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PREPRINTSERIES

No. 293 ,.,.

ACCRETIONSHOCKGEOMETRIESIN THE MAGNETIC _

VARIABLES _,

H. S. Stockman :.;

?

l

_,1_',: _ '_' i.,. ,-L," _, ('_.! ,_-;;

/

, July 1988

(NASA-CU-lc,,joj) ACc_:Ti_3t'i .bHL;CIk _.r_j_l[Td,[t_-_ NOO-IAO6_ '_

j [,,j Tt,ir MAnN[IIC V^_TAqLr_ (3[)acc folpscope '_

_cionc¢ Inst.) Z3 n C;Cl OJ" _,

Uncl _s '._'

C3/90 0757i_

SPACETELESCOPESCIENCE INSTITUTE

_700 San MartinDrive Baltimore, MD 21218

Opecatedby U_eAmocl_*,_ of U_Ue_ for Re_u'ch in A_'lmomy, Inc., forthe N_kma[Am'o_utt(_ _ Space_o_ :'
¢,

Accretion Shock Geometries in the Magnetic Variables

' by

II. S. Stockman

Space Telescope Science Institute 1

3700 San Martin Drive

Baltimore, Maryland 21218

Presented at the 1987 Vatican Workshop on Circumstellar Polarization.

To be published in "Polarized Radiation of Circumstellar Origin", 1988 in press

2,

1. Operated by the Association of Universities for Research i_ Astronomy, Inc., for

NASA.
/

_'$11_ • 'i, "

1 , - ORIGINAL PAGE IS 1/I, i , OFPOORQUALITY f

ABSTRACT _

i The first self-consistent shock models for the AM Herculls-type systems were successful in

identifying the dominant physical processes and their signatures. These homogenous shock I

_ models predict unpolarized, Rayleigh-Jeans optical spectra with sharp cutoffs and rising

, polarizations as the shocks become optically thin in the ultraviolet. However, the observed

em.rgy distributions are generally flat with intermediate polarizations over a broad optical

band. These and other obServatioI,al evidence support a non-homogenous accretion profile

' which may extend overa Considerablefractionof thestellarsurface.We reviewboth the

_ fundamentalassumptions_nderlyingthe canonicalI-D shock model and the extensionof

this model to inhomogenouS accretion shocks, for both radial and linear structures. We also

examine the observational e_ridence for"'tall" shocks and find little evidence for relative shock i

heights in excess of, h/Rl _ 0.1. For several systems, upper limits to the shock height can

be obtained from either )(-ray or optical data. These lie in the region h/t_f <_)0.01 and are

in general agreement with the current physical picture for these systems. The quasi-periodic

optical variations observed in several magnetic variables may eventually proveto be a major

aid in further understanding their accretion shock geometries. ,-

t

1.0 INTRODUCTION

The discovery of circular polarization in AM Iterculis twelve years ago (Tapia 1977)

introduced a new observational field of time-resolved circular polarimetry of the Cataclysmic

Variables and new opportunities for studying the physics of accretion binary systems. While

tentative observations of circular polarization in DQ Her by Kemp et al (1974) had suggested

the presence of a magnetic white dwarf primary, the strong circular polarizations found in

the AM Her systems (V ,-- 8-40%) could only originate from harmonic cyclotron emission in

strong magnetic fields, B > 10 - 20 x 106 gauss (MG, see Lamb 1985, and previous papers in

this volume.) Two remarkable properties of these systems make them particularly valuable

for studying high energy astrophysics: the white dwarf rotation is perfectly synchronized

with the companion star's orbital motion and the magnetic field controls the accretion flow,

funneling it into a small area of the white dwarf's surface (see Lamb, 1988). As a result,

the synchronous variations of the optical flux, its polarization properties, and the hard !

and soft X-ray flux promised to elucidate the physics of the interface between the infalling

material and the stellar surface: the accretion shock. Indeed, the initial shock models ii

by Lamb and Masters (1979) successfully explained the qualitative features of the early

observations. However, the improved X-ray and optical coverage and more quantitative I

analysis progressively indicated that the actual accretion geometries and shock conditions !

were more complicated (Wickramasinghe, 1988). !

In this paper, we discuss several modifications to the canonical shock model which are i

both reasonable and appear necessary to explain the observational data summarized by

Bailey, Piirola, Schmidt, Beuermann and others in this volume. Since a rigorous exposition t

" of the shock physics is unjustified by the simplifying assumptions required to complete our

mental picture of these shocks, we emphasis both the feasibility of the various effects and

the evidence which supports their presence.
.y

2.0 CANONICAL SIIOCK MODEL

Lamb and Masters (]979) created a series of energetically self-consistent models of accretion shocks for field strengths between 0.1 MG _< B < 100 MG and a range of white

dwarf masses, 0.4Mo _< 3/1 < 1.4M®. These models assumed a cylindrically symmetric,

homogenous accretion flow and a "thin" shock, ie. the height of the shock, h, was small

compared to the half-width of the shock, r,, which in turn was small compared to the radius

of the white dwarf, R1, or h << rs << R1 (see Figure la). While the model results often fell

outside these regimes, this approximation permitted the solution and extension of the local

shock characteristics without recourse to a specific, multidimensional model. As we indicate

in Sections 3-4 and is foreshadowed in Figures lb&lc, the physical consequences of relax;.-._

these and other assumptions are significant and may resolve current discrepancies between

the canonical model predictions and observations.

