2a2 - ve2/2 + v (v - 2Vr A/A^ = 0 .
Using equation (10) gives
2
(23) 2a2 -- ~ (1-D + Q2 r2 = 0 to 0(62)
as terms of O(6) exactly cancel.
XXXII-7
Using the non-dimensional parameters in HM,
2 2 3Z = rs/r0f ft = a GM/r0 / gnd the non-dimensii
Parker critical point Z = » , we f ind
(24) zs -^ + a2 Zp Zs3 - Zp (1-r) = 0
ao
which reduces to equation (9) in HM for the isothermal
limit. Equation (24) does not explicity depend on 'the
radiation parameter, A. The distance to the slow-mode
point is the same in the radiative wind (except for the
hottest stars where r is not negligible) as in the non-
radiative MHD wind, except that the radial velocity in
the radiative case is greater by the factor I/ 1 - A.
Another point is that the calculation of Z assumes
v. is given by the corolation value at rs. This is
accurate to 0(M. ), yet due to a fortuitous cancellation,
2the equation for Z is accurate to 0(M ) so that these
results and HM's are more accurate than anticipated.
2 2
To solve (24) for Zc, we must know a /a^ which
2 2depends on v . In the isothermal limit, a /a = 1
and this problem does not arise. However,
(25) - = - . ] where
oe>\ R - 2( 2 6 ) 8
A second equation in Z and v can be found by
examining the energy/kg in the flow.
i -5 i " -> =.2
(27) E . (1 - A) V2 + 1 22 "* Y-l 2 vr
X X X I I - 8
The Poynting flux term can be calculated from
equation (9) so that
2
(28) E = ^  (1 - A) v 2 + \ v^2 + ^ 5y |- + QL - nr v
Consider the two terms in v. using v, = Qr(l - 6)3
so that terras of O(Ma ) have again cancelled.
A L
Setting E = E we find after some work
^ o s
' i . v 2 i ,t 9cn (-*-~A} rQ _ (±. 4. __L_"\
1. 2 J 2 ^ Y T'
ao
2zp HI -f-) (i-D - Y~ (zs2 - 1}} -^T
s
This reduces to equation (13) in HM with y = 1 and A = 0
a2 2provided we replace —r- by the isothermal term a inp.
Equations (24) and (29) can be self-consistently
iterated between to find v and Z using HM's initial
guess, modified for radiation
1+3 a2/3 Z (1-D2/3
2 2
which assumes a /ao * 1. This value for Z allows the
calculation of v in (29). The iteration procedure
rapidly converges to a solution.
XXXII-9
C. Optically Thick Fast-Mode Point
•A,
+\
We have shown that for M
 ar » 1 v f « - •. J
which together with equation (12) gives
vrf = 7f=f- --- 1/2
-
where vffl is the usual Michel velocity, Michel (1969).
2 2n f
(30) v 3 = - -- so that
m
 m
(31) v - —--
rt
In order to calcuate zf, we need to know v,,; which
can be calcuated from (11):
v L {i - ei , k (i - IEA, ra x
<)>f r l J r *• v F
which is formally correct with or without radiation.
Setting the numerator of (16) equal to zero at
(32) 2a2 - ve2/2 (1-r) + v^ (v + 2 (v - L/r)) = 0
L2 -1The term in v reduces to —=• (1 - 46) to O(M ). to
Equation'''t 32) now redaces to r
X X X H - i o
af o 5 a. •> vm(33) -4 Zf2 - Z Zf (1-D + a2 Zp ZA4 - 2ZA2 -Sj (1-A) = 0
ao ao
Equation (33) reduces to (17) in HM in the
vm
4
isothermal limit with A = 0 except for the term in — ^j .
This is 0(M ~ ) and should not appear as other terms of
Pi. L
the same size have been neglected in calculating Z^.
As ZA appears in the calculation of Zf we must
generate another equation for Z^ and Z^. The Alfven
radius is not important except insofar as it is necessary
in the calculation of 7,^ . If the wind passes through Zg
and Zf, it must of necessity pass through ZA, Goldreich
and Julian (1970) .
