The Dependence of the Core Properties of Solar-like Main Sequence Stars on
Hydrogen Mass Fraction
The VMI Summer Undergraduate Research Institute
Written by
Cadet Thananart Klongcheongsan
Faculty Mentor Dr.Bruce R. Boller
Department of Physics and Astronomy
1. Introduction
It is believed that main sequence stars derive their energy production from the
fusion of hydrogen nuclei into helium nuclei. The chemical composition of our sun consisted of 70.6% hydrogen, 27.3% helium, and 2.1% heavier elements (also called “metal” in this context) when nuclear fusion of hydrogen into helium began some 4.5 billion years ago. The values of luminosity and effective temperature at that time placed the sun on a curve that is known as the zero-age main sequence (ZAMS) when log is plotted versus -log Teff. This curve is the locus of points of stars with different mass that have just begun to produce energy by the fusion of hydrogen into helium. Those stars with mass greater than the sun are found upward and to the left while those with mass less than the sun are found and to the right.
As nuclear fusion continues, the chemical composition changes within the star. The hydrogen becomes depleted while the helium is enhanced. For stars with mass like our sun there are two principal nuclear fusion reactions that consume hydrogen and generate helium. The first is the proton-proton (PP) cycle while the second is the carbon-nitrogen-oxygen (CNO) cycle. This means that, as the star ages, the abundance of hydrogen, helium, and the metal change. It is clear that the nature of main sequence stars must change as their hydrogen content diminishes. The reason that this must be so is that the rate of the nuclear fusion reaction depends on temperature and concentration (or low density) of hydrogen.
The purpose of the research is to write a program to examine the dependence of the luminosity on the central temperature, mass density, and the hydrogen content in the cores of the main sequence stars. A simple model of a main sequence star will be used which consists of a set of five equations known as the stellar structure equations. In addition, the equation of state of an ideal gas will be used to supply the needed relationship between the thermodynamic variables.
The stellar structure equations will be numerically integrated using FORTRAN programming methods. These equations will be integrated from the core outwards to a fitting point. This will be done for different hydrogen contents so that the dependence of the luminosity on the central temperature, mass density, and hydrogen content may be determined.
2. The stellar structure equations
The five stellar structure equations consist of the following:
1. Hydrostatic equation:
2. Mass-density relation:
3. Luminosity-Energy production rate:
4. Radiative heat transfer:
5. Convective heat transfer:
The equation of state for an ideal gas together with the hydrostatic equation enable us to estimate the very high temperature and the enormous pressure in the interior of a star of given mass and radius.
The mass-density relation is used to provide a relationship between each mass shell and the mass density within that shell.
The energy production rate equation is used to obtain the incremental change in luminosity in term of mass-density, radius, and rate of nuclear fusion energy production. The nuclear energy production rate is the amount of available energy generated per gram per second. Each reaction in the proton-proton cycle produces 13.34 MeV of thermal energy while each complete reaction in the carbon-nitrogen-oxygen cycle produces 25.01 MeV of thermal energy.
The heat transport consists of two parts, the radiative heat transfer and the convective heat transfer. At the given mass, radius, temperature, pressure, and the luminosity from the above equations, both radiative and convective temperature gradient equations will be added together and become the total change of the temperature dT. Another crucial factor will be involved in this part, the opacity or the Rosseland mean absorption coefficient.
There are some additional equations used in the program such as the following:
1. Equation of state:
2. Modified gas constant:
3. Mean molecular weight per particle:
4. Hydrogen fusion energy generation rate:
5. Carbon cycle fusion energy generation rate:
6. Heat capacity at constant volume
;= 13.54 eV = 2.169x10-11 erg
7. Effective temperature
; ;
3. Physical constants and variables
3.1 Physical constants
The following constants are used in the program. All symbols are the same as appearing in the program.
QUANTITY SYMBOL VALUE
Solar mass M○1.991 x 1033 gm
Hydrogen mass mH1.673 x 10-24 gm
Newton’s gravitational constant G6.673 x 10-11dyne cm2 g-2
Speed of lightc2.998 x 1010 cm sec-1
Radiation pressure constanta7.57 x 10-15 erg cm-3 K-4
Boltzmann’s constantk 1.381 x 10-16 erg K-1
Ionization of hydrogenχ2.16 x 10-11 erg
Planck’s constanth6.626 x 10-27 erg sec
Stefan-boltzmann constantσ5.670 x 10-5 erg sec-1 cm-2 K-4
3.2 Variables
The following variables are used in the program. Some symbols are the same as appearing in the program.