The canonical model shock characteristics are parameterized by the white dwarf mass,

M1, and implied radius, R1, a constant magnetic field strength, B, and _he local accretion

density, _, which is often stated in relation to the normalized accretion hminosity, L/.f,

where f is the fractional coverage of the stellar surface by the accretion shock:

L 4GM 1]tIR 1

f -- rs2 (1)

and

_I = _rr_. C2)

The pre-shock density of the infalling gas, Pl, can be obtained from vh = plY1, where

v 1 is approximately the free-fall velocity, vl = (2GM1/R1) _ 7.3 × lOakm/s(M/R) 1/2.

(M = M1/M® and R = R1/5 x 10s cm are convenient scaling factors.) In the canonical

model, the density behind the shock, P2, is given by the jump conditions for a strong shock

while the velocity behind the shock, v2 is obtained from the continuity equation.

3

P2 = 4pl (3)

and

p2v2 = PlY1 (4)

yielding,

1

V2 ---- _V 1. (5) :

In the spirit of the homogeneous shock approximation and, therefore, neglecting temperature structure and thermal conduction, the shock height can be determined by the

requirement that a parcel of shocked gas must cool by the time it reaches the base of the

shock or

GM1 / R1

teool _,, ¢(p2, Te, h) (6) '

and

3 . _z
J

I

tooi=--. (7)

' v2

1

Here ¢(p, Te, h) is the specific cooling rate at an electron temperature Te.

i

' 2.1 Bremsstrahfimg-Domin_ted Shocks

For magnetic white dwarfs, with MI < ].0Mo, the dominant cooling mechanisms are

, t,remsstrahlung emission and optically thick cyclotron emission. For a fixed field strength, '

the two cooling regimes and resultant structures depend on the accretion rate. For high rh,

or L/f > L/fcrit, bremsstrahlung is the d,,minant coolant (Regime I in Lamb and Masters

1979). Typically in these shocks, tcoot >> tie , tile exchange time between the electrons

and the shocked nuclei, and Ti = Te = 7'8 = 64keV(M/R) (Imamura et al 1987). Since

! ebrerns c<p2Te 1/2, for a given M/R,

' tcool ¢x h ¢x p21 OC(L/f) -1. (8) |'

The cyclotron opacity is a strong function of the magnetic field strength, the temperature,

and the frequency. _snor!.ng angular beaming effects, the emerging flux from an isothermal

thin shock will appear as a Rayleigh-Jeans spectrum up to a critical frequency, refit , above

i which the spectrum is truncated. While a quantitative calculation of the cyclotron emission

, requires an integration over angles, polarizations, and the inevitable temperature structure

in the shock, we will use the simplest approximations of Dulk and Marsh(1982) for an

emergent angle of 45° to illustrate the general beh_,vlor of the cyclotron cooling. Using

the nomenclature already introduced, they obtain the critical frequency of the emergent

radiation of:

Vcrit "_ 2.l × 1015BO.9 30 kTO.T50 r_• 0.1 Hz. (9)

Here, B30 = B/30 MG, kTso = kTe/50 keV, and re is the optical depth of the shock to

Compton scattering. Note the strong dependence on temperature and field strength and the

extremely weak dependence on surface density. For the bremsstrahlung-dominated shocks, '

re _ 0.3p2h --_0.3(M/R) 1/2. Thus, with our assumptions, the emergent cyclotron flux and

spectrum for a bremsstrahlung-dominatcd shock has a characteristic brightness temperature,

Ta, and cutoff frequency, t]crit, given above with no dependence on accretion density.

Integrating the Rayleigh-Jeans spectrum to the critical frequency, we obtain the local

cyclotron luminosity as: i :,

2_" 3 (10) i

lcyc = ._c2kT vcrit ,

and the tota! cyclotron luminosity for a brcmsstrahlung-dominated shock as: i

L / refit(B, M It) ... 0.1L Edd(M/R) 2"25R1B_b 7. (11) ,
J,,

"1

_._ Cyclotron-Dominated Shocks

For field strengths much greater than typical AM tier fiela strengths, B >> 30 MG, or

for accretion flows substantially below the local EddmotOn limit, LEd d 1.4 × 1038M erg/s,

the dominant cooling mechanism is optical cyclotron emission. (_bcx TeV3crit/re). Generally,

in the cyclotron-dominated regime, tcoot('l's) < tie and the electron and ion temperatures are

driven out of equilibrium. This corresponds to R.egime II in Lamb and Masters (1979). To

obtain the electron temperature, ,-e extend tile cooling time argument to cover the electronion cooling.

h

tcoot(Te) = tie(Te) = --. (12)

v2

Here tie is obtained from the Coulomb exchange time. For Te/Ti >> me/mi and Te << Ti,

(cflmamura et al 1987 but also Section 2.3),

T3e/2 _.

tie(Te) oc ---- (13)

P2 _,

Thus, in this regime, the electron temperature is regulated by the cyclotron cooling and

ion-electron exchange. The post-shock velocity and density are still given by the strong

shock condition but the height and temperature are given by the following approximate

scaling relations:

~ T, [

(L/f)

0.3

(I.//)crit (14)

[ l-°'°

h/R ,_ 0.003 [L---_dd] (M/R)2B;l'2M°'56. (15) ,

Despite the coupled nature of these shock equations, the strong temperature dependence

C" the cyclotron opacity essentially adjusts the temperature to obtain the refit required

to radiate the accretion luminosity, while the resulting ion-electron coupling time sets the i

height.

