Compton Scattering of Fe K alpha Lines in Magnetic Cataclysmic Variables
Mon. Not. R. Astron. Soc. 000, 1–11 (2006) Printed 22 November 2018 (MN LATEX style file v2.2)
Compton Scattering of Fe Kα Lines in Magnetic
Cataclysmic Variables
A. L. McNamara1⋆, Z. Kuncic1⋆, K. Wu1,2, D. K. Galloway3† and J. G. Cullen4
1School of Physics, University of Sydney, NSW 2006, Australia
2Mullard Space Science Laboratory, University College London, Holmbury St Mary, Surrey, RH5 6NT, UK
3School of Physics, University of Melbourne, Victoria 3010, Australia
4Thales, Garden Island, Cowper Wharf Rd, Potts Point, NSW 2011, Australia
Accepted ... . Received ...; in original form ...
ABSTRACT
Compton scattering of X-rays in the bulk flow of the accretion column in magnetic cat-
aclysmic variables (mCVs) can significantly shift photon energies. We present Monte
Carlo simulations based on a nonlinear algorithm demonstrating the effects of Comp-
ton scattering on the H-like, He-like and neutral Fe Kα lines produced in the post-
shock region of the accretion column. The peak line emissivities of the photons in
the post-shock flow are taken into consideration and frequency shifts due to Doppler
effects are also included. We find that line profiles are most distorted by Compton
scattering effects in strongly magnetized mCVs with a low white dwarf mass and high
mass accretion rate and which are viewed at an oblique angle with respect to the ac-
cretion column. The resulting line profiles are most sensitive to the inclination angle.
We have also explored the effects of modifying the accretion column width and using
a realistic emissivity profile. We find that these do not have a significant overall effect
on the resulting line profiles. A comparison of our simulated line spectra with high
resolution Chandra/HETGS observations of the mCV GK Per indicates that a wing
feature redward of the 6.4 keV line may result from Compton recoil near the base of
the accretion column.
Key words: accretion – line: profiles – scattering – binaries: close – white dwarfs –
X-rays
1 INTRODUCTION
Magnetic cataclysmic variables (mCVs) are close interact-
ing binaries consisting of a magnetic white dwarf (WD) and
a low mass red dwarf (Warner 1995). Near the white dwarf
surface the accretion flow in mCVs is confined by the mag-
netic field of the WD and is channelled to the magnetic
pole region(s) of the WD, forming an accretion column (see
Warner 1995; Cropper 1990; Wu 2000; Wu et al. 2003, for
reviews).
Near the base of the accretion column, material in su-
personic free-fall is brought to rest on the WD surface, form-
ing a standing shock which heats and ionizes the accreting
plasma. The shock temperature Ts depends mainly on the
mass Mwd and radius Rwd of the WD and is given by (e.g.
Wu 2000)
kTs =
GMwdµmH
Rwd + xs
, (1)
⋆ E-mail:aimee, kuncic@physics.usyd.edu.au
† Centenary Fellow
where µ is the mean molecular weight and xs is the
shock height. For WD masses Mwd ≈ (0.5 − 1.0)M⊙
and typical mCV parameters, the shock temperature is
kTs ≈ (10 − 40) keV ≈ (1 − 4) × 10
8 K. The plasma
in the post-shock region of the column cools by emitting
bremsstrahlung X-rays and optical/IR cyclotron radiation
(Lamb & Masters 1979; King & Lasota 1979). Since cooling
occurs along the flow, the post-shock region is stratified in
density and temperature. The height of the post-shock re-
gion is determined by the cooling length. For a flow with
only bremsstrahlung cooling, the shock height is given by
(Wu, Chanmugam & Shaviv 1994)
xs ≈ 3× 10
1 g cm−2 s−1
0.5M⊙
109cm
)−3/2
cm (2)
where ṁ is the specific mass accretion rate.
A plasma temperature of kT ≈ 10 keV is sufficient to
fully ionize elements such as argon, silicon, sulphur, alu-
minium or calcium. Heavier elements such as iron can be
c© 2006 RAS
http://arxiv.org/abs/0704.1516v1
2 McNamara et al.
highly ionized, resulting in H-like Fe XXVI and He-like Fe
XXV ions. K-shell transitions in Fe XXVI and Fe XXV ions
give rise to Kα lines at 6.97 keV and 6.675 keV, respectively
(see e.g. Wu, Cropper & Ramsay 2001). The irradiation of
low ionized and neutral iron by X-rays above the Fe K edge
produces fluorescent Kα emission at approximately 6.4 keV
inside the WD atmosphere, beneath the accretion column
and in surrounding areas. The natural widths of the Fe Kα
lines are small, but the lines can be Doppler broadened by
the bulk and thermal motions of the emitters in the post-
shock flow. The bulk velocity immediately downstream of
the shock is ≈ 0.25 (GMwd/Rwd)
∼ 1000 kms−1 for typ-
ical mCV parameters. Lines can also be broadened by scat-
tering processes. Compton (electron) scattering is expected
to be more important than resonance (ion) scattering for the
Kα transitions (Pozdnyakov, Sobol & Sunyaev 1983). For
an mCV with specific accretion rate ṁ ∼ 10 g cm−2 s−1
the electron number density is ne ∼ 10
16 cm−3 for a shock
heated region of thickness xs ∼ 10
7 cm, giving a Thompson
optical depth of τ ∼ 0.1. Thus, one in every ten photons
would encounter an electron before leaving the post-shock
region. The relative importance of Doppler shifts, thermal
Doppler broadening and Compton scattering depends on the
ionization structure in the post-shock flow.