QUANTITY SYMBOL
Central temperatureTc
Central densityρc
Hydrogen mass fractionX
Helium mass fractionY
Metal mass fractionZ
Temperature in Kelvin T
Pressure in dynes per square centimeterP
Mass in grams within the radius rsmr
Luminosity in ergs per second at radius r L
Radius in centimetersr
Opacity in square centimeters per gramκ
Nuclear fusion energy generation rate ep
Ratio of molar heat capacity at constant pressureγ
to the molar heat at constant volume
Gas constantR*
The degree of ionizationx
Specific heat of hydrogen gasCvg
Radiative temperature gradientdTrad
Convective temperature gradientdTconv
4. The main program
FORTRAN77 is used to write several programs to calculate and display various physical parameters within a star of one solar mass from the center outwards in equal mass shell fractions. Values of central temperature, Tc, and central mass density, ρc, are picked and input into the program. Each program uses specific values of hydrogen mass fraction (X), helium (Y), and metal mass fraction (Z) in one or more zones of the star. Once the central values of temperature and mass density are picked, the program calculates the change in all physical variables for the next mass shell. These are then added to the previous values in an iterative manner (finite difference equations) to obtain the values for the new mass shell. Only one combination of effective temperature for a star of given mass on the zero-age main sequence.
A test program was written to verify its ability to correctly evaluate all physical parameters by checking them against values previously determined by Iben (Novotny, p.375) for the sun approximately 4.5 billion years ago when it was on the zero-age main sequence (X=0.706, Y-0.273, Z=0.021).
Programs for other models were written for stars of one solar mass that have evolved to a lower hydrogen mass fraction (X=0.350). This mass fraction of hydrogen represents stars of one solar mass after evolving for billions of years. In that time the luminosity and the effective temperature will have changed. Each of the models used in this research shows that the luminosity and effective temperature increase in agreement with some very recent work by Kippenauer and Wiegert (pp. 271-272). Unfortunately exact values for all parameters are not certain because precise conditions within the sun or similar stars are not currently known.
The program integrates the stellar structure equations as finite difference equations from the center outwards to the fitting point at the mass fraction equal to about 0.5 of the solar mass using boundary values of temperature and mass density at the center. It is normal to integrate the stellar structure equations from the surface inwards to the fitting point using outer boundary values of luminosity and mass. However, because of limitations associated with running times, this program was written only to integrate from the center outwards toward the surface. The values of two boundary parameters at the center: central temperature and central mass density are adjusted in order to track five physical variables: temperature, density, radius, pressure, and luminosity; as functions of mass fraction to the selected fitting point. As mentioned earlier, these variables are known from previous research work and are being used as a check for the current program.
The program is divided into three parts. The first part is the initial integration of the first shell with the total mass of one ten-thousandth of the total mass of the present sun. Some results from the first part are stored in a data file: “result(x).dat” where (x) is the number of the model. The second part is the integration of 4,999 loops where each loops has 900 steps. It is written separately from the first program because of the different geometry to be used as the first “shell”, which is a sphere. The second program reads some significant results from the data file written by the first program which are needed as initial variables for the second part of the integration. At the end of the program of the second part, all results are stored into the data file “result(x).dat” again so that they are to be used as initial variables for the next loop of the integration. The last part is a program written to print out the important final results of the integration such as the mass, radius, pressure, luminosity, temperature, and density. These become the results at the fitting point.
As a compromise between computing time and accuracy, each set of the stellar structure equations is numerically integrated in steps of one five-thousandth of the total stellar mass fraction at the fitting point with each of the integrations consisting of 900 minor steps.
5. Solar models based on Rosseland opacity tables
Three solar models are created with different interior properties. Different hydrogen content, central temperature and density of each model result in different values of the radius, pressure, luminosity, temperature, and the density at the fitting point. One other problem is the radiative temperature gradient based upon the Rosseland opacity tables. It is added to the convective temperature gradient after the integration beyond 0.3 of the mass of the present sun.
5.1 The tested model
The first model called X1 is the model interior for the star near the initial main sequence or zero age main sequence (ZAMS). It is constructed to compare with the sun, approximately 4.50 billion years ago. The stellar structure equations are integrated to the fitting point of about 0.5039 of the total solar mass.
The model listed as X1 was written to reproduce the physical variables such as temperature, pressure, mass-density, and luminosity at various mass fractions from the center outwards. The results were matched against some well-established work by Iben (Novotny, p. 375-376).
The central temperature of 1.39 x 107 K and central mass-density of 85.15 g/cm3 are consistent with a hydrogen mass fraction of 0.706, helium mass fraction of 0.273, and a metal mass fraction of 0.021. Under these conditions the central pressure is 1.617 x 1017 dyne/cm2.