_.3 Eztensions of the Canonical Model

As pointed out by Lamb and Masters(1979) and Masters (1978), the two-temperature

model breaks down at very low accretion densities when teool < tii , the ion-ion coupling

time. In this regime (Regime III), the electron-ion coupling becomes stronger than the ionion coupling and the shock is no longer self-supporting. Stockman and Lubenow (1987)

have extended the canonical model into this regime by picturing the infalling ions as an )

interpenetrating fluid into a cooling layer. In this scenario, the relationship between Pt and i

P2 is given by a pressure relation: 1

pl(Vl) 2 "_ni2kTi2 + ne2kTe2 (16) |

|

and

tii "" tcoot = h/v2 (17)

.d

5
Here tii is the momentum exchange time between the incoming ions and the ions in the

cooling layer. To obtain a self-consistent calculation, a full network of energy exchange between the infalling ions and the species in the cooling layer is required. In practise, however,

the electron temperature continues to bca primary function of the accretion luminosity and

closely follows the temperature relation for Regime lI. The shock height is a mor..- complicated function of the accretion rate as can be seen in Figure 2. In this figure, the model

results for M 1 = 0.4 and 1.0M O and B = 10 and 30 MG are displayed. The non-monotonic

behavior in the shock height is due to the initial onset of Regime II, where the ion-electron

coupling time is inversely proportional to &Tie = Ti - Te, which increases rapidly with lower

I,/f. Indeed, the shock never reaches the assymptotic formula given above for Regime II

before the onset of the non- hydrodynamic shock regime (Regime III). This result differs

somewhat from that of Lamb and Masters (1979), who assumed Ti " 2Ts for Regime II.

While the integrated energy distribution is not sensitive to the shock height, self- eclipses by

the white dwarf limb will produce qualitatively different light curves for the two models.

Also shown in Figure 2 are lines indicating the criteria, L = L/f x (h/R1)2. For radial, t

"pillbox" type shocks (Fig.la), shock heights exceeding these contours for a given luminosity , ,

violate the thin-shock assumption (see Section 3.1). We note that the combination of strong

radiation transfer effects, coupled with the rapid change in the shock height near the onset

of Regime II may be a strong mechanism for a global shock instability (see Section 4.3).

3.0 NON-HOMOGENEOUS SHOCK MODELS

While the dominant coolants and plasma effects vary in a complicated fashion according

to the local accretion rate as described in Section 2, the fundamental characteristics of the

emergent radiation are easily _tated. For all regimes excepting the directly heated pt'otos'_here, the critical cyclotron frequency decreases as the local accretion rate declines. For

homogeneous shock models (as illustrated in Figure la), the observed cyclotron flux should

drop precipitously at frequencies above the critical frequency and show strong polarizations

characteristic of an optically thin plasma. The energy distribution at lower frequencies

(longer wavelengths) should be approximately Rayleigh-Jeans and show relatively little po- i

larization. The difficulty of reconciling these predictions with the flatter observed energy

distribution, f(v) ¢x v -1 and moderate broadband polarization of AM Her was recognized i

soon after Lamb and Masters 1979 publication and the confirmation of the fundamental role )

of cyclotron harmonics as evidenced by the low field strength in VV Pup (Visvanathan and

Wichramasinghe 1979) and AM IIer (Schmidt, Stockman and Margon 1981).

In general, the conflict between the simple homogeneous model predictions and the ob- ','

served characteristics was addressed by assuming a site for the optical radiation separate from t ,

the origins of the X-ray flux: either above the bremsstrahlung-dominated shock or in a much

larger and cooler region. Liebert and Stockman (1985) first suggested a n,)n-homogeneous

shock geometry and noted that power-law energy distributions could be obtained if the local

. accretion rate were to decline as a power- law away from the "core" of the shock. This picture :

wa_ developed further by Schmidt, Sl,,,ckman, and Grandi (1983), who also considered expo- I

nential and Gaussian declines in the local accretion rate. In numerical simulations covering

broad class of exponential and power-law dependencies, Stockman and Lubenow (1987) 1

a

have confirmed the simple analytical results of Schmidt, Stockman, and Grandi (1983). In

tt

° l"

......... I- ----., . =JO
i

i.t

i this section, we summarize the elements of tile non-homogeneous shock models and some of

,i their far-reaching consequences.

i 3.1 Variable Accretion Densities

i

In the simplest extension of tile homogeneous shock model, we consider the local accretion

; rate _(r) to be a monotonic_dly declining function of the radial distance,r, from the "core"

of the shock. We can extend the "thin-shock" assumption so that .h(r) _ I 4;::R1, where I is

a characteristic scale for change, l ... (dln(rh(r))/dr) -1. With this ass,lmption, the structure

of the shock at a given radius r is simply that of the homogeneous model and the observed_

characteristics of the shock can be obtained by integrating over the characteristics from the

core of the shock to its outermost boundary, rmaz. In particular, the total luminosity of the

shock is given by e

L=['"" 2,,rat. (17)

The observed cyclotron flux at a frequency, u, is proportional to

t

t

f(u) cx v 3 --fr0(u) T(r)27rrdr. (18)

where r(v) is the radius at which the shock becomes optically thin to cyclotron radiation

at the frequency, y. Similar relations can be obtained for the integrated hard and soft X-ray

spectra using the formulae presented in Section 2. Before discussing the qualitative results !

for various accretion rate "profiles," it is wduable to consider the limitations imposed by the

thin-shock assumption.

As shown in Section 2, the shock height can be comparable to the shock width or characteristic scale length, I, for accretion luminosities characteristic of the AM Her variables.