Setting EQ= Ef
i«i
 7 2(35) Z
2a* Z a * r x 2a 'p o o
with
(36) n = i (1-A)1/3 + 17
XXXII-ll
A good first approximation to Z^ neglects 1/Zj and
2 2
sets af /a » 0. This allows the calculation of Zg in
(33) and the iteration proceeds as usual.
XXXH-12
III. CONCLUSIONS
While the results of this study are preliminary,
several important conclusions can be drawn from the
analytical techniques that have been developed. The
strong-magnetic field, optically thick limit of the MHD
equations produces physical solutions through a tractable
technique that shows both how the critical points are to
be treated and gives the values of the velocities at
these points as well as the distances to the critical
points.
The streamline constants are determined once the
solution has been generated at the critical points. In
particular, the distance to the slow-mode point and the
radial velocity at the base must be iterated on to
produce physical solutions. Once this is done, the mass-
loss rate can be determined from the value of the density
in the photosphere together with the radius of the star.
Additionally, the total energy/kg is also determined.
The angular momentum/kg is determined after the iteration
procedure to determine the Alfven radius and fast-mode
radius is completed.
Preliminary numerical solutions have been generated
and are being compared to the work of Friend and
MacGregor, 1984, although it is still too early to report
these results. These solutions have been found by using
an IBM-PC and, in principle, could have been done on a
hand calculator.
XXXII-13
REFERENCES
1. Castor, J., D. Abott, and R. Klein, Astrophys. J.
195, 157.
2. Friend, D. and K. MacGregor, 1984, Astrophys. J.
282, 591.
3. Goldreich, P. G. and W. H. Julian, Astrophys. J.,
1970 160, 971.
4. Hartmann, L. and K. B. MacGregor, 1982, Astrophys.
J. 259, 180.
5. Hundhausen, A. J., 1972, Coronal Expansion and Solar
Wind, Springer-Verlag.
6. Michel, F. C., 1969, Astrophys. J. 158, 727.
7. Mihalas, O., 1978, Stellar Atmospheres, second
edition, University of Chicago Press.
8. Nerney, S. F. and S. T. Suess, 1975, Astrophys. J.
196, 837.
9. Nerney, S. F., 1980, Astrophys. J. 242, 723.
10. Pizzo, V., R. Schwenn, E. Marsch, H. Rosenbauer, K.
Muhlhauser, and F. Neubauer, 1983, Astrophys. J.
271, 335.
11. Weber, E. J. and L. Davis, 1967, 148, 217.
XXXli-u


Radiatively driven winds from magnetic, fast-rotating stars

SEE ParentDocumentRecord|Ntt=19860015036 "NASA. Marshall Space Flight Center Research Reports: 1985 NASA(ASEE Summer Faculty Fellowship Program"; p. 16 pAn analytical procedure is developed to solve the magnetohydrodynamic equations for the stellar wind problem in the strong-magnetic field, optically thick limit for hot stars. The slow-mode, Alfven, and fast-mode critical points are modified by the radiation terms in the force equation but in a manner that can be treated relatively easily. Once the velocities at the critical points and the distances to the points are known, the streamline constants are determined in a straight-forward manner. This allows the structure of the wind to be elucidated without recourse to complicated computational schemes.Approved for public release; distribution is unlimited

Calhoun, Institutional Archive of the Naval Postgraduate School

