Ray Scattering on Dust Grains

The physical situation we consider is the small-angle scattering of X-rays emitted by a bright X-ray source from dust grains in the ISM near the line of sight, which will create an X-ray halo around the point source. The scattered angles are small enough that, for astronomical purposes, we can assume that the path traveled by the unscattered primary X-rays is nearly identical to the paths of scattered photons, in terms of their distances and environments.

Traditionally, the scattering is treated by the so called Rayleigh-Gans theory, a simplification of the general Mie theory. This approximation makes two assumptions: First, the reflection from the surface of the grain is negligible, i.e., that \m - 1| C 1 with m is the complex index of refraction of the grain. Second, the phase of the incident wave is not shifted, i.e., that koa\m - 1\ C 1, where ko is the wave number of the X-ray and a is the radius of the grain. Here, we restrict ourselves to the Rayleigh-Gans approximation, although it has been shown that it holds for X-ray energies above 1 keV only [56]. The differential cross section for a single grain is given by [36]

d°sca(®sca) = c (2Z )2( p )2a(nm)6rF (E) i2 02 ) (181)

where c1 = 1.1cm2sr-1, Z is the mean atomic charge, M the mean molecular weight (in amu) p the mass density, E the energy in keV, F (E) the atomic scattering factor as, for instance, given by [28], 0sca the scattering angle, and 0(0sca) a "form factor," which depends on the shape of the particle. For spherical particles, the form factor is well approximated by a Gaussian, which yields, simplified, for the differential cross section [35]

with a the radial distance and x the fractional distance between grain and observer (see Fig. 18.6). With integrating the differential cross section over all solid angles we obtain the total scattering cross section, again simplified:

The total scattering cross section depends roughly on E-2 and on the grain size with a4. Polycyclic hydrogen carbonates (PAHs), which are considered to be ultra

Fig. 18.6 Scattering geometry with X-ray source S, observer O, and grains G. Typical values for the angular distance a from the direct line of sight toward the source are of the order of 10arcmin

10 100 1000 radius [arcsec]

Fig. 18.7 The radial brightness distribution of an X-ray source with bright halo (from [46]). The brightness distribution (solid line) is a composite of a dust scattering halo (dotted line) and the instrument's point response function (dashed line)

10 100 1000 radius [arcsec]

Fig. 18.7 The radial brightness distribution of an X-ray source with bright halo (from [46]). The brightness distribution (solid line) is a composite of a dust scattering halo (dotted line) and the instrument's point response function (dashed line)

small particles [23] play no role in small angle X-ray scattering. On the other hand, the grain size distribution follows approximately a powerlaw —a-3'5, which means that over a size range around the mean value a = 0.1^m all grain sizes contribute similarly to the scattering halo.

The observed radial brightness profile around a source is derived by integration of the differential cross section over all grains along the line of sight and a large solid angle, the size distribution of grains and the photon spectrum. According to (18.2) large grains, high energies, and dust in the vicinity of the source produce narrow brightness profiles, while small grains, low energies, and dust close to the observer produce wide profiles. The scattering cross section is also a strong function of the scattering angle. Since scattering close to the observer occurs under smaller scattering angles (at given angular distance), it contributes much more than scattering in the vicinity of the source. The fractional scattered intensity is Ihalo = Itotai(1 - exp(-Tsca)), with the optical depth in scattering Tsca = ascaN, with N the dust column density between source and observer. Multiple scattering can be neglected in most cases because the absorption cross section is much greater than the scattering cross section.

Fig. 18.8 In rare cases using the moon as a "shutter," a halo can be made directly visible without disturbing instrumental effects. This halo around Sco X-1 was observed with ROSAT in 1998. The image is centered with respect to the moon while Sco X-1 is moving; therefore, the halo appears to be asymmetric. The soft photons from the moon are coded in red, the harder photons from Sco X-1 in green

Fig. 18.8 In rare cases using the moon as a "shutter," a halo can be made directly visible without disturbing instrumental effects. This halo around Sco X-1 was observed with ROSAT in 1998. The image is centered with respect to the moon while Sco X-1 is moving; therefore, the halo appears to be asymmetric. The soft photons from the moon are coded in red, the harder photons from Sco X-1 in green