In the bremsstrahlung regime, the principle effect will be a slightly larger and somewhat

cooler soft X-ray region underlying the shock. In the cyclotron dominated regime, however, cyclotron cooling will be enhanced by radiation channeling through the sides of the

column; both radiall:' inward above the higher density core and outward through the cooler

shock region. The primary effect of the enhanced cooling will be to suppress the electron

.. temperature and, subsequently, to reduce the shock height. ;,

In accretion geometries with large shock regions, l ,.- R1, both the variation in the surface

magnetic field strength and the surface and field curvature should be considered. The latter _

effects are described in Section 4.2 and have been modeled by Wickramasinghe and Ferrario

(98s).

" 3._ Power-law Accretion Profiles

While the observed flux from exponential or gaussian accretion profiles are essentially

identical to those for the appropriate "tfilllmx" geometry, power-law accretion profiles can

produce dramatically different energy distri butions. These can be illustrated by the following

example from Stockman and Lubenow (1987) which approximates the active state of AM

Her: L = 1033erg/s, (L/[) = 1038erg/s, M = 0.8, and B = 20 MG. The adopted accretion

profile is rh(r) ¢x 1/(1 + rZ). Figure 3 shows the integrated energy distribution for this model

compared to that for a gaussian accretion profile, while Figure 4 shows the variation in shock

characteristics versus radius for the power-law model, tI "

[

ql
., Several features illustrate the nature of the accretion profile models. First, the brems-

'_;! strahlung-dominated "core" of the shock provides most of the hard and soft X-ray flux. Since

;i the electron temperature and surface density are relatively constant, the optical cyclotron

from this region is characterized by a constant critical frequency and area-llmited flux. Sec-

¢ ! ond, in the "wings" or "halo" of the shock, the peak wavelength and shock height increase

• for larger radii and smaller rh(r). A qualitative treatment of the radiation channeling has

_ i been included in these models and yields a smoother variation with shock height than that

: displayed in Figure 2. For differing power-law profiles, the resulting energy distributions ex-

: _ hibit different power law characteristics-both in the optical cyclotron portion of the spectrum

. and in the reprocessed ultraviolet portion of tile spectrum. The predictions of the quadratic

form are particularly interesting siztce they provide both a red, f(v) ,.x u-1 spectrum in the

j visible and an "ultraviolet excess" in theIIJE band-pass. This agrees qualitatively with the

_ bright-phase observations of AMIIer. Steeper power-iaw profiles result in steeper, gaussian-

: ; like spectra, while flatter power-laws produce flat optical spectra. Recen'Jy, Wickramasingb e

and Ferrario (1988, and this volume) have studied the cyclotron intensity and polarization

from various shock profiles, u_ing 3-D radiation transfer calculations. They also obtain "flat"

i optical spectra, confirming our pseudo-2D calculations, and find good agreement with the

i observed broadband polarizations. (The quadratic accretion profile best compares to their

:. model with//_ 4 with the linearly declining temperature.)

3.4 Eztcnsion to Linear Structures

Recent analyses of x-ray data (Beuerrnann et al. 1987), and optical data by Schmidt

(1988) suggest that the accretion shocks may be more linear or arc-l_ke than circu!ar. This

structure is shown schematically in Figure lc. The non-homogeneous model can be extended

to these structures by choosing a fu,,ctlo,al fi)rm for rh(z) which depends on the perpendicular distance, z, from the cettter of tl,: arc. Using the same thin-shock assumptions, we

would obtain virtually identical energy d;st ributions for the AM Her model using the linear

form of the power-law profile, tit(z) _ 1/( t + a:). lake the quadratic form, the linear form

must be truncated for z -.. R1, either by the onset of direct-photospheric heating by the

dilute accretion stream, or, perhaps, by physical limits to the out-of-plane dispersion in the

unthreaded accretion stream.

_. 8.5 Implications of Power.Law Accretion Profiles

The qualitative agreement between the results of the inhomogeneous shock model and

X-ray and optical observations, in addition to many independent analyses which support

an X-ray core and optical halo structure for the accretion shock (e.g., Beuermann 1988),

_" provide strong support for diffuse accr,"",u over much of the surface of the white dwarf.

"_ In Figure lb, we indicate schematically ,ow such an accretion shock might appear. The

:. ! "inevitable" downturn in the visible shock would be due to the onset of direct-heating of the

.2_ white dwarf photosphere. While the cyclotron emission may be confined to a fraction of the

-,;, stellar surface, accretion and photospheric reradiation in the mid-ultraviolet may occur over

• much of the entire star. To explain the UV "bump" in AM Her, the derived coverage for

_! the halo radiation, f > 0.1, implies that significant portions of the accretion stream remain

"' unthreaded until very near the white dwarf surface, Rthread -_ 3 - 5R 1. Since the ballistic

trajectory of the accretion stream has a typical minimum radius of approach of Rb _- 10Rt

"_ (Lubow and Shu 1975), significant out-of plane dispersion must occur before the accretion

stream is completely threaded.

8
W

.J

-j

4.0 OBSERVATIONAl'_, CONSTRAINTS ON SHOCK STRUCTURES

The discussions in Sections 2-3 suggest that the shock structures for the AM Herculis

magnetic variables may be more complicated than those described in the initial models.

Not onlyareseveralregimesofshock-physicspotentiallimp y ortant,but many orallofthese

regimesmay contributtoe theobservedopticalu,ltraviolesto,ftX-rayand hard X-rayfluxes.