X-ray observations by Chandra/HETGS and
ASCA/SIS have revealed significant broadening of some
Fe Kα lines in mCVs (Hellier, Mukai, & Osborne, 1998;
Hellier & Mukai 2004). It was suggested that Compton
scattering in the accretion column is largely responsible for
the broadening in the observed lines. Doppler broadening
should only be significant in lines emitted close to the
shock. The absence of Doppler shifts in the observed H-like
and He-like lines suggests that these photons may be
emitted predominantly from regions of lower velocity near
the base of the accretion column (Hellier & Mukai 2004).
The observed line profiles have yet to be fully interpreted
with a quantitative model that takes into account Compton
scattering effects in a complex ionization structure.
In this paper, we study the effects of Compton scat-
tering in the accretion column of mCVs using a non-
linear Monte Carlo algorithm (Cullen 2001a,b) that self-
consistently takes into account the density, velocity and tem-
perature structure in the column (Wu et al. 2001). The ef-
fects of dynamical Compton scattering are also included. In
a preliminary investigation (Kuncic, Wu & Cullen 2005), it
was found that Fe Kα line photons emitted from the dense
base of the accretion column undergo multiple Compton
scatterings and as a result, the base of the line profile is
substantially broadened. Photons emitted near the shock
can also undergo scatterings with hot electrons immediately
downstream of the shock, as well as cold electrons in the
pre-shock flow before escaping the column. The resulting
line profiles display a shoulder-like feature redward of the
line centre. More significant broadening is observed when cy-
clotron cooling is sufficiently strong to produce a dense, com-
pact post-shock region (see Wu 2000, for example). Here, we
make three substantial improvements to the previous study:
(i) we include Doppler effects; (ii) the photon source regions
in the post-shock column are determined from the ioniza-
tion structure, rather than specified arbitrarily; and (iii) the
effects of different viewing angles are fully explored. The
paper is organized as follows: the theoretical outline and ge-
ometry of the model are described in Section 2. Numerical
results for cases where cyclotron cooling is negligible and
when it dominates are presented and discussed in Section 3.
A summary and conclusions are presented in Section 4.
2 THEORETICAL MODEL
2.1 Physical Processes
Line photons can undergo energy changes when scattering
with electrons, resulting in distortions in the line profile. The
energy change of a photon per scattering is given by (e.g.
see Pozdnyakov et al. 1983),
1− µβ
1− µ′β + E
γmec2
(1− cosα)
where E is the initial photon energy, γmec
2 is the electron
energy, with γ = (1 − β2)−1/2 and where β = v/c includes
both thermal and bulk motion, θ = cos−1 µ is the incident
photon propagation angle measured relative to the electron’s
direction of motion, and the scattering angle is α. The prime
superscript denotes quantities after a scattering event. Al-
though the energy change per scattering is typically small,
the line profile can be broadened considerably as a result of
multiple scatterings if the optical depth is large.
Photons scattering with hot electrons (i.e. kTe > Ec
where Te is the electron temperature and Ec is the line centre
energy) will gain energy, while photons scattering with cold
electrons (kTe ≪ Ec) will lose energy due to recoils. In the
mCV context, Compton recoil can be important for photons
scattering with electrons in the cold pre-shock flow and also
near the base of the column, where β rapidly decreases and
where the optical depth is high. From equation (3), the frac-
tional energy change due to Compton recoil
E ≫ γmec
(1− cosα) . (4)
In the post-shock accretion column in mCVs, line pho-
tons undergo thermal Doppler broadening as well as Doppler
shifts. The bulk velocity of the accreting material immedi-
ately downstream of the shock is ∼ 1000 km s−1. For all
inclination angles i (see Figure 1), the bulk flow is moving
away from our line of sight, so the line centre energy is red-
shifted by an amount ∆E/E ∼ β cos i. However, since the
bulk velocity in the post-shock flow in mCVs is always less
than a few × 1000 kms−1, giving β < 10−2, Doppler shifts
are expected to be negligible. Thermal Doppler broaden-
ing, on the other hand, is expected to be of order ∆E/E ∼
(2kTs/mic2) ≈ 0.002 for lines emitted in the hottest re-
gions of the post-shock flow (i.e. immediately downstream
of the shock), where mi is the mass of the ion.
In strongly magnetized mCVs, cyclotron emission is the
dominant cooling process. The effect of this additional elec-
tron cooling process in the post-shock region is to reduce
the shock height and modify the density and temperature
structure of the region (Wu et al. 1994). This can enhance
Compton scattering features in line profiles (Kuncic et al.
2005).
c© 2006 RAS, MNRAS 000, 1–11
Compton Scattering of Fe Kα Lines in mCVs 3
Figure 1. Schematic illustration of the geometry of the magne-
tized white dwarf accretion column showing approximate loca-
tions of Fe Kα source regions.
2.2 Geometry of the Accretion Column
The geometry of an mCV accretion column is shown in fig-
ure 1. The column is modelled as a cylinder and is divided
into a shock heated region and a cool pre-shock region. In
the pre-shock region, the density and velocity of the accret-
ing material are constant. The velocity of the pre-shock flow
is approximated by the free-fall velocity at the shock height,
vff(xs) = [GMwd/(Rwd+xs)]
1/2, and the plasma is assumed
to be cold (kTe ≪ 1 keV). The electron number density in
the pre-shock flow is ne = ṁ/(µmHvff), where ṁ is the spe-
cific mass accretion rate. In the post-shock region, the den-
sity, velocity and temperature profiles are calculated using
the hydrodynamic solution described in Wu et al. (1994),
for bremsstrahlung and cyclotron cooling. The parameters
that determine the structure of the post-shock region are
the WD mass Mwd, the WD radius Rwd, the specific mass
accretion rate ṁ and the ratio of the efficiencies of cyclotron
cooling to bremsstrahlung cooling ǫs.
The accretion column is viewed from an inclination an-
gle i, measured relative to the column axis (see Fig. 1). An
inclination angle i = 0◦ corresponds to viewing the column
along its axis, towards the WD, while viewing the column
from an inclination angle i = 90◦ is equivalent to viewing
the column from the side.
