Constant-Density Model, Energy Generation/Transport, Dimensional Analysis, and the Main Sequence

Constant-Density Model, Energy Generation/Transport, Dimensional Analysis, and the Main Sequence#

Wednesday, Jan. 22, 2025

astrophysics of stars and planets - spring 2025 - university of arizona, steward observatory

Today’s Agenda#

  1. Announcements / Recap (2m)

  2. Dicuss Readings/ICA (5m)

  3. Reading Overview/Key Points (10m)

  4. In-Class Assignment (Groups of 2) (20m)

  5. Debrief on ICA (5m)

The Constant-Density Model#

For the following analysis we will make the assumption that the central density is the density throughout the entire star, \(\rho = \rho_c = \) constant and the \(c\) subscript denotes the center.

In HKT 1.4, we saw that making this approximation allows us to, with some “trivial” algebra come up with an equation for the central pressure (under the above assumption)

Definition 5

central pressure \(P_c\) at M_r = 0

\[ P_c = \frac{3}{8 \pi} \frac{GM^2_{\star}}{R^2_{\star}} = 1.34 \times 10^{15} \left ( \frac{M_{\star}}{M_{\odot}} \right )^2 \left ( \frac{R_{\star}}{R_{\odot}} \right )^{-4} (\textup{dyne \ cm}^{-2})~. \]

Calculation of Molecular Weights#

Some definitions and vocabulary:

  • \(n\) - the number of free particles in a gas

  • \(Z\) - nuclear charge

  • \(A\) - nuclear mass number in (amu)

  • \(X\) - fraction by mass of a species (aka mass fraction)

  • \(n_{I,i}\) - the ion number density in units of cm\(^{-3}\) of a given species \(i\)

The ion number density takes the form of (HKT 1.40)

Definition 6

\[ n_{I,i} = \frac{(\textup{mass / unit volume of species } i)}{(\textup{mass of 1 ion of species } i)} \equiv \frac{\rho X_{i} N_{\textup{A}}}{A_{i}} \]

Taking a sum over all ions gives the and defining \(\mu_{I}\) as the total mean molecular weight of ions, we have the ion number density for all ions as

Definition 7

the ion number density for all ions

\[ n_{I} = \frac{\rho N_{A}}{\mu_{I}} \]

Here, \(N_{A} = 6.022137\times10^{-23}\) mole\(^{-1}\).

As such, we can rewrite

Definition 8

mean molecular weight of ions

\[ \mu_{I} = \left [ \Sigma_{i} \frac{X_{i}}{A_{i}} \right ]^{-1} \]

Following a similar derivation, we can also define the mean molecular weight per free electron,

Definition 9

mean molecular weight per free electron

\[ \mu_{e} = \left [ \Sigma_{i} \frac{Z_{i} X_{i} y_{i}}{A_{i}} \right ]^{-1} \]

where \(y_{i}\) is the ionization fraction, a value of \(y_{i}=1\) means the gas is completely ionized while \(y_{i}=0\) implies the gas is completely neutral.

We finally arrive at expressions for the total mean molecular weight,

Definition 10

total mean molecular weight

\[ \mu = \left [ \frac{1}{\mu_{I}} + \frac{1}{\mu_{e}} \right ]^{-1} \]

with \(n = n_{I} + n_{e} = \frac{\rho N_{\rm{A}}}{\mu}\)

Some Naming Conventions and Approximations#

The mass fractions \(X_{i}\) for often defined as \(X\) for \(^{1}\)H and \(Y\) for \(^{4}\)He and all else referred to as “metals” denoted by \(Z\) (not ion charge!).

This leads to the following result for the composition of a star, \(X+Y+Z=1\).

Now, lets consider some scenarios where we can approximate values for the molecular weights.

Definition 11

When \(X\),\(Y\), and \(Z\) are completely ionized \(y_{i}=1\) and \(Z \ll 1\) (metals are a small mass fraction)

\[ \mu_{e} \approx \frac{2}{1+X} \]

Definition 12

When using the same assumptions as above and the fact that Z is small compared the \(\left < A \right >\)

\[ \mu_{I} \approx \frac{4}{1+3X} \]

Combining these results leads to an approximate total mean molecular weight

Definition 13

\[ \mu \approx \frac{4}{3+5X} \]

Energy Generation and Transport#

Some definitions:

  • \(\epsilon\) - energy generation rate (erg g\(^{-1}\) s\(^{-1}\))

  • \(\kappa\) - lets you know how the flow of radiation is hindered by the medium through which it passes (cm\(^{-2}\) g\(^{-1}\))

The energy equation

Definition 14

\[ \frac{d L}{dr} = 4 \pi r^2 \rho \epsilon \]

or in Lagrangian form:

Definition 15

\[ \frac{d L}{dm} = \epsilon \]

Stellar Dimensional Analysis#

In this section, we found the Mass-Luminosity relationship for stars with masses greater than about one solar mass,

\[ \frac{R}{R_{\odot}} \approx \left ( \frac{M}{M_{\odot}} \right )^{0.75} \]
\[ \frac{L}{L_{\odot}} \approx \left ( \frac{M}{M_{\odot}} \right )^{3.5} \]

Evolutionary Lifetimes on the Main Sequence#

\[ t_{\rm{nuc}} \approx 10^{10} \left ( \frac{M}{M_{\odot}} \right ) \left ( \frac{L}{L_{\odot}} \right )^{-1} (\rm{years}) \]

or using the mass-luminosity relationship,

\[ t_{\rm{nuc}} \approx 10^{10} \left ( \frac{M}{M_{\odot}} \right )^{-2.5} (\rm{years}) \]

In-Class Activity#

Head over to ICA2, work with the person next to you.

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