To explainthe relativsetrengthsand time behaviorof thesewavebands,one may resortto

complicatedstructureswithlittl,emderstandlngcftheirunderlyingup- streamoriginsIt. is

important,therefore,toobscrvationallte:it y some oftheunderlyingassumptionsand results

of thesemodels beforecontinuingadditionalparamcterlzedstudiesor usingmodel fitsto

draw strongconclusionrsegardingtheaccrctlonstreamor threadingregion.In thissection,

we reviewthe currentobservationaevid l encerelativtoe a fundamentalresultof the shock

modelsoutlinedinSection2:theshockhclght,h. Ina number ofsources,thehardX-rayand

opticalvariation(spolarimetricand photometric)generallysupportthe qualitativpic e ture

develol:ed here. In other objects, the quasi-stable oscillations offer great potential as a probe

of the shock structure.

4.1 Compact Bremsstrahlun9 AccretionCores

Several lines of evidence (Beuermann, 1988a), as well as the results of model fits to the

observed energy distributions of AM Her and E2003+225 (Stockman and Lubenow 1987)

indicate that the central high density accretion luminosities approach the local Eddington

luminosity, L/f ",, LEdd "_ 1 - 2 x 103s erg/s. In this regime, Compton cooling becomes

a dominant cooling mechanism, lowering the shock temperature and height (Imamura et

al 1987). Radiation pressure will not play a major role if the core of the shock is small

compared to the stel!ar radius, hL/(R 1LEdd) _ 1. ttowever, as suggested by Beuermann

(1988b), the ram pressure of the accretion stream can compress the photosphere of the white

dwarf by many scale heights. Coupled with the low X-ray shock heights and the power-law

accretion profiles, this can create "dimples" or "trenches" in the white dwarf atmosphere

which can absorb and collimate the X-ray and the beamed, ultraviolet cyclotron emission

from the core. Figure 5 shows a cross section for such a trench. The criteria for significant !

collimation is that the central X-ray shock height is less than the ram depth or

hzC_h(,'))< dC,_Cr)), (19)

and that the change _.nthe ram depth across the core of the shock is greater than the

half-width of the core, ra. Using the more accurate results of Imamura et al (1987) for this

regime (L/f .., 0.1 - 1 LEDD, M _ 1), we can approximate the X-ray shock half-height

(appropriate for 3-10 keV band) as:

hz/R1 "_ 1.5 x IO-4LEdd/L. (20)

The ram depth is approximated by equating the accretion ram pressure to the underlying

photospheric pressure, or

~ 2n kT, ~ 2nphotkTphot. (21) ,

Calculating the characteristic photospheric pressure scale height, h.L.,r..,.."_ 2kTphotR2t ]

GMImp, and finding the depth in reference to an X-ray photosphere where re -.. 1, we

obtain

!

I
J _ h..4_ ' t *

t

Y

{

, |

--hpt, ln(nphothphotC=,) In (22)

or

,--(Tn,od?,)lnO',Rzlh). (23) ,..

For obscuration to he significant, h= < hphot In_'phot and 4rs < dCrh(r)). These conditions

are relatively straightforward to satisfy if L/f (core) > 0.1LEdd with a characteristic core

width of rs/R1 ": 10-4 - 10-3. However, for the azim'lthaily symmetric accretion profiles ;,

such as that shown in Figure lb, the corresponding maximum core luminosity is Lc_e "-

0.1LEdd(lO-S)2/4 ,_ 1030 - 1031 ergs/s. For the "trench" geometry, on the other hand,the i'

maximum core luminosity is approximately Lcore ~ 0.1LEdd(lO-3)/4 _ 1033 - 1034 ergs/s.

Clearly, the photospheric obscuration of arc-like structures, such as shown in Fig lc, might f

explain a wide variety of phenomena associated with the hard X-ray flux: asymmetric and

variab!e eclipse profiles, the "missing" hard X-rays in AM Iler, and a consistent excess of ,:

soft X-ray flux observed in many AM IIer sources (Beuermann, 1988a). Indeed, although

the concept of photospheric obscuration post-dates the X-ray data, the sharp and irregular ¢

X-ray eclipses are strong evidence that the X-ray shock heights are comparable to those 1

predicted for the bremsstrahlung-dominated Regime I. i -

4._ Self-eclipse of the Optical Shock

EcTipses of the accretion shock by the phase-locked companion star are only confirmed in 0.

one source, El114-182. Elsewhere in this volume, Schmidt describes two-color eclipse data

which supports the notion of a large optical emission region, possibly a linear structure like

Fig. lc. However, the rotation of the wMte dwarf provides another eclipsing mechanism if

the accretion region is small compared to the radius of the star, h ,_ R1. As the optical

region passes over the llmb, the optical light will slowly diminish and the observed circular

polarization may change sign. While the finite width of the accretion region complicates

interpretation of both photometric and polarimetric eclipse data, it is often possible to set

upper limits on the height of the accretion shock.

The relationship between the pol_trimctrlc and photometric horizons for a shock height, ]_

h _ 0.1RI, is shown in Fig. 63. ht this panel, the magnetic field is normal to the stellar !

surface, as it would be :tear the magaetic poles. The eclipse horizon is depressed below the I

polarimetric horizon or the tangent plane to the stellar suface by an angle, 8, given by

z+ g/ z "

If h _ R1, a good approximation is ,_ i

i '

~ (2h/n ) (25) ! ;

Thus, a small shock height, h/R t ~ 0.01, can result in a substantial depression of the i

eclipse horizon, 6 -,- 0.14 rad -,- 8°. As the cyclotron-emitting shock transits the limb of

the star, the optical light will rapidly decline and the observed circular pola:ization will be

observed to "reverse" for a portion of the 1-3 hr period,

A__._,_p_p(2h/Rl)l/2 (26) { ; P 2_r

10

xx_
<7:

!

