An Introduction to Modern Astrophysics · Problem 18.22

Physical Processes in the Solar System — Carroll & Ostlie

Physics — Astrophysics · xray-pulsar-spin-up-rate-graph

Problem

(a) Use Eq. (18.24) to show that the spin-up rate can be written as log⁡10 ⁣(−P˙P)=log⁡10 ⁣(PLacc6/7)+log⁡10 ⁣[α2πI(22π Bs2R12μ0G3M3)1/7].\log_{10}\!\left(-\frac{\dot{P}}{P}\right) = \log_{10}\!\left(PL_{\rm acc}^{6/7}\right) + \log_{10}\!\left[\frac{\sqrt{\alpha}}{2\pi I}\left(\frac{2\sqrt{2\pi}\,B_s^2 R^{12}}{\mu_0 G^3 M^3}\right)^{1/7}\right]. The term on the left and the first term on the right consist of quantities that can be measured observationally. The second term on the right depends on the specific model (neutron star or white dwarf) of the X-ray pulsar. (b) Make a graph of log⁡10(−P˙/P)\log_{10}(-\dot{P}/P) (vertical axis) vs. log⁡10(PLacc6/7)\log_{10}(PL_{\rm acc}^{6/7}) (horizontal axis). Use the values from Example 18.6.3 to plot two lines, one for a neutron star and one for a white dwarf. Let log⁡10(PLacc6/7)\log_{10}(PL_{\rm acc}^{6/7}) run from 25 to 29. (c) Use the data in Table 18.2 to plot the positions of six binary X-ray pulsars on your graph. (You will have to convert −P˙/P-\dot{P}/P into units of s−1\mathrm{s}^{-1}.) (d) Which model of a binary X-ray pulsar is in better agreement with the data? Comment on the position of Her X-1 on your graph.

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