In-Class Assignment 8

In-Class Assignment 8#

Exploring Heat Transfer via Convection with MESA#

Learning Objectives#

  • identify convective and radiative regions using MESA models and analytical expressions

  • explain convective behavior in advanced burning massive stars and Solar-like stars

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd

a. Convection in Massive Stars#

Download the following model files locally. These data were produced using the MESA 20m_ms_convection_profile.data test suite.

  1. Plot the adiabatic gradient grada, the radiative gradient gradr, and the Ledoux term (gradL) as a function of radius (in units of Rsun) using the MESA data.

Hint: You may want to set your ylim to 0.2,1

  1. Locate the approximate fractional radius and annotate where the Schwarzchild Criterion $\( \nabla < \nabla_{\rm{ad}} \)$ is volated and fill in the convective region of the star using plt.fill_betweenx().

Hint: You may want to set your yrange to (0,1)

  1. We found that the limiting cases for \(\nabla_{\rm{ad}}\) is: 0.4 for an ideal gas without radiation and 0.25 for a radiation-dominated gas. Label these two limits on the same plot using plt.axhline

  2. Describe in a few sentences where is the model convective and way? Also, describe where \(\nabla_{\rm{ad}}\) is closer to one limit or the other and why? What does it mean when \(\nabla_{\rm{L}}=0\)?

Hint: Compare with Pols Fig. 5.4.

## load MESA data here
#massive_star_ms_profile = pd.read_csv('data/20m_ms_convection_profile.data',sep=r'\s+',header=4)
#list(massive_star_ms_profile)

rsun_cgs = 6.96e10  # cm

# example reading a variable from the profile data
#massive_star_ms_profile_gradrad = massive_star_ms_profile['###
#massive_star_ms_profile_radius_cm = massive_star_ms_profile['##
#massive_star_ms_profile_gradad = ##
#massive_star_ms_profile_gradL = ##

## 1-3 result here
#plt.title('20$M_\odot$ Main-Sequence MESA Model')

#plt.plot(##
#         ##,label=r'$\nabla_{\rm{ad}}$')


#plt.axhline(##,ls='--',lw=1,color='cyan') # ideal gas nabla ad
#plt.axhline(##,ls='--',lw=1,color='cyan') # radiation dominated gas nabla ad

#plt.fill_betweenx((0,1),#,##,color='dodgerblue',alpha=0.5) # fill in convective zone

#plt.axvline(##,color='k',ls='--',lw=1) # convective zone boundary

#plt.ylim(
#plt.xlim(

#plt.legend()
#plt.xlabel(r'$r/R_{\odot}$')
#plt.ylabel(r'$\nabla$')

4. result here#

Radiation dominated, lower \(\nabla_{\rm{ad}}\) in the atmosphere and it is in fact convective. When \(\nabla_{\rm{L}}=0\), the composition is uniform.

b. Convection in a Solar-Like Star#

Download the following model files locally. These data were produced using the MESA 1m_pre_ms_to_wd.data test suite.

  1. Plot the adiabatic gradient grada and the radiative gradient gradr as a function of radius (in units of Rsun) using the MESA data.

  2. Locate the approximate fractional radius and annotate where the Schwarzchild Criterion $\( \nabla < \nabla_{\rm{ad}} \)$ is volated and fill in the convective region of the star using plt.fill_betweenx().

Hint: You may want to set your yrange to (0,1)

  1. We found that the limiting cases for \(\nabla_{\rm{ad}}\) is: 0.4 for an ideal gas without radiation and 0.25 for a radiation-dominated gas. Label these two limits on the same plot using plt.axhline

  2. Describe in a few sentences where is the model convective and way? Also, describe where \(\nabla_{\rm{ad}}\) is closer to one limit or the other and why?

Hint: Compare with Pols Fig. 5.4.

## load b data here
#massive_star_ms_profile = pd.read_csv('data/1m_pre_ms_to_wd.data',sep=r'\s+',header=4)
#massive_star_ms_profile_gradrad = massive_star_ms_profile['##
#massive_star_ms_profile_radius_cm = massive_star_ms_profile['##
#massive_star_ms_profile_gradad = massive_star_ms_profile['##
# this data set doesnt have gradL :(                                                        

## 1-3 result here
#plt.plot(##,label=r'$\nabla_{\rm{ad}}$')

#plt.plot(##,label=r'$\nabla_{\rm{rad}}$')

#plt.title('1$M_\odot$ Main-Sequence MESA Model')

#plt.fill_betweenx(#

#plt.axvline(#

#plt.legend()
#plt.xlabel(r'$r/R_{\odot}$')
#plt.ylabel(r'$\nabla$')

c. Convection in an Advanced Massive Star#

Download the following model files locally. These data were produced using the MESA 15m_core_O_burning_profile.data test suite.

  1. Plot the adiabatic gradient grada and the radiative gradient gradr as a function of mass mass (in units of Msun) using the MESA data from 0 to 7 Msun.

  2. On the same plot, plot the normalized nuclear energy generation rate eps_nuc and the non-normalized mean molecular weight. Set the ylim to 0,2.

  3. Locate the approximate fractional radius and annotate where the Schwarzchild Criterion $\( \nabla < \nabla_{\rm{ad}} \)$ is volated and fill in the convective region of the star using plt.fill_betweenx().

Hint: There are at least two in this model.

Hint: You may want to set your yrange to (0,2)

  1. Describe in a few sentences where is the model convective and way? Also, describe where \(\nabla_{\rm{ad}}\) is closer to one limit or the other and why?

Hint: Compare with Farmer et al. 2016

Stars might be like onions after all?? Not parfaits..

## load c data
#massive_star_ms_profile = pd.read_csv('data/15m_core_O_burning_profile.data',sep=r'\s+',header=4)
#massive_star_ms_profile_gradrad = massive_star_ms_profile['##
#massive_star_ms_profile_mass = massive_star_ms_profile['##
#massive_star_ms_profile_gradad = massive_star_ms_profile['##
#massive_star_ms_profile_eps_nuc = ##
#massive_star_ms_profile_mu = ##

# 1 - 3 result here
#plt.plot(##,label=r'$\nabla_{\rm{ad}}$')

#plt.plot(##,label=r'$\nabla_{\rm{rad}}$')

#plt.plot(##,label=r'$\epsilon_{\rm{nuc}}$') # dont forget to normalize by max

#plt.plot(##,label=r'$\mu$')

#plt.title('15$M_\odot$ Core O-Burning MESA Model')

#plt.legend()
#plt.xlabel(r'$m/M_{\odot}$')
#plt.ylabel(r'$\nabla,\mu,\epsilon_{\rm{nuc,normalized}}$')
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