Y

Lieb, rt et al (1978) interpreted a particularly abrupt self- eclipse of VV Pup (Warner

and Nather 1972) using this argument to obtain an upper limit to the optical shock of

fi/R1 < 0.01. On the other hand, the .,omewhat longer polarization reversals observed

in that source (AP/P ".. 0.07, l,iebert a,d Stockman 1979) yield a !after shock height,

h/R 1 ... 0.1. While both shock heights are compatible with the model results described

in Section 2 and displayed in Fig. 2, the interpretation of the polarimetric eclipse may be

seriously flawed. As anticipated by Stockman (1977) and modeled by Cropper and Warner

(1986) and Wickramasinghe and Ferrario (1988, this volume), the magnetic field may not

be normal to the stellar surface at tl,e accretion shock. This situation is shown in Fig. 6b.

Even for a shock of negligible height, the polarimetric horizon may be elevated substantially

above the plane tangent to the stellar surface, yielding an observed polarimetric "reversal"

at ingress, egress or both.

It is interesting to consider how large an effect we might expect. If the accretion stream

is fully threaded at a diq'.ance comparable to the ballistic radius of closest approach, R b,

the accreted material will form a shock within an angle, $ of the magn • pole given by the {

dipole approximation,

o < (ettt/] b) (27)

And the imbedded magnetic fichl will deviate less than 6b from the stellar normal,

-<(R/2nb) (28) '

From the models of Lubow and Shu (1975), the radius of closest approach is Rb/R1 "" 10.

Thus, the resulting polarization "reversal" may be significant, AP/P -., 0.04,and lead to an

erroneous estimate cf the shock height for hill 1 < 0.03. We note, however, that multicolor

polarization observations of VV Pup (Piirola et al. 1988), silow no reversal ; , the U band

compared to a large reversal in V. These data support the "height" interpretation rather

than off-normal magnetic fields.

Despite the difficulties, we may be able to set reasonable upper limits to ttle optical

shock height by noting the lack of an t,bcrved polarization "reversal" in certain geometries.

In Fig. 7, we indicate the expected duratiot_ of the "bright", uneclipsed phase for a pointlike accretion shock for the complete range of magnetic colatitude and binary inclination

(eg Brainerd and Lamb 1985, also Bailey _tn¢! Axon 1980). In dotted lines, the duration

of the bright phase is given for a sl,ock height of h/R 1 = 0.1. The difference between the

two geometries predicts the duration of the observed polarization "reversal" for the tv.ller

shock. Two qualitative effects of a significant shock height are evident: polarization "reversals" should be common, particularly wltelt the active hemisphere is "up"; the fraction of

unobservable systems (no bright period) becomes insignificant. Unfortunately our statistical

L

sample is too small to address the second effect and most of the sources are either at too

,l

high or too low a colatitude to produce tl,e polarization signature which would confirm a

moderate shock height. AM lh'r, for instaHce, displays a prolonged polarization reversal in

its active state which couhl be interpreted as either two active poles (Piirola ,t al. !985)

or a shock height, h/Rl ". 0.05. During its faint state, however, AM ller shows no polarization reversal (Latham et al 1979), implying a very low upper limit to the shock height,

h/R t < 0.01, similar to the VV Pup photometric upper limit. DP Leo (El114-1-182), with a

magnetic colatitude of 103° anti an inclination of 76 ° may be the best candidate for future

study (Biermann et al 1985). |

II
i There is no unequlvocablc evidence for substantial heights (h/Rt > 0.1) of the optical I

• shock. Some shocks might be expected to reach such a thickness (Section 2), particularly ..

for lower field strengths. I[owever, rediation transfer effects can genezally be expected to '

reduce the shock thickness (Stockma,, and Lubenow 1987). Thus, the lower limits found in

the bright state of VV Pup and the hint _tate of AM Her are consistent with the physical

results and tb!a-shock approximation which are presented in S-.ction 2 and which form the '

basis for the non.homogeneous shock models described in St :ion 3.

4.8 Seismolog!¢ ".

Since the discovery of the AM Ilcr systems in 1976, their strong photometric flickering

has been often observed but seldom analyzed in depth. It is generally thought that the 10-

100 s flicker noise discovered by Berg and Duthie (1975) and studied both in photometric,

spectroscopic, and polarimetric detail by Bailey et al (1977), Margon and Szkody (1978), ,.

and Stu__kman _tnd Sargent (1078) originates in the accretion shock but is due to instabilities

in the upstream accretion flow- either near the secondary star or in the accretion threading

region. However, without a quantitative theory describing either region, cur progress "_n _.

understanding these relatively slow variati:ms has been slow.

Middleditch (1982) discovered a qualitativel) different behavior in E1405-451 and AN

UMa. In the former object, 5-10% variations can be observed with periods of approximately

P -.- 1 - 3s. More remarkably, these oscillations can be quite distinct, often displaying one or

two Fourier components with a coherence time of minutes or Q > 30 (Larsson 1984). Larsson

has also discovered similar variatio,s in VV l'up (Larsson 1987a) and EF Eri (Larsson

1987b). For all four objects, the range of fluctuations is similar and can be bounded by 0.3

tIz< f <1.3 IIz. Larsson (1984) found that the pulsed light in E1405-451 was typically red

in color, although on one occasion, the color was n:ui.ra! or blue in color (Larsson 1987a).

