Master thesis - UBC Physics & Astronomy

M. Alexandersen : Master thesis 13
Figure 3. The X-rays from the source is scattered by the dust at the
small angle θ, thereby being observed as a halo around the source. The
scattering angle is greatly exaggerated in this diagram, for clarity. The
broken lines going to the source indicate that these distances are much
longer than the other distances in the diagram, and resultantly are virtually
parallel.
distance is proportional to the difference in travel time:
h d + h ds − D s = ct. (1)
From the geometry of Fig. 3 it is clear that
h d = D d
(2)
√
cos θ
h d s = Dd 2 tan2 θ + Dds 2 . (3)
The distance to the dust is much smaller than the distance to the source,
and θ ≪ 1, so collectively D d tan θ ≪ D ds , so
h ds ≃ D ds . (4)
Using Eq. (1), (2), (4) and the fact that D ds = D s − D d we can derive
D d
cos θ + D ds − D s = ct, (5)
D d
cos θ − D d = ct, (6)
D d (1 − cos θ)
cos θ
D d =
= ct, (7)
ct cos θ
1 − cos θ . (8)