Calhoun: The NPS Institutional ArchiveDSpace RepositoryFaculty and Researchers Faculty and Researchers' Publications1986-01Radiatively driven winds from magnetic,fast-rotating starsNerney, S.http://hdl.handle.net/10945/60121This publication is a work of the U.S. Government as defined in Title 17, UnitedStates Code, Section 101. Copyright protection is not available for this work in theUnited States.Downloaded from NPS Archive: CalhounN86-245391985NASA/ASEE SUMMER FACULTY RESEARCH FELLOWSHIP PROGRAMMARSHALL SPACE FLIGHT CENTERTHE UNIVERSITY OF ALABAMARADIATIVELY DRIVEN WINDS FROM MAGNETIC,FAST-ROTATING STARSPrepared By:Academic Rank:University and Department;NASA/MSFC:Division:Branch:MSFC Counterpart:Date:Contract No.:Steven Nerney, Ph.D.Associate ProfessorNaval Postgraduate SchoolPhysics DepartmentSpace Science LaboratorySolar-Terrestrial PhysicsSolar PhysicsS. T. SuessAugust 28, 1985NGT 01-008-021The University ofAlabama in HuntsvilleXXXII-ihttps://ntrs.nasa.gov/search.jsp?R=19860015068 2018-09-26T21:25:37+00:00Z' ' *' "\ - "̂ <;*„"* ~" Jf» -• \; "̂  .̂ "*" O *> 'fRADIATIVELY DRIVEN WINDS FROM MAGNETIC,FAST-ROTATING STARSBYSteven NerneyABSTRACTAn analytical procedure is developed to solve themagnetohydrodynamic equations for the stellar windproblem in the strong-magnetic field, optically thicklimit for hot stars. The slow-mode, Alfven, and fast-mode critical points are modified by the radiation termsin the force equation but in a manner that can be treatedrelatively easily. Once the velocities at the criticalpoints and the distances to the points are known, thestreamline constants are determined in a straight-forwardmanner. This allows the structure of the wind to beelucidated without recourse to complicated computationalschemes.XXXIl-iiI. INTRODUCTIONSince the study of Weber and Davis (1967, WD) showedthat the '"sun could be spun down by the solar wind overits main sequence lifetime, work has continued on angularmomentum loss in the solar wind and in stellar winds(Pizzo, et al., 1983; Nerney and Suess, 1975; Nerney,1980; Goldreich and Julian, 1970; and Friend andMacGregor, 1984).Early-type stars present additional problems inunderstanding angular momentum loss as stellar winds from0 and B stars are primarily driven by their intense fluxof ultraviolet radiation. The radiation has anassociated outward force due to Thomson scattering byphotospheric electrons as well as a line radiation forcethat acts on metals in the wind. Unfortunately,analytical solutions are not possible as the basicstream-line constants are not known until a fullnumerical solution is generated. The complexity arisesbecause the force due to line radiation dependsnonlinearly on the acceleration in the wind. Thecritical point analysis shows that there is a focus ofsingular points which the numerical solution must betangent to. This is accomplished through an iterationscheme that produces an acceptable physical solution.The purpose of this study is to attempt toanalytically model radiatively driven stellar winds withstrong magnetic fields from fast-rotating stars in theoptically-thick limit. This will allow a simplificationof the problem so that it becomes readily apparent howthe streamline constants determine the solution. Also,the modified magnetohydrodynamic critical points aretreated and a procedure is developed for constructingwind models which does not require a complicatedinteraction scheme.XXXII-1II. EQUATIONS OF MOTIONThe steady, spherically-symmetric', dissipationless,magnetohydrodynamic (MHD) equations, together with termsthat give rise to radiative bulk forces may be written asfollows:A(1) 4irr pv = M, which is the continuity equationwhere M is £he mass-loss rate in kg/sec, r is radialdistance from the center of the star in the usual r,8,<f>spherical polar coordinate system, p is the mass densityin kg/m , and vr is the radial velocity of the wind.rwhich is the radial momentum equation where P is thepressure, G is the gravitational constant, M is the massof the star, r is a constant (see below) related to theradiative force due to Thomson scattering of continuumphotons by electrons, v is the rotational (azumuthal)velocity of the wind, B^ is the azimuthal component ofthe magnetic field, and f.. is the force/kg due to lineradiation. The term in B is the radial component ofthe 5 x ^ force. ^(3) r = . p -- where L is the stellar luminosity, c isthe speed