this week's question was suggested by R. Schmidt (a fellow cosmologist):

$X$-ray observations suggest that the centres of clusters of galaxies are sometimes colder than the rest of the cluster. what's the physical reason for these cold cores? Would cold cores show up in the thermal Sunyaev-Zel'dovich effect?

bonus question: can you show that $\ln\det(A) = \mathrm{tr}\ln(A)$, or equivalently $\exp\mathrm{tr}(A) = \mathrm{det}\exp(A)$ for any non-singular square matrix A?

bonus question 2: what do you think, do you have more or less friends than your friends have on average?

bonus question: the relation is easiest shown in the eigenframe of $A$, because both the trace and the determinant are invariants under orthogonal transformations. the trace is the sum of all eigenvalues, and the determinant the product of the eigenvalues. Clearly, the logarithm of the determinant would then be the sum of the logarithms of the eigenvalues.

ReplyDeletebonus question 2: surprisingly, most people have fewer friends than their friends on average. this is known as the friendship paradox and has first been observed by S.L. Feld. the distribution of the number of friends is shaped in such a way that there are a small number of people with very large number of friends that lift the average, but at the same time it is unlikely that one falls in this category.

ReplyDeleteself-gravitating systems are very interesting because the sign of the specific heat (describing the proportionality between thermal energy change and temperature change) depends on how fast thermal energy is removed in comparison to the dynamical time scale of the object. if you remove energy slowly from a cluster, it heats up (as predicted by the virial theorem), but if you remove energy fast, the system cools down. that's what's happening in the cold core: thermal energy is radiated away by bremsstrahlung, and the system can not replenish the thermal energy by tapping gravitational binding energy. the plasma is not in hydrostatic equilibrium anymore, and these pressure differences would show up in the SZ.

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