The line photons are injected into the post-shock region
at a specific dimensionless height ζ ≡ x/xs at or above the
WD surface, where ζ = 0 and ζ = 1 correspond to the
WD surface and shock surface, respectively (see Fig. 1). For
each set of mCV parameters, the injection point of the line
corresponds to the location in the post-shock region where
the emissivity of the specific line peaks, according to the
ionization structure determined by Wu et al. (2001). The
bulk and thermal velocities at the line injection height ζ are
used to calculate the Doppler shift and broadening of the
lines.
The Monte Carlo technique is used to model Comp-
ton scattering effects on photon propagation in the column.
The distance to a tentative scattering point is determined
using an algorithm based on a nonlinear transport tech-
nique (Stern et al. 1995) which integrates the mean free path
over the spatially varying electron density (Cullen 2001a,b).
The scattering cross-section is determined from the Klein-
Nishina formula and the momentum vector at the scattering
point is drawn from an isotropic Maxwellian distribution at
the local temperature. A rejection algorithm is used to de-
cide whether the scattering is accepted (Cullen 2001b). For
an accepted event the energy and momentum changes of
the photon are calculated as described in Pozdnyakov et al.
(1983). In each simulation photons are followed until they
leave the column and binned to form a spectrum. A full de-
scription of the numerical algorithm can be found in Cullen
(2001a).
In the simulations described below, we make the follow-
ing simplifying assumptions: a fixed number of 108 photons
are used to simulate each line; a single energy is used for
each line (whereas in reality, the neutral and Lymann α
transitions are doublets, with energies 6.391/6.404 keV and
6.952/6.973 keV, and the He-like transition has both reso-
nant, inter-combination and forbidden components); a fixed
accretion column width is used and a single line injection
site is used.
3 RESULTS AND DISCUSSION
We present simulations of Compton scattering of Fe Kα
lines in an mCV accretion column for two different WD
mass-radius values: Mwd = 0.5M⊙, Rwd = 9.2 × 10
and Mwd = 1.0M⊙, Rwd = 5.5× 10
8 cm (Nauenberg 1972).
For each mass, we consider two different specific mass ac-
cretion rates, ṁ = 1 g cm−2 s−1 and ṁ = 10 g cm−2 s−1,
and inclination angles i = 0◦, 45◦ and 90◦ with ±5◦ range.
The cross-sectional radius of the accretion column is fixed
at 0.1Rwd. We investigate the effect of varying the column
width in section 3.3.
Figure 2 shows the temperature, velocity and density
profiles of an accretion column for an mCV with WD mass
Mwd = 0.5M⊙ and ṁ = 10 g cm
−2 s−1, for cases where the
ratio of cyclotron to bremsstrahlung cooling at the shock is
ǫs = 0, 10, and 100. The temperature decreases monotoni-
cally from a shock temperature kTs ≈ 14 keV (ζ = 1), to a
small finite value at the base of the column (ζ = 0).The
mass density in the post-shock region is determined by
ρ(ζ) = 4ṁ/vb(ζ) and is a minimum at the shock and reaches
a maximum at the base of the column. The bulk velocity of
the accreting material at the shock is 0.25vff and decreases
to zero at the base of the column, where the plasma settles
on the WD surface.
Figures 3 and 4 show the simulated line spectra
for the case where cyclotron cooling is negligible and
bremsstrahlung cooling dominates (ǫs = 0). Figure 5 shows
the simulated profiles for Mwd = 0.5M⊙ and Mwd =
1.0M⊙ when cyclotron cooling dominates bremsstrahlung
(ǫs = 10). In these cases, the neutral Fe Kα line is emitted
at the WD surface (ζ = 0), where the bulk velocity is ap-
c© 2006 RAS, MNRAS 000, 1–11
4 McNamara et al.
0.0 0.2 0.4 0.6 0.8 1.0
εs = 0
εs = 10
εs = 100
0.0 0.2 0.4 0.6 0.8 1.0
εs = 100
εs = 10
εs = 0
0.0 0.2 0.4 0.6 0.8 1.0
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
εs = 0
εs = 10
εs = 100
Figure 2. The profiles of (a) electron temperature Te, (b) mass density ρe and (c) bulk velocity vb = βbc in the post-shock accretion
column of an mCV with Mwd = 0.5M⊙ and ṁ = 10 g cm
−2 s−1. ζ = 0 corresponds to the base of the accretion column and ζ = 1 to
the shock. The solid line shows the profile for the case where the ratio of cyclotron to bremsstrahlung cooling at the shock is ǫs = 0, the
dotted line for ǫs = 10 and the dashed line for ǫs = 100. In (a) the corresponding mean thermal electron velocity βe = ve/c is also shown
as a function of ζ.
proximately zero and the thermal velocity of the plasma is
small. The 6.675 keV Fe Kα line is emitted from the lowest
few percent of the column (ζ ∼ 0.003) where the velocity
of the infalling material and the thermal electron velocity
are still relatively small. These lines thus show very little
Doppler broadening. The 6.97 keV line, however, is emit-
ted much closer to the shock (ζ ∼ 0.16), in regions where
the temperature of the accreting material is considerably
higher and thus suffers more substantial Doppler broadening
(see Wu et al. 2001). The simulated line spectra for the case
where the accretion column radius is fixed at 4.6 × 107 cm
is shown in Figure 6. Figure 7 shows the simulated pro-
files for the case where the line photon injection along the
flow is specified according to the emissivity profile model of
Wu et al. (2001). In figure 8 we compare our simulated line
spectra with an observation of the mCV GK Per detected
by Chandra/HETGS.