of light, and a is the electron scatteringopacity per unit mass.which is an approximate law derived in the originaltheory of Castor, Abbott, and Klein (1975, CAK) where kis a constant which is a measure of the mixture of thenumber of strong radiation lines in the wind, t is anoptical depth parameter, and a is a constant whichdepends on the importance of optically thick and thinradiation lines and is model-dependent.XXXII-2ay'1(5) t = a p v , (-j—) and vfcn is the thermal velocityof the ions which absorb and scatter radiation. CAR usedvalues of k = 1/30 and a = .7, but I will use a = 1 andchoose various values for the constant that k appearsin. This Sobolev high-velocity gradient approximation tothe line radiation force is discussed in Mihalas, 1978,p.561.The azimuthal equation of motion and the inductionequation (a form of Faraday's law) are the same as inWeber and Davis (1967, WD) and Friend and MacGregor(1984), namely:(7) (V x E)^ = 0 - p [ras V • § = 0 gives2(8) B = fb/r for spherical symmetry and as B /pv is aconstant, (b) integrates toBr(9) rv, - -: - rB. = L where L is the stream constant<f> 4-irpv^ <J>total angular momentum per unit mass which is the sum ofthe angular momentum carried in particles, rv , as wellas that in the electromagnetic field. Faraday's lawintegrates to2(10) r(v B, - v.B ) = constant = - n ffe and (9) and (10)together can be solved for v and B as in Weber andDavis (1967). 9 *r A l - M A rXXXIl-3r 2 - r2(12) BA = - B_ ̂  ~where Q is the rotation rate of the region where thefield lines are rooted (usually taken to be thephotospheric value), f. is the conserved radial magneticflux, v A is the radial velocity at the Alfvenradius (r.) where vr equals the speed of an Alfven wavein the meaium, and NL is the radial Alfven mach number.AIT(13) M 2 = 4*p -£=. = p /pA r l aClosure is obtained by using the polytropicapproximation.(14) P = P (p/p )Y where P and p refer to values atthe base of°the w?nd (whether the base is in a corona orphotosphere), and y is the polytropic index.This approximation is often used when the heatingsources in the wind are not well-known. Generally,modellers replace this with an isothermal approximationfor hot stars and use the stellar effective temperature.This is formally incorrect as it leads to inappropriateasymptotic states of the wind, Hundhausen (1972, p. 9).However, the use of effective temperatures of even 50,000K as for 0 stars does not lead to significant problems asit would for stars with coronae. Most early-type starsshow x-ray emission. Therefore, I will examine a hot-wind solution and the polytropic approximation isappropriate.The radial momentum equation, (2), can be simplifiedusing a = YP/P (a is the sound speed) together with- Using a " lfreduces to PXXXII-4•} O O O VOdv (v - A ) (2a + v/ 1- (1-r)) + 2 v v A A,i, , . r r_ _ __ r r ^ 9 ± r 9 r 9lib; , - ^ s ^ = - —sVr dr (vr2- Ar2) [vr2 (1-A) - a2] - v/ A^22 2 2 2where A = B /4irp, A. = B, /4 i rp , , the radial andr r 9 9^ O * /"'Mazimuthal Alfven speeds, v = —^— , the escape velocityat r, and r(17) A = (t ) F GMk or, for a = 1,M a v. ,e th(18) A = kLCM vfchA is the essential radiation parameter in this windmodel and can be estimated for a particular star. Forinstance, Friend and MacGregor (1984) model the 06efstar X Cephei with a = .7, M * 5.2 x 10~6Mo/year,T = 6.5xl04K, L = 6.76xl05L0, and k was specified byusing the CAK value of 1/30. These numbers give A = .66in the current model. It is difficult to estimate Abecause k is model dependent, vfcn depends on both thebase temperature as well as the mass of the ions thatabsorb and scatter radiation, and mass-loss rates are notwell-known for many stars.Therefore A will be specified for a range of valuesand a variety of models will be calculated.The general results of this study are based on thestrong magnetic field limit