Thus, the general assumption that the Iluctuating light is due to cyclotron radiation has not

been convincingly confirmed.

Instability studies of bremsstrahlung-dominated shocks have suggested that the observed

pulsation range is feasible for low-mass white dwarfs (MI < 0.4M® ) and shock heights of

0.03 < h/R 1 < 0.2 (Larsson 1987c). Tiffs connection, if correct, would be very important since it would perm'_t ae. independey:t, detailed study of the AM Her accretion shocks i

in optical light, ttowever, this interpretation of the observed fluc*uations is not without

difficulty: I

a) The coherence of one or two Fourier modes and the strength of the fluctuations in _,

E1405-451 requires that a major portion of the shock participates in the instability in a coherent fashion. However, in the context of a two-dimensional problem, the necessary coupling

or synchronizing mechanisms may Ire either _bsent or critical to the correct interpretation.

For instance, the instability m_ty I,e oec,,rring in a central, high density flow which, by

fluctuating Compton cooling, affects the surrounding lower density regions.

b) Since the fluctuations are generally seen in red optical light, it i,J likely that the '

portion of the shock responsible f,)r the red emission is cyclotron-dominated. As shown in !

Section 2.2, both important cooling rates, cyclotron radiation and electron-ion coupling, are

very temperature dependent. For this reason, cyclotron-dominated shocks at," not thought

to be unstable to the same mechanism, although they may have instabilities due t_, other

processes.

!

im
]

¢;

] must be examined. Ite suggests, for inst_mce, that mabnetohydrodynamic processes in the ', _'

threading region may have the appropriate frequencies. The d'fllculty for all these theories

is explaining the high quality factor, or h)ng coherence time, particularly in the presence of

strong, low frequency flickering. The best candidates will be those that evoke both a global

' synchronizing mechanism anti an intrinsically sensitive physical characteristic, such as an

photoionization edge. The current theories fail this criteria.

Clearly, further observations in both the optical and x-ray bands are required to establish

the origins of the fast optical fluctuations and their promise for studying the structure of the

magnetic variable accretion shocks.

REFERENCES

Bailey, J., Mason, K., and Parkes, G. E. 1977, M.N.R.A.S., 166,113. [

Bailey, J. and Axon, D. J. 1980, M.N.R.A.S., 194, 187 f

Bailey, J., 1988, tiffs volume.

Berg, R. A. and Duthie, J. G. 1977, Ap..L, 211,859.

Beuermana, K., Stella, L., and Patterson, J. 1987 Ap. J., 316, 360. i'

Beuermann, K. 1988a, this volume.

Beuermann, K. 1988b, in The Physics of Compact Objects: Theory Versus Observations, !

Reidel, in press.

Biermann, P., Schmidt, G. D., Liebert, J., Stockman, H. S., Tapia, S., Kuhr, H., Strittmatter, _

P. A. S., West, S., and Lamb, D. Q. 1!)8.5, Ap. J., 293,303.

Brainerd, J. J., and Lamb, D. Q. 1985, in Cataclysmic Variables and Low-Mass X-ray

Binaries, ed. J. Patterson and D. Q. Lamb (Cambridge: Reidel), p. 247.

Cropper, M., and Warner, B. 1986, M.N.ILA.S., 220,633.

Dulk, G. A., and Marsh, K. A. ]982, Ap. ,I., 259, 350.

Imamura, J. N., Durisen, R. tI., Lamb, D. Q., and Weast, G. J. 1987, Ap. J._ 313,298.

Kaluzny, J. and Semenlnk, I. 1987, Acta Astron., 37, in press. !

•. Kaluzny, J. and Chlebowski 1988, Ap. J. (Letters), in press. _

Kemp, J., Swedlund, J. and Wolstencroft, R. 1974, Ap. J. (Letters), 192, L15.

Lamb, D. Q., and Mast=_.rs, A. R. 1979, ,lp. J. (Letters), 234, Ll17. _'

Lamb, D. Q. 1985, in Cataclysmic Variables and Low-Mass X- ray Binaries, ed. J. Patterson ,.

and D. Q. Lamb (Cambridge: R.eidel), p. 179.

Lamb, D. Q., 1988, this volume _

Lanning, H. H. and Semeniuk, I. 1981, Acla A iron., 31,175.

Larsson, S., 1985, Astron. Astrophys, 145, LI. _ '

Lars_on, S., 1987a, Astrophys. and Space Sci, 130, ed. Remeis-Sternworte, Reidel. 187.

Larsson, S., 1987b, Astron. Aslrophys,181, L15. 1

Larsson, S., 1987c, "Proceedings of the COSPAR/IAU Symposium on the Physics of

Compact Objects: Theory versus Obscrvatmns." i •

Latham, D. W., Liebe_t, J., and Steiner, .]. E. 1981, Ap. J., 246,919. !i

Liebert, J., Stockman, II. S., Angel, J. R. P., Woolf, N. J., Hege, E. K., aad Margon, B.

1978, Ap. J, 219,201.

Liebert, J. and Stockman, It. S. 1979, Ap. J., 229, 652.

13

I
Liebert, J. and Stockman, It. S. 1985, in Cataclysmic Variables and Low-Mass X-ray

Binaries, eel. J. Patterson and D. Q. Lamb (Cambridge: Reidel), p. 151.

Lubow, S. H., and Shu, F. H. 1975, Ap. J., 198, 383.