Table 1 shows a comparison of the FWHM of the Fe Kα
lines for Mwd = 1.0M⊙ and Mwd = 0.5M⊙ with low and
high ṁ and for ǫs = 0 and ǫs = 10.
3.1 No Cyclotron Cooling
When cyclotron cooling is negligible (ǫs = 0), the 6.4, 6.675
and 6.97 keV lines are mostly emitted at heights above the
WD surface of x ≈ 0, 0 and 1.3× 106 cm for Mwd = 1.0M⊙
and x ≈ 0, 1.0 × 105, and 5.1 × 106 cm for Mwd = 0.5M⊙,
respectively (Wu et al. 2001). Note that Doppler broaden-
ing is most significant for the 6.97 keV line which is emit-
ted further up in warmer regions of the post-shock column.
Figures 3 and 4 show that in all cases, there is substantial
Compton broadening near the base of the line profiles due to
scatterings in the post-shock region where photons may lose
and gain energy. For example, for the 6.4 keV line emerging
at i = 90◦ in Fig. 3a, the Compton broadened wings con-
tain ≈ 28% of the total line photons. The blue wing extends
up to ∆E ≈ 0.09 keV above the line centroid, while the
red wing extends down to ∆E ≈ 0.16 keV (see Appendix).
Compton features are generally more prominent at higher
accretion rates since this gives higher optical depths in the
column and hence, a higher scattering probability. For com-
parison, the average number of scatterings per photon in the
Table 1. FWHM (eV) values of the (a) 6.4 keV, (b) 6.675 keV
and (c) 6.97 keV lines calculated for two different values of ṁ and
for Mwd = 1.0M⊙ and Mwd = 0.5M⊙. The ratios of cyclotron
to bremsstrahlung cooling at the shock are ǫs = 0 and ǫs = 10.
(a) Mwd = 1.0M⊙ Mwd = 0.5M⊙
6.4 keV ṁ = 1 (10) g cm−2 s−1 ṁ = 1 (10) g cm−2 s−1
ǫs = 0 5 (7) 4 (5)
= 10 4 (5) 3 (4)
(b) Mwd = 1.0M⊙ Mwd = 0.5M⊙
6.675 keV ṁ = 1 (10) g cm−2 s−1 ṁ = 1 (10) g cm−2 s−1
ǫs = 0 6 (7) 4 (5)
= 10 5 (6) 3 (5)
(c) Mwd = 1.0M⊙ Mwd = 0.5M⊙
6.97 keV ṁ = 1 (10) g cm−2 s−1 ṁ = 1 (10) g cm−2 s−1
ǫs = 0 8 (9) 10 (10)
= 10 7 (15) 8 (10)
ṁ = 1g cm−2 s−1 case is 0.5 but this increases to 2 in the
ṁ = 10 g cm−2 s−1 case.
For Mwd = 0.5M⊙ (Fig. 4), a recoil tail redward of the
line centroid can be seen in the cases i = 45◦ and 90◦ specif-
ically for high ṁ (Fig. 4b) The shock temperature for these
cases is kTs ≈ 14 keV, while for the Mwd = 1.0M⊙ case, the
shock temperature is much higher (kTs ≈ 34 keV for low ṁ
and kTs ≈ 46 keV for high ṁ). The plasma temperature in
the 1.0 M⊙ case is too high to produce any downscattering
features and a large Mwd (small Rwd) has an accretion col-
umn with a lower optical depth, hence the absence of these
features in Fig. 3. Again the downscattering features are
more enhanced in the high ṁ cases since the optical depth
is higher (c.f. Figs. 4a and 4b for i = 90◦). Note also that
the 6.97 keV line centre is Doppler shifted slightly redward
which is most noticeable for the 0.5 M⊙ case.
In addition to the recoil signatures, upscattering fea-
tures are also seen in the line spectra for the high ṁ cases
(Figs. 3b and 4b) when i = 90◦. For these inclination an-
c© 2006 RAS, MNRAS 000, 1–11
Compton Scattering of Fe Kα Lines in mCVs 5
(a) Mwd = 1.0M⊙, ṁ = 1 g cm
−2 s−1, kTs = 34 keV, no cyclotron cooling
i = 0
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 45
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 90
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
(b) Mwd = 1.0M⊙, ṁ = 10 g cm
−2 s−1, kTs = 46 keV, no cyclotron cooling
i = 0
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 45
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 90
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
Figure 3. Profiles of Fe Kα lines scattered by electrons in the accretion column of an mCV with Mwd = 1.0M⊙ and with (a)
ṁ = 1 g cm−2 s−1 and (b) ṁ = 10 g cm−2 s−1. The shock temperature kTs is indicated and the ratio of cyclotron to bremsstrahlung
cooling is ǫs = 0.
gles, photons emerge from the column with a final scattering
angle cosine µ′ ≈ 0. Equation 3 then gives
≈ 1− µβ . (5)
Since head on collisions (µ < 1) have a higher probability,
E′ ≃ (1+|µ|β)E. The upscattering features are more promi-
nent in Fig. 3b than in Fig. 4b because the shock temper-
ature, and hence β, is higher. The sharpness of the upscat-
tering features seen in the 6.4 keV profiles in particular can
be attributed to the maximum possible energy gain when
µ ≈ −1 and to very few hot electrons with energies beyond
a few standard deviations of the mean thermal (Maxwellian)
energy.
Scattering features are less prominent in the low ṁ
cases (Figs. 3a and 4a), since the overall optical depth is
smaller. For Mwd = 1.0M⊙ and ṁ = 1 g cm
−2 s−1 for in-
stance, the optical depth across the column at the shock
is τ ≈ 0.04, which is a factor 10 smaller than that for the
ṁ = 10 g cm−2 s−1 case (since τ ∝ ne ∝ ṁ). For higher
WD masses (i.e. smaller Rwd), the optical depth across the
column is smaller, so fewer photons are scattered, especially
for low ṁ cases. This explains the difference in scattering
features in the profiles shown in Figs. 3 and 4 (in particular,
the high ṁ, i = 90◦ cases).