Scattered radiation has to travel along a slightly longer path than the direct, un-scattered light. Any intensity variation of the source, therefore, occurs somewhat delayed in the halo. Although proposed as method to measure the distance of variable X-ray sources already in 1973 [68], it took almost 30yrs, before this effect could be measured for the first time directly. The attractiveness of the "halo-method" is that it yields a geometrical rather than an physical distance as other methods like the 21 cm absorption, visual extinction, or X-ray absorption. The time delay is given by d xa2 , xa2 dt =--= 1.15 d --(18.4)

2c 1 — x 1 — x if d is given in kpc and a in arcs (c is the speed of light).

18.4.3 Observations and Results 18.4.3.1 Physical Constitution of Dust Grains

The shape of the halo profile is primarily determined by the grain size or size distribution, respectively, and the dust distribution along the line of sight. Various studies show that the derived grain size distributions are consistent with common dust

Fig. 18.9 Optical depth in X-ray scattering vs. X-ray absorption (expressed as equivalent hydrogen column density (left) and vs. visual extinction of the optical counterpart (right). These diagrams led to the conclusion that LMC X-1, Cas A, and GX 5-1 are, in addition to the interstellar absorption, locally absorbed. GX 17+1 was optically misidentified

Fig. 18.9 Optical depth in X-ray scattering vs. X-ray absorption (expressed as equivalent hydrogen column density (left) and vs. visual extinction of the optical counterpart (right). These diagrams led to the conclusion that LMC X-1, Cas A, and GX 5-1 are, in addition to the interstellar absorption, locally absorbed. GX 17+1 was optically misidentified models [34,75]. In some cases, the halo profile reflects also the inhomogeneous structure of the milky way, e.g., [11].

From the clear correlation between dust scattering and visual extinction (Fig. 18.9), one derives that both visual extinction as well as X-ray scattering must essentially be due to the same dust grains.

Since absorption is a measure for the total amount of heavy elements in the interstellar medium, and scattering a measure for the amount of those elements locked in grains, a correlation between both quantities should reveal the degree of depletion of metals in the ISM. A simple calculation shows in first place that either the depletion is much less than already supposed or dust is not composed of solid grains but more of fluffy grains [45]. This result has later been debated in the light of a revised scattering theory [12], and updated elemental abundances [76].

18.4.3.2 Chemistry of Dust Grains

Making use of the intrinsically good energy resolution of imaging semiconductor X-ray detectors, the pure scattered radiation, i.e., after eliminating external contribution (instrument and source spectrum) can be studied. The moderate ISM absorption toward Cyg X-2 for instance allowed a precise study of the halo spectrum down to 0.4keV [12] (Fig. 18.10). Scattering features from oxygen, magnesium, and silicon could be fitted with dust model containing a major contribution from olivines (Fe2-xMgxSiO4) and pyroxenes (Fe1-xMgxSiO3) [75].

18.4.3.3 Distance Determination of Variable X-Ray Sources

Although the idea is fascinating, its realization is difficult to perform; distances of only a few sources could be measured using this method. Since usually the scattering

Fig. 18.10 Spectrum of the halo around Cyg X-2 between 2.1' and 4' angular distance [12]. Data taken with the XMM-NewtonEPIC-pn camera compared with the best fit dust models WD01 ([75], dashed) containing 27% pyroxenes and 73% olivines, and the MRN model ([34], solid line)

does not happen at one particular point between source and observer, the time delay is accompanied by a damping of the lightcurve variation, which becomes more pronounced at larger angular distances. Best suited are sources with a sharp drop in intensity (eclipses) seen by a telescope with very good angular resolution. A few attempts have been made even without an imaging telescope by modeling the lightcurve at different energies: the lightcurve at higher energies is intrinsically produced, at lower energies (with halo) the time delay and damping effects become dominant. The direct, imaging method worked for the first time with Cyg X-3 [47], recently also with other sources [2].

An interesting consequence arise by reversing the method: if the source is at "infinite" or known distance, the distances toward dust clouds in the foreground can be measured [11]. Recently, an expanding halo could be detected around a GRB. This is due to the afterglow scattered on dust in our galaxy. Thus, the distance to two different dust clouds could be well determined [73].

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