Margon, B. and Szkody, P. 1978, prlvate communication.

Mason, K. O., Middleditch, J., Cordova, F. A., Jensen, K. A., Reichert, G., Murdin, P. G.,

Clark, D. and Bowyer, S., 1983, Ap. J., 264, 575.

Masters, A. 1l. 1978, Ph.D. Dissertation, Univ. Ill.

Middleditch .T., 1982, Ap. J. (Letters),257, L71.

Piirola, V., Vilhu, O., Kyrolainen, J., Sh'xkhovskoy, N. M., and Efimov, Y. S. 1985, ESA

SP-236, 256.

Piirola, V. 1988, this volume.

Piirola, V., Relz, A. and Coyne, G. V. 1988, Asb'on. Astrophys. submitted.

Schmidt, G. D., Stockman, H. S., and Margon, B. 1981, Ap. J.(Letters}, 243, L157.

Schmidt, G. D., Stockman, H. S., and Grandi, S. A. 1983, Ap. J., 271,735.

Schmidt, G. D. 1988, this volume.

Stockman, H. S., 1977,Ap. J.(LeUers), 218, L57.

Stockman, H. S. and Sargent, T. A. 1979, Ap. J., 227, 197.

Stockman, H. S., and Lubenow, A. F. 1987, Astrophysics and Space Science, 131,607.

Tapia, S. 1977, Ap..1.(Letters}, 212, 1,125.

Visvanathan, N. and Wickramasinghe, D. T. 1979, Nature, 281, 47.

Warner, B., and Nather. R. E. 1972, M.N.Id.A.S., 156, 305.

Wickramasinghe, D. T., and Ferrario, L. 1988, Ap. J., in press.

Wickramasinghe, D. T., 1988, this volume.

ql

14
:!

'!l FIG U RE CAPTION S ."

! Figure 1. Schematic i]lustratlons of three different shock geometries. The upper panel "_:

,, (a)represents the typical "pillbox" geornetry of the homogeneous shock and the basic shock

terminology. The middle panel (b) represents the radially symmetric, non-homogeneous

shock. High density flow _n the core transitions smoothly to a low-density accretion flow _'

in the wings which produce much of the observed optical cyclotron flux. TIle bottom panel

(c) illustrates a more linear structure in which the the accretion shock forms a sheath or

arc-like structure.

Figure 2. The relative shock height for a range of accretion luminosities. The four heavy

lines indicate the relative shock height for 0.4 and 1.0M® white dwarfs and two field values,

B =10,30 MG. Contours labeled 1032, 1033_ and 1034 indicate the upper limits to the

thin shock approximation for radially symmetric shocks with the corresponding luminosity

(e,gs/s).

Figure 3. The complete energy (listributi,)rl for a gaussian and quadratic accretion profile.

The model parameters are )lf 1 = 0.8./1[®, L =: 10:_3erg/s, B = 20 MG, < L/f >= 1038 erg/s t:

for both models. From Stockman and Lubcnow (1987). 1'

Figure 4. The shock characteristics w'rsus radius, r, for the quadratic accretion profile i

described in Figure 3. From Stockmal; and Lubcnow (1987).

Figure 5. A cross section of a high density accretion core. The depression of the photosphere '

is due to accretion ram pressure.

Figure 6. Two scenarios which produce polarization "reversals." The left panel (a) indicates t

the depression of the eclipse horizon due to significant shock height. The right panel (b) shows

the effect of non-normal magnetic fichl lines. Note the beaming of the polarization horizon

into the stellar surface on one side of the shock.

Figure 7. The effect of column height on shock visibility. The solid lines indicate the fraction

of the period in which a one-pole accrel, iou shock would be visible. The dotted lines indicate

the "bright" phase fraction for a shock height of h/R 1 = 0.1. Note that, in this case, accretion

on the "South Pole" can be visible for a large fraction of the period.

i

i

1

2

i

!

I

0

17 !

Figure 2. I.

I
20 _

18 _

16 _

_. 14 _

12 _ Gau_i_ A=cn_on

o

10 I J I _,_I

12 14 16 18 20

Io9 frequency (Hz)

18.5 _ +

Pcm_r_ AccratonProlh

18 _ I "" _.

-r 17.5 / _"

-- \

u / \

_. 17 _ /

_. 16.5 !

• /

o _,

16 .

o i 15.5

13.5 14 14.5 15 15.5 '

• log frequency (Hz)

#

ll"

t

10

Figure 3.
eoI- T;2.T.2 ,!

• 20

0

1500

o I I I

0.1

_ 0.05

Q

o ,. I " ! I I

1.5

_ 0.5

?

3

_._ 2 I _Peak wavelength

.u I

E

0 I i I I

40 101JL/f

Q

_ 35

30 I _ I I I

5 6 7 8 9

Io9R (cm)

19 Figure 4.
-|

Figure 5.

|

/

' 0

/

\ --h/R. = 0

10 \ -- -- h/R. =

' 20 Always Visible \ \\Never Visible

\ \

30 \ \ ,

\ \

= \ \

(1)

"o.....40 \ '

tO \ \

/ /

•=- 50 0.8 0.7 0.3 0.0

/ , \ \

-= / / i oo \

1.0 0.7 / / 0.5 \ 0.4 0.3 _ ; ;,

70 / k _/. :

/ io

i"

80 ' " :i

o.7 _

90 / k

'I, ',. I , _ , , I ,/, I /

0 30 60 90 120 150 180 '

Magnetic Colatitude (deg.) i

i

22 i.

Figure 7.