Photons observed at an inclination angle of i = 0◦ prop-
agate through the entire length of the column before es-
caping. These photons propagate through a thick section of
cold pre-shock flow and can thus downscatter, resulting in
broadening redward of the line centre. However, the electron
number density, and hence, optical depth, in the pre-shock
flow is small so recoil effects are correspondingly small. Fur-
thermore, fewer photons are detected at i = 0◦ because the
solid angle centered around i = 0◦ is considerably smaller
than that centered around larger inclination angles.
3.2 Cyclotron Cooling Dominated Flows
Figure 5 shows the profiles of Fe Kα lines emitted from
mCVs with Mwd = 1.0M⊙ and Mwd = 0.5M⊙ where
i = 90◦, ṁ = 10 g cm−2 s−1 and the ratio of cyclotron to
bremsstrahlung cooling at the shock is ǫs = 10. Also plotted
is the thermal Doppler broadened line profile before scat-
tering (dotted curve). The additional cooling of the plasma
in the post-shock flow enhances the density and results in
larger scattering optical depths. Because the electron tem-
perature of the shock-heated region decreases as ǫs increases,
the peak emissivities of the 6.675 keV and 6.97 keV Fe lines
are found at larger values of ζ, as can be seen in Fig. 2, which
shows the temperature, density and velocity profiles for dif-
ferent cases of ǫs. The additional cyclotron cooling results in
a decrease in the electron temperature and bulk velocity in
the post-shock column at a fixed ζ.
For ǫs = 10, the 6.4, 6.675 and 6.97 keV lines are emit-
ted from heights above the WD surface of x ≈ 0, 3.1 × 105
and 2.4 × 107 cm for Mwd = 0.5M⊙, and x ≈ 0, 0 and
5.3 × 107 cm for Mwd = 1.0M⊙, respectively (Wu et al.
2001). The emission height of the 6.97 keV line is signifi-
cantly higher than in the ǫs = 0 case and consequently,
c© 2006 RAS, MNRAS 000, 1–11
6 McNamara et al.
(a) Mwd = 0.5M⊙, ṁ = 1 g cm
−2 s−1, kTs = 14 keV, no cyclotron cooling
i = 0
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 45
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 90
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
(b) Mwd = 0.5M⊙, ṁ = 10 g cm
−2 s−1, kTs = 14 keV, no cyclotron cooling
i = 0
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 45
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
i = 90
6.0 6.2 6.4 6.6 6.8 7.0 7.2
Energy (keV)
Figure 4. Profiles of Fe Kα lines scattered by electrons in the accretion column of an mCV with Mwd = 0.5M⊙ and with (a)
ṁ = 1 g cm−2 s−1 (top panels) and (b) ṁ = 10 g cm−2 s−1 (bottom panels). The shock temperature kTs is indicated and the ratio of
cyclotron to bremsstrahlung cooling is ǫs = 0.
6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2
Energy (keV)
6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2
Energy (keV)
Figure 5. Simulated profiles of Compton scattered Fe Kα lines in the accretion column of an mCV with (a) Mwd = 1.0M⊙ and (b)
Mwd = 0.5M⊙. In both cases the inclination angle is i = 90
◦, the specific mass accretion rate is ṁ = 10 g cm−2 s−1 and the ratio of
cyclotron to bremsstrahlung cooling at the shock is ǫs = 10. The dotted curves show the contribution to each line profile by thermal
Doppler broadening only.
thermal Doppler broadening is more prominent. The addi-
tional cooling in the accretion column also results in a lower
plasma temperature at the base of the column, ζ = 0 (see
Fig. 2), near where irradiation of neutral iron occurs and the
fluorescent 6.4 keV line is emitted. This line thus undergoes
less thermal Doppler broadening than in the ǫs = 0 case. As
a result of Compton scattering, the line profiles show addi-
tional broadening and recoil tails redward of the line centre
(Fig. 5b). Upscattering features are also evident in the 6.4
and 6.675 keV lines especially for Mwd = 1.0M⊙ (Fig. 5a).
Overall, Compton scattering features are more promi-
nent for cyclotron cooling dominated accretion columns,
especially for high accretion rates. For photons viewed at
i = 90◦, upscattering and recoil features that are only
marginally seen in the ǫs = 0 case are considerably more
pronounced in the ǫ = 10 case. Thus, we expect Compton
scattering features to be most conspicuous in Fe Kα lines
emitted in strongly magnetized mCVs accreting at a high
rate.
c© 2006 RAS, MNRAS 000, 1–11
Compton Scattering of Fe Kα Lines in mCVs 7
6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2
Energy (keV)
6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2
Energy (keV)
Figure 6. Simulated profiles of Compton scattered Fe Kα lines in an accretion column of an mCV with (a) Mwd = 1.0M⊙ and (b)
Mwd = 0.5M⊙ both with a specific mass accretion rate of ṁ = 10 g cm
−2 s−1, a ratio of cyclotron to bremsstrahlung cooling of ǫs = 10,
an inclination angle of i = 90◦ and accretion column radius of 4.6×107 cm . The dotted curves show the profiles when the column radius
is fixed at 0.1Rwd (equivalent to the profiles shown in Fig.5).