Matching the conditions for the interior properties, the following results were calculated and displayed on the screen by the file “test11.for”, “test12.for”, and “test.for”, the percent agreement with Iben’s table is indicated following each number.
Mass = 1.003265 x 1033 g or 1.003265 x 1030 kg (100%)
Radius = 1.770574 x 1010 cm or 1.770574 x 105 km (97.9%)
Pressure = 1.990754 x 1016 dyn/cm2 (88.5%)
Luminosity = 2.99113 x 1033 erg sec-1 (107.4%)
Temperature = 6958500 K (91.9%)
Density = 21.2563 g/cm3 (95.4%)
5.2 The models
The results from the test program are close enough to say that the program is correct. Now, a table is constructed for a variety of combinations of central temperature and mass density ranging from 1.20x107 K to 1.70x107 K and 75 g/cm3 to 95 g/cm3 respectively for model X1 and three other models with different zones and evolved mass fractions of hydrogen, helium, and metals.
These models are:
Model X1 (ZAMS), One zone.
0 M 0.50 Msun X = 0.700, Y = 0.275, Z = 0.025
Model X2 (Evolved), Two zones.
0 M 0.25 Msun X = 0.35, Y = 0.625, Z = 0.025
0.25 Msun M 0.50 Msun X = 0.70, Y = 0.275, Z = 0.025
Model X3 (Evolved), One zone.
0 M 0.50 Msun X = 0.35, Y = 0.625, Z = 0.025
Model X4 (Evolved), Three zones.
0 M 0.10 Msun X = 0.15, Y = 0.825, Z = 0.025
0.10 Msun M 0.25 Msun X = 0.35, Y = 0.625, Z = 0.025
0.25 Msun M 0.50 Msun X = 0.70, Y = 0.275, Z = 0.025.
The table below illustrates a variety of central temperatures and mass-densities for each of these models. Central pressure is also listed in addition to luminosity and radius.
Tc / ρc / Mass / PX1 / X2 / X3 / X4 / X1 / X2 / X3 / X4
1.70 / 95.0 / 9.955 / 9.955 / 9.955 / 9.955 / 4.9173 / 2.9979 / 1.6369 / 2.5967
1.50 / 95.0 / 9.955 / 9.955 / 9.955 / 9.955 / 2.9745 / 1.7636 / 0.5233 / 1.2011
1.39 / 95.0 / 9.955 / 9.955 / 9.955 / 9.955 / 1.7997 / 0.8739 / 0.1651 / 0.7427
1.20 / 95.0 / 9.955 / 9.955 / 9.955 / 9.955 / 0.7608 / 0.3585 / 0.0367 / 0.1701
1.70 / 85.0 / 9.955 / 9.955 / 9.955 / 9.955 / 4.5553 / 3.0056 / 1.6282 / 2.0263
1.50 / 85.0 / 9.955 / 9.955 / 9.955 / 9.955 / 2.8927 / 1.7224 / 0.5630 / 1.3007
1.39 / 85.0 / 9.955 / 9.955 / 9.955 / 9.955 / 2.0216 / 0.9285 / 0.2012 / 0.8892
1.20 / 85.0 / 9.955 / 9.955 / 9.955 / 9.955 / 0.8303 / 0.4080 / 0.0984 / 0.1995
1.70 / 75.0 / 9.955 / 9.955 / 9.955 / 9.955 / 4.4533 / 2.7948 / 1.6330 / 2.5945
1.50 / 75.0 / 9.955 / 9.955 / 9.955 / 9.955 / 3.1535 / 1.5920 / 0.5839 / 1.2949
1.39 / 75.0 / 9.955 / 9.955 / 9.955 / 9.955 / 2.3925 / 1.0107 / 0.3152 / 0.9606
1.20 / 75.0 / 9.955 / 9.955 / 9.955 / 9.955 / 1.1128 / 0.5078 / 0.1297 / 0.3946
Tc / ρc / L / Density
X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4
1.70 / 95.0 / 10.17 / 7.855 / 6.972 / 2.596 / 34.647 / 23.478 / 18.194 / 20.688
