2.3: Homology Transformations

The term homology has a wide usage, but in general it means "proportional to" and is denoted by the symbol ~ . One set is said to be homologous to another if the two can be put into a one-to-one correspondence. If every element of one set, say z_i, can be identified with every element of another set, say z'_i, then z_i ~ z'_i and the two sets are homologous. Thus a homology transformation is a mapping which relates the elements of one set to those of another. In astronomy, the term homology has been used almost exclusively to relate one stellar structure to another in a special way.

One can characterize the structure of a star by means of the five variables \(\mathrm{P}(\mathrm{r}), \mathrm{T}(\mathrm{r}), M(\mathrm{r}), \mu(\mathrm{r}), \text { and } \rho(\mathrm{r})\) which are all dependent on the position coordinate r. In our development so far, we have produced three constraints on these variables, the ideal-gas law, hydrostatic equilibrium, and the definition of \(M(\mathrm{r})\). Thus specifying the transformation of any two of the five dependent variables and of the independent variable r specifies the remaining three. If the transformations can be written as simple proportionalities, then the two stars are said to be homologous to each other. For example, if \[r^{\prime}=C_1 r \quad \rho^{\prime}\left(r^{\prime}\right)=C_2 \rho(r) \quad \xi^{\prime}\left(r^{\prime}\right)=C_3 \xi(r)\label{2.3.1}\]

then \[\zeta^{\prime}\left(r^{\prime}\right)=C_4 \zeta(r) \quad \eta^{\prime}\left(r^{\prime}\right)=C_5 \eta(r) \quad \chi^{\prime}\left(r^{\prime}\right)=C_6 \chi(r)\label{2.3.2}\]

where \(\xi, \zeta, \eta, \text { and } \chi\) stand for any of the remaining structure variables. However, because of the constraints mentioned above, C₄, C₅, and C₆ are not linearly independent but are specified in terms of the remaining C's. Consider the definition of \(M(\mathrm{r})\) and a homology transformation from \(\mathrm{r\rightarrow r’}\). Then \[\frac{M^{\prime}\left(r^{\prime}\right)}{M(r)}=\frac{\int_0^{r^{\prime}} 4 \pi\left(x^{\prime}\right)^2 \rho^{\prime}\left(x^{\prime}\right) d x^{\prime}}{\int_0^r 4 \pi x^2 \rho(x) d x}=C_2 C_1^3\label{2.3.3}\]

In a similar manner, we can employ the equation of hydrostatic equilibrium to find the homology transformation for the pressure P, since \[\frac{P^{\prime}\left(r^{\prime}\right)}{P(r)}=\frac{\int_0^{r^{\prime}}\left[G M^{\prime}\left(x^{\prime}\right) \rho^{\prime}\left(x^{\prime}\right) / x^{\prime 2}\right] d x^{\prime}}{\int_0^r\left[G M(x) \rho(x) / x^2\right] d x}=C_2^2 C_1^2\label{2.3.4}\]

If we take \(\mu\) to be the chemical composition m, then the remaining structure variable is the temperature whose homology transformation is specified by the ideal-gas law as \[\frac{P^{\prime}\left(r^{\prime}\right)}{P(r)}=\frac{\rho^{\prime}\left(r^{\prime}\right) k T^{\prime}\left(r^{\prime}\right) / \mu^{\prime}\left(r^{\prime}\right)}{\rho(r) k T(r) / \mu(r)}=C_2^2 C_1^2\label{2.3.5}\]

so that \[\frac{T^{\prime}\left(r^{\prime}\right)}{T(r)}=C_3 C_2 C_1^2\label{2.3.6}\]

Should we take \(\xi\) to be T, then the homology transform for µ is specified and is \[\frac{\mu^{\prime}\left(r^{\prime}\right)}{\mu(r)}=\frac{C_3}{C_1^2 C_2}\label{2.3.7}\]

We can use the constraints specified by equations \ref{2.3.3}, \ref{2.3.4}, and \ref{2.3.6} and the initial homology relations [equation \ref{2.3.1}] to find how the structure variables transform in terms of observables such as the total mass M and radius R. Thus, \[\begin{aligned}
\frac{\rho^{\prime}\left(r^{\prime}\right)}{\rho(r)} & =\frac{M^{\prime}}{M}\left(\frac{R}{R^{\prime}}\right)^3 \\
\frac{P^{\prime}\left(r^{\prime}\right)}{P(r)} & =\left(\frac{M^{\prime}}{M}\right)^2\left(\frac{R}{R^{\prime}}\right)^4 \\
\frac{T^{\prime}\left(r^{\prime}\right)}{T(r)} & =\frac{\mu^{\prime} M^{\prime} R}{\mu M R^{\prime}}
\end{aligned}\label{2.3.8}\]

Since homology transformations essentially represent a linear scaling from one structure to another, it is not surprising that the dependence on mass and radius is the same as implied by the integral theorems [equations \ref{2.2.8}].

The primary utility of homology transformations is that they provide a "feel" for how the physical structure variables change given a simple change in the defining parameters of the star, "all other things being equal." An intuitive feel for the behavior of the state variables P, T, and p which result from the scaling of the mass and radius is essential if one is to understand stellar evolution. Consider the homologous contraction of a homogeneous uniform density mass configuration. Here the total mass and composition remain constant, and we obtain a very specific homology transformation \[\frac{\rho}{\rho_0}=\left(\frac{R_0}{R}\right)^3 \quad \frac{P}{P_0}=\left(\frac{R_0}{R}\right)^4 \quad \frac{T}{T_0}=\frac{R_0}{R}\label{2.3.9}\]

which is known as Lane's Law¹ (p.47) and has been thought to play a role in star formation. In addition, certain phases of stellar collapse have been shown to behave homologously. In these instances, the behavior of the state variables is predictable by simple homology transformations in spite of the complicated detailed physics surrounding these events.