3.3 Effect of Accretion Column Radius
The width of the accretion column in mCVs is poorly known
and may vary significantly from system to system. In the
results presented so far, we have fixed the accretion col-
umn radius to be 0.1Rwd. Here, we investigate the effect
of relaxing this assumption. Figure 6 shows the resulting
profiles of Fe Kα lines emitted from two mCVs of different
masses (Mwd = 1.0M⊙ and Mwd = 0.5M⊙) but with the
same absolute accretion column radius of 4.6× 107 cm. This
corresponds to 0.08Rwd for the 1.0M⊙ case (Fig. 6a) and
0.05Rwd for the 0.5M⊙ case (Fig. 6b). The other parame-
ters used to produce the profiles are the same as those used
for Fig. 5, namely ṁ = 10 g cm−2 s−1, ǫs = 10 and i = 90
Fig. 6 shows that the changes in the accretion column width
generally result in changes near the base of the line profiles,
which is broadened by multiple scatterings. The effect of in-
creasing (decreasing) the width of the accretion column is
to increase (decrease) the overall scattering optical depth.
The Thompson optical depth across the base of the ac-
cretion column is ∼ 20 for the 1.0M⊙ case (Fig. 6a) and
∼ 40 for the 0.5M⊙ case (Fig. 6b). In comparison, the
Thompson optical depth across the column base for the
same lines when the column has a width of 0.1Rwd (dot-
ted curves in Fig. 6) are ∼ 25 for Mwd = 1.0M⊙ and ∼ 80
for Mwd = 0.5M⊙. Fig. 6 indicates that due to the nonlin-
ear nature of multiple scatterings in the accretion column,
particularly near the base, small changes in optical depth
associated with the accretion column geometry can result
in significant changes in line profiles. These changes mostly
effect the base of the line profiles.
3.4 Emissivity Profile Effects
Realistically, the photon source regions are determined by
an emissivity profile along the post-shock region of the ac-
cretion column. In the results presented so far, all the pho-
tons were injected at a single height in the accretion column
corresponding to the location of the peak in the emissiv-
ity profile, as calculated by Wu et al. (2001). Here, we in-
vestigate the effect of spreading the photon injection site
over a finite range of heights in the post-shock region ac-
cording to the calculated emissivity profiles. Only the 6.675
6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2
Energy (keV)
Figure 7. Simulated profiles of 6.675 keV and 6.97 keV Fe lines
in an mCV accretion column with Mwd = 1.0M⊙, a specific
mass accretion rate of ṁ = 1 g cm−2 s−1, no cyclotron cooling
and viewed at an inclination angle of i = 45◦ (solid lines). An
emissivity profile is used to disperse the photon injection site
over a finite range of heights along the post-shock column. The
dotted curves show the corresponding profiles for photons injected
at a single height where the emissivity peaks for the same mCV
parameters (equivalent to the profiles shown in Fig 3a, middle
panel).
keV and the 6.97 keV line are studied in this manner since
much of the fluorescence 6.4 keV yield derives from beneath
the column, where X-ray irradiation is strongest so the in-
jection site is always at the base of the column. Figure 7
shows an example of the emissivity profile effect for the
6.675 keV and 6.97 keV Fe Kα lines emitted in an mCV
with Mwd = 1.0M⊙, ṁ = 1 g cm
−2 s−1, ǫs = 0 and i = 45
The dotted curves show the corresponding profiles for the
case where photons are injected at a single height (where the
emissivity peaks) for the same mCV parameters. These pro-
files are the same as those shown in Fig. 3a for the i = 45◦
case (middle panel).
The line profiles calculated using a realistic emissivity
profile (solid curves in Fig. 7) show some additional smear-
ing, particularly redward of the line centre, compared to
the single injection site case (dotted curves in Fig. 7). This
can be attributed to a small fraction of photons now being
emitted from regions closer to the shock, where the tem-
perature is ≈ 34 keV and the flow speed is ≈ 1230 km.s−1.
c© 2006 RAS, MNRAS 000, 1–11
8 McNamara et al.
Thus, these photons are more affected by Doppler effects
than other photons emitted from regions further away from
the shock. The small smearing redward of the line centre is
therefore due to a combination of Doppler shift and ther-
mal broadening. The effect is not prominent because only
a small fraction of photons are emitted close to the shock,
as predicted by the emissivity profile (Wu et al. 2001). The
FWHM for the 6.675 keV and 6.97 keV line is ≈ 8 eV and ≈
9 eV respectively. This corresponds to an additional broad-
ening of approximately 25% for the 6.675 keV line and 11%
for the 6.97 keV line.
3.5 Comparison with Observations
Chandra/HETGS observations of a number of mCVs were
reported by Hellier & Mukai (2004). The highest signal-to-
noise spectrum was obtained in two observations of the mCV
GK Per during its 2002 outburst. This system is classified
as an intermediate polar (IP); the WD accretor is thought
to have a magnetic field of ≈ 1MG (Hellier & Mukai 2004),
and a mass of Mwd > 0.87±0.24M⊙ (Morales Rueda et al.
2002). Its typical X-ray luminosity in the 0.17−15 keV band-
pass is Lx ≈ 7.14 × 10
33 erg s−1 (Vrielmann et al. 2005),
which places a lower limit on the specific mass accretion
rate, ṁ > 7.3 g cm−2 s−1 (adopting Rwd ≈ 6.7× 10
8 cm, in-
ferred using the Nauenberg (1972) mass-radius relation, and
assuming that accretion proceeds onto a fraction ≈ 10−3 of
the WD surface area; see e.g. Frank, King & Raine 2002).
Here we present a pilot study of GK Per and produce a
simple comparison between the profiles of the simulated and
observed spectra. A more detailed analysis of the data is left
for future study. The summed first-order HETGS spectrum
from the 2002 Chandra observations of GK Per is shown
in Figure 8. The He-like and fluorescence Fe Kα lines are
detected, and there is a possible excess at the expected en-
ergy for the H-like line at 6.97 keV. Also shown in Fig. 8 are
the upper and lower mass simulated HETGS spectra from
a 10 ks Chandra observation of an mCV with parameters
appropriate for GK Per: an upper mass of Mwd = 1.11M⊙
with Rwd = 5×10
8 cm, a lower mass ofMwd = 0.63M⊙ with
Rwd = 8.2× 10
8 cm and for both masses ṁ = 10 g cm−2 s−1
and ǫs = 4 × 10
−5. We adopted inclination angles aver-
aged over an orbital period of GK Per (50◦ 6 i 6 90◦)
(Hellier, Harmer & Bendmore 2004). We also added a nor-
malized bremsstrahlung continuum with an electron tem-
perature kTe = 11 keV (Vrielmann et al. 2005).