of the MHD equations. Thisallows the explicit calculation of the position of thecritical points based upon a similar prescription fornon-radiative strong-field winds in the study of Hartmannand MacGregor (1982, HM).XXXH-5III. THE CRITICAL POINTSA. Critical VelocitiesThe WD theory gives rise to three critical points inthe radial momentum equation. These points correspond todistances from the star where the speed of the stellarwind equals the speed of the characteristic MHDdisturbances that occur in the fluid approximation;namely, the slow-mode wave, Alfven wave, and fast-modewave. Setting A = r = 0 for the moment, the denominatorof equation (16) goes to zero for three different windspeeds and, hence, the numerator must be zero at thesecritical points. These boundary conditions partiallyspecify the physical characteristics of the wind. In thecurrent study, radiation changes the nature of thecritical points in a manner that can be explicitly shown.2 2The strong-field limit requires that v « A atthe modified slow-mode point, rg. This is equivalentto mA r « 1 at rs.This will be true for sufficiently strong magneticfields and is justified a posteriori. Setting thedenominator of (16) equal to zero:(1 - A) - - v YAr2} = 0Faraday's law can be written asand as v, is equal to the corotation value at rg to9 2 2 20(M z A r ) we can neglect v A /A in the strong-fieldlimit . Therefore,(19) vrsXXXII -6which reduces to a well-known result whenA = °' vrs = as'At the Alfven point, the radial velocity is assumed to bemuch greater than its value at r so thatSvrA2 (1 - A) » a2ASetting the denominator of (16) equal to zero.2 2 d>(20) v = A + j-?-r at r at which reduces to the well-known r r i-A aresult v = A at ra when A = 0.IT AThe modified^fast-mode velocity is found byassumming v > A (M > 1) at r^. In a similar manner,B. Optically-Thick Slow-Mode PointExpanding Equation (11) for v.(22) v = nr(l - M + — MAr + ...)rAFor clarity, let v = nr(l - 5) where 6 < 1.Setting the numerator equal to zero at rco2a2 - ve2/2 + v (v - 2Vr A/A^ = 0 .Using equation (10) gives2(23) 2a2 -- ~ (1-D + Q2 r2 = 0 to 0(62)as terms of O(6) exactly cancel.XXXII-7Using the non-dimensional parameters in HM,2 2 3Z = rs/r0f ft = a GM/r0 / gnd the non-dimensiiParker critical point Z = » , we f ind(24) zs -^ + a2 Zp Zs3 - Zp (1-r) = 0aowhich reduces to equation (9) in HM for the isothermallimit. Equation (24) does not explicity depend on 'theradiation parameter, A. The distance to the slow-modepoint is the same in the radiative wind (except for thehottest stars where r is not negligible) as in the non-radiative MHD wind, except that the radial velocity inthe radiative case is greater by the factor I/ 1 - A.Another point is that the calculation of Z assumesv. is given by the corolation value at rs. This isaccurate to 0(M. ), yet due to a fortuitous cancellation,2the equation for Z is accurate to 0(M ) so that theseresults and HM's are more accurate than anticipated.2 2To solve (24) for Zc, we must know a /a^ which2 2depends on v . In the isothermal limit, a /a = 1and this problem does not arise. However,(25) - = - . ] whereoe>\ R - 2( 2 6 ) 8A second equation in Z and v can be found byexamining the energy/kg in the flow.i -5 i " -> =.2(27) E . (1 - A) V2 + 1 22 "* Y-l 2 vrX X X I I - 8The Poynting flux term can be calculated fromequation (9) so that2(28) E = ̂  (1 - A) v 2 + \ v^2 + ̂ 5y |- + QL - nr vConsider the two terms in v. using v, = Qr(l - 6)3so that terras of O(Ma ) have again cancelled.A LSetting E = E we find after some work^ o s' i . v 2 i ,t 9cn (-*-~A} rQ _ (±. 4. __L_"\1. 