The measured equivalent widths of the 6.4, 6.675 and
6.97 keV lines in the Chandra/HETGS spectra of GK Per
are 260, 117 and 80 eV, respectively. The relative strength
of the lines in the simulated spectra is fixed by the number
of photons used in each simulation and the assumed contin-
uum flux level. We used 108 photons for all three lines, so
that their relative strengths are comparable. We intend to
relax this condition in future work. There is a remarkable
similarity between the observed and the simulated spectra
for the 6.4 keV line in particular. Realistically, the actual
strengths of the 6.675 and 6.97 keV lines depend on the ion-
isation structure of the flow (Wu et al. 2001), since this de-
termines the emissivity of the lines. The H-like and He-like
lines are thus potentially powerful diagnostic tools that can
be used to deduce the physical properties of the post-shock
accretion column in mCVs. The 6.4 keV line strength, on the
other hand, depends on the flux of the illuminating X-rays,
which is largest near the WD surface. Fluorescent lines may
also be produced in the surface atmospheric region around
the accretion column (Hellier & Mukai 2004), and this may
contribute to the observed equivalent width of the 6.4 keV
line.
The observed fluorescent line in GK Per exhibits a red
wing extending to 6.33 keV which Hellier & Mukai (2004)
attribute to Doppler shifts. In general we find that bulk
Doppler shifts have a negligible effect on Fe Kα lines. Our
simulated spectra show a weaker red wing arising from
downscatterings near the base of the accretion column. The
observed spectrum also exhibits a shoulder extending 170 eV
redward of the 6.4 keV line centre, which Hellier & Mukai
(2004) suggest may be due to Compton downscattering. Our
simulations confirm that recoil can indeed affect the fluores-
cent line when it is emitted at the base of the accretion
column, although in our spectra (Fig. 8) there is no clear
evidence that recoil extends down to 170 eV redward of the
6.4 keV line.
4 SUMMARY AND CONCLUSION
We have investigated line distortion and broadening effects
due to Compton scattering in the accretion column of mCVs
using a nonlinear Monte Carlo technique that takes into ac-
count the nonuniform temperature, velocity and density pro-
files of the post-shock column. Scattered Fe Kα lines were
simulated for a range of different physical parameters: white
dwarf mass, accretion rate, magnetic field strength and in-
clination angle. The photon source regions in the post-shock
flow were determined from the ionization structure and the
effects due to the bulk velocity Doppler shift and thermal
broadening are also considered.
We find that line profiles are most affected by Compton
scattering in the cases of low white dwarf mass, high specific
mass accretion rate, strong cyclotron cooling and oblique in-
clination angles. Both a lower white dwarf mass (or equiv-
alently a larger white dwarf radius) and a higher accretion
rate result in a higher optical depth, hence more pronounced
scattering effects. Strong cyclotron cooling associated with
a high magnetic field strength results in a higher density
in the post-shock flow and hence, a correspondingly larger
optical depth. Recoil signatures are evident for oblique view-
ing angles when cyclotron cooling becomes important. These
are due to scatterings near the base of the flow, where both
the dynamical and thermal velocities are small. Sharp up-
scattering features are also seen in the line profiles for large
viewing angles. These upscattering features are attributed to
head-on collisions with hot electrons near the shock. Pho-
tons emitted close to the shock in the accretion column dis-
play more thermal Doppler broadening. This is generally
the case for the Fe XXVI 6.97 keV line. The Fe 6.4 keV and
Fe XXV 6.675 keV lines are generally emitted at or near the
base of the accretion column respectively, and show a rel-
atively small amount of thermal Doppler broadening. Bulk
velocity induced Doppler shifts are negligible compared with
scattering.
We also investigated the effects of dispersing the 6.675
and 6.97 keV line photons using an emissivity calculated
self-consistently from the ionization structure of the post-
c© 2006 RAS, MNRAS 000, 1–11
Compton Scattering of Fe Kα Lines in mCVs 9
6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1
Energy (keV)
0.0001
0.0002
0.0003
0.0004
0.0005
Observed Spectrum
Low Mass Simulated Spectrum
High Mass Simulated Spectrum
Figure 8. Fe Kα emission lines for the mCV GK Per detected by Chandra/HETGS (bottom) and simulated by our model for a lower and
upper white dwarf mass Mwd = 0.63M⊙ (middle) and Mwd = 1.11M⊙ (top) both with a specific mass accretion rate ṁ = 10 g cm
−2 s−1.
The simulated spectra have been convolved with the Chandra/HETGS response function. Cyclotron cooling is negligible and we have
averaged the simulated lines over all inclination angles in an orbital period. An offset has been added to the simulated plot to allow a
comparison between the spectra. See text for further details.
shock column. We find that, compared to injecting the line
photons at a single height where the emissivity peaks, us-
ing an emissivity profile does not significantly change the
overall profiles of the 6.675 and 6.97 keV lines. The result-
ing line profiles show a small degree of smearing redward
of the line center due to additional dynamical and thermal
Doppler effects associated with a small fraction of line pho-
tons emitted close to the shock. The effect is most important
for the 6.675 keV line. We estimate a fractional increase in
the FWHM of no more than ≈ 25%.
In general, our simulations predict that recoil in scat-
tering with cool electrons near the base of the column as
well as upscattering by hot electrons near the shock can im-
print signatures on the profile of lines emitted near the base
of the flow. Such Compton signatures may thus be used to
determine the primary source region of the fluorescent line.
We predict that when 6.4 keV photons are emitted in the
dense, cool plasma at the base of the accretion column, they
can suffer strong downscattering when propagating through
the electrons streaming downward in the flow. The scatter-
ing probability is governed by the effective scattering optical
depth, which increases with the mass accretion rate of the
system.
We convolved simulations with the Chandra response
function and compared them to Chandra/HETGS observa-
tions of GK Per. Our results indicate that the red wing seen
in the 6.4 keV line in GK Per could be attributed to Comp-
ton recoil near the base of the flow.
Finally we remark that tighter constraints on the dy-
namics and flow geometry in magnetized accreting com-
pact objects can be obtained by considering the polariza-
tion properties of the lines (see Sunyaev & Titarchuk 1985;
Matt 2004). There is currently considerable effort to develop
X-ray polarimeters which can detect degrees of polarization
of the order of one percent (Costa et al. 2001). The spectral
resolution of these detectors should be adequate to search
for different polarization degrees in emission iron lines. It is
possible to include a polarization treatment in our calcula-
tions, but the computational algorithm in our Monte Carlo
code will need to be revised, which we leave for a future
study.
ACKNOWLEDGMENTS
ALM thanks a University of Sydney Denison Scholarship.
KW’s visit to Sydney University was supported by a NSW
State Expatriate Researcher Award. We thank an anony-
mous referee whose comments helped improve the paper
considerably.
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APPENDIX A: THE WIDTH OF THE
COMPTON SHOULDER
Here, we derive an estimate for the width of the prominent
wings near the base of the line profiles. Let the accretion
flow be along the z-axis and consider an observer on the
x−z plane. Let i be the line-of-sight inclination angle of the
accretion column, and let β be the local velocity of the flow,
normalized to the speed of light c, which can expressed as
~β = −β ẑ .
The normalized vector of a scattered photon of energy E′
propagating in the direction to the observer is
= sin i x̂ + cos i ẑ .
Suppose that the normalized vector of the incident photon,
with energy E, is
k̂ = sin θ cosφ x̂ + sin θ sinφ ŷ + cos θ ẑ .
Then the change in energy of the photon, after scattering
with an electron in the flow, due to bulk motion is given by
1− µβ
(1− µ′β) + E
γmec2
(1− k̂ · k̂′)
where γ = (1−β2)−1/2, µβ = k̂·~β, and µ′β = k̂′ ·~β . In terms
of the viewing inclination angle and the photon propagation
vectors,
1 + β cos θ
(1 + β cos i) + E
γmec2
(1− sin i sin θ cosφ− cos i cos θ)
Then for β ≪ 1 and E ≪ mec
≈ 1 + β(cos θ − cos i)− λ(1− sin i sin θ cos φ− cos i cos θ) .
It follows that
≈ β(cos θ − cos i)− λ(1− sin i sin θ cosφ− cos i cos θ) ,
where ∆E = E′ − E and λ = E/mec
The maximum energy downshift is caused by the recoil
process when the photons are scattered by “cold” electrons
(with β ≈ 0). This occurs when φ = π, θ = π − i, which
yields
max down
= −2λ .
This result is practically independent of the WD mass
and the viewing inclination angle. For the 6.4 keV line,
∆E/E|max down ≈ −2.50× 10
The maximum energy upshift is caused by a Doppler
shift when the photons are scattered by the fastest available
downstream electrons (i.e. β is no longer a negligible factor).
The condition for its occurrence can be derived as follows.
= 0 ,
which gives two conditions for extrema with respect to the
azimuthal coordinate: φ = 0 and φ = π, corresponding re-
spectively to
= β(cos θ − cos i) − λ(1∓ sin i sin θ − cos i cos θ) .
Differentiating the above expression with respect to θ and
setting the resulting expression to zero yields the following
condition for the extrema:
sin θ
cos θ
= ±ξ ,
where
λ sin i
β + λ cos i
The first case (φ = 0) leads to a maximum energy upshift,
which requires
sin θ =
1 + ξ2
cos θ =
1 + ξ2
At the viewing inclination angle i = π/2,
sin θ =
β2 + λ2
cos θ =
β2 + λ2
Hence, the maximum energy upshift is given by
max up
β2 + λ2 − λ .
If we assume that the maximum β takes the free-fall veloc-
ity at the WD surface, then β is simply the reciprocal of the
square root of the WD radius in the Schwarzschild unit, i.e.
2GMwd/Rwdc2. For Mwd = 1.0M⊙, β = 2.31×10
and for Mwd = 0.5M⊙, β = 1.23 × 10
−2 (assuming the
Nauenberg mass-radius relation (Nauenberg 1972)), this
gives ∆E/E|max up ≈ 1.36 × 10
−2 and 5.06 × 10−3 respec-
tively for the 6.4 keV line.
The Compton shoulder of an Fe line is due to a single
scattering event. For the 6.4 keV line, the broadening ex-
tends over 6.24 keV 6 E 6 6.49 keV for a 1.0 M⊙ WD and
6.24 keV 6 E 6 6.43 keV for a 0.5 M⊙ WD (omitting ther-
mal broadening and flow Doppler broadening), when viewed
at i = π/2.
c© 2006 RAS, MNRAS 000, 1–11
Compton Scattering of Fe Kα Lines in mCVs 11
This paper has been typeset from a TEX/ L
ATEX file prepared
by the author.
c© 2006 RAS, MNRAS 000, 1–11
	Introduction
	Theoretical Model
	Physical Processes
	Results and Discussion
	No Cyclotron Cooling
	Effect of Accretion Column Radius
	Emissivity Profile Effects
	Comparison with Observations
	Summary and Conclusion
	The width of the Compton shoulder