2 J 2 ^ Y T'ao2zp HI -f-) (i-D - Y~ (zs2 - 1}} -^TsThis reduces to equation (13) in HM with y = 1 and A = 0a2 2provided we replace —r- by the isothermal term a inp.Equations (24) and (29) can be self-consistentlyiterated between to find v and Z using HM's initialguess, modified for radiation1+3 a2/3 Z (1-D2/32 2which assumes a /ao * 1. This value for Z allows thecalculation of v in (29). The iteration procedurerapidly converges to a solution.XXXII-9C. Optically Thick Fast-Mode Point•A,+\We have shown that for M ar » 1 v f « - •. Jwhich together with equation (12) givesvrf = 7f=f- --- 1/2-where vffl is the usual Michel velocity, Michel (1969).2 2n f(30) v 3 = - -- so thatm m(31) v - —--rtIn order to calcuate zf, we need to know v,,; whichcan be calcuated from (11):v L {i - ei , k (i - IEA, ra x<)>f r l J r *• v Fwhich is formally correct with or without radiation.Setting the numerator of (16) equal to zero at(32) 2a2 - ve2/2 (1-r) + v^ (v + 2 (v - L/r)) = 0L2 -1The term in v reduces to —=• (1 - 46) to O(M ). toEquation'''t 32) now redaces to rX X X H - i oaf o 5 a. •> vm(33) -4 Zf2 - Z Zf (1-D + a2 Zp ZA4 - 2ZA2 -Sj (1-A) = 0ao aoEquation (33) reduces to (17) in HM in thevm4isothermal limit with A = 0 except for the term in — ̂j .This is 0(M ~ ) and should not appear as other terms ofPi. Lthe same size have been neglected in calculating Z^.As ZA appears in the calculation of Zf we mustgenerate another equation for Z^ and Z^. The Alfvenradius is not important except insofar as it is necessaryin the calculation of 7,̂ . If the wind passes through Zgand Zf, it must of necessity pass through ZA, Goldreichand Julian (1970) .Setting EQ= Efi«i 7 2(35) Z2a* Z a * r x 2a 'p o owith(36) n = i (1-A)1/3 + 17XXXII-llA good first approximation to Z^ neglects 1/Zj and2 2sets af /a » 0. This allows the calculation of Zg in(33) and the iteration proceeds as usual.XXXH-12III. CONCLUSIONSWhile the results of this study are preliminary,several important conclusions can be drawn from theanalytical techniques that have been developed. Thestrong-magnetic field, optically thick limit of the MHDequations produces physical solutions through a tractabletechnique that shows both how the critical points are tobe treated and gives the values of the velocities atthese points as well as the distances to the criticalpoints.The streamline constants are determined once thesolution has been generated at the critical points. Inparticular, the distance to the slow-mode point and theradial velocity at the base must be iterated on toproduce physical solutions. Once this is done, the mass-loss rate can be determined from the value of the densityin the photosphere together with the radius of the star.Additionally, the total energy/kg is also determined.The angular momentum/kg is determined after the iterationprocedure to determine the Alfven radius and fast-moderadius is completed.Preliminary numerical solutions have been generatedand are being compared to the work of Friend andMacGregor, 1984, although it is still too early to reportthese results. These solutions have been found by usingan IBM-PC and, in principle, could have been done on ahand calculator.XXXII-13REFERENCES1. Castor, J., D. Abott, and R. Klein, Astrophys. J.195, 157.2. Friend, D. and K. MacGregor, 1984, Astrophys. J.282, 591.3. Goldreich, P. G. and W. H. Julian, Astrophys. J.,1970 160, 971.4. Hartmann, L. and K. B. MacGregor, 1982, Astrophys.J. 259, 180.5. Hundhausen, A. J., 1972, Coronal Expansion and SolarWind, Springer-Verlag.6. Michel, F. C., 1969, Astrophys. J. 158, 727.7. Mihalas, O., 1978, Stellar Atmospheres, secondedition, University of Chicago Press.8. Nerney, S. F. and S. T. Suess, 1975, Astrophys. J.196, 837.9. Nerney, S. F., 1980, Astrophys. J. 242, 723.10. Pizzo, V., R. Schwenn, E. Marsch, H. Rosenbauer, K.Muhlhauser, and F. Neubauer, 1983, Astrophys. J.271, 335.11. Weber, E. J. and L. Davis, 1967, 148, 217.XXXli-u

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Radiatively driven winds from magnetic, fast-rotating stars

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