1.50 / 95.0 / 5.635 / 1.452 / 1.2185 / 0.609 / 26.887 / 18.156 / 9.191 / 13.722
1.39 / 95.0 / 3.095 / 0.688 / 0.617 / 0.266 / 21.720 / 12.268 / 4.231 / 10.383
1.20 / 95.0 / 1.234 / 0.252 / 0.235 / 0.070 / 14.269 / 7.4398 / 1.525 / 4.070
1.70 / 85.0 / 20.77 / 7.537 / 6.609 / 2.753 / 31.768 / 22.993 / 17.41 / 16.235
1.50 / 85.0 / 5.192 / 1.395 / 1.142 / 0.626 / 24.932 / 17.103 / 9.378 / 13.835
1.39 / 85.0 / 2.985 / 0.649 / 0.578 / 0.273 / 21.470 / 12.527 / 4.831 / 11.319
1.20 / 85.0 / 1.143 / 0.242 / 0.222 / 0.070 / 14.660 / 7.856 / 3.025 / 4.533
1.70 / 75.0 / 0.999 / 71.47 / 6.179 / 3.394 / 30.104 / 20.747 / 16.000 / 19.635
1.50 / 75.0 / 5.183 / 1.298 / 1.057 / 0.606 / 26.030 / 15.790 / 9.629 / 13.227
1.39 / 75.0 / 2.916 / 0.628 / 0.534 / 0.277 / 23.338 / 12.477 / 6.233 / 11.494
1.20 / 75.0 / 1.105 / 0.228 / 0.206 / 0.078 / 15.930 / 8.4546 / 3.647 / 6.571
Tc / ρc / Radius / T
X1 / X2 / X3 / X4 / X1 / X2 / X3 / X4
1.70 / 95.0 / 1.627 / 1.751 / 1.850 / 1.845 / 10.56 / 9.55 / 10.06 / 9.39
1.50 / 95.0 / 1.648 / 1.897 / 2.102 / 2.042 / 8.23 / 7.27 / 5.84 / 6.55
1.39 / 95.0 / 1.739 / 2.121 / 2.324 / 2.237 / 6.17 / 5.33 / 4.00 / 5.35
1.20 / 95.0 / 1.928 / 2.303 / 2.727 / 2.588 / 3.97 / 3.60 / 2.46 / 3.13
1.70 / 85.0 / 1.666 / 1.791 / 1.871 / 1.952 / 1.07 / 9.78 / 9.57 / 9.33
1.50 / 85.0 / 1.691 / 1.927 / 2.148 / 2.034 / 8.63 / 7.53 / 6.15 / 7.03
1.39 / 85.0 / 1.765 / 2.136 / 2.312 / 2.249 / 7.07 / 5.54 / 4.27 / 5.88
1.20 / 85.0 / 1.939 / 2.318 / 2.573 / 2.563 / 4.21 / 3.89 / 3.33 / 3.29
1.70 / 75.0 / 1.717 / 1.854 / 1.913 / 1.946 / 11.00 / 10.07 / 9.22 / 9.88
1.50 / 75.0 / 1.744 / 1.952 / 2.200 / 2.081 / 9.01 / 7.54 / 6..22 / 7.32
1.39 / 75.0 / 1.768 / 2.142 / 2.293 / 2.232 / 7.67 / 6.06 / 5.18 / 6.25
1.20 / 75.0 / 1.909 / 2.366 / 2.546 / 2.502 / 5.20 / 4.49 / 3.64 / 4.49
Tc / ρc / PC
X1 / X2 / X3 / X4
1.70 / 95.0 / 2.1636 / 1.5691 / 1.5691 / 1.2358
1.50 / 95.0 / 1.9076 / 1.3830 / 1.3830 / 1.0890
1.39 / 95.0 / 1.7669 / 1.2808 / 1.2808 / 1.0083
1.20 / 95.0 / 1.5241 / 1.1044 / 1.1044 / 0.8691
1.70 / 85.0 / 1.9634 / 1.4044 / 1.4044 / 1.1062
1.50 / 85.0 / 1.7072 / 1.2378 / 1.2378 / 0.9747
1.39 / 85.0 / 1.5813 / 1.1463 / 1.1463 / 0.9025
1.20 / 85.0 / 1.3640 / 0.9884 / 0.9884 / 0.7796
1.70 / 75.0 / 1.7091 / 1.2397 / 1.2397 / 0.9766
1.50 / 75.0 / 1.5068 / 1.0926 / 1.0926 / 0.8604
1.39 / 75.0 / 1.3956 / 1.0118 / 1.0118 / 0.7967
1.20 / 75.0 / 1.2038 / 0.8725 / 0.8725 / 0.6873
Tc = Central temperature ( x 107 K ) ρc = Mass density ( g/cm3)
Mass = x 1033 gm Pressure = x 1016 dyn/cm2 Radius = x 1010 cm
Luminosity = x 1033 erg/secDensity = g/cm3 Temperature = x 106 K
PC = Central pressure (x1017 dyne/cm2)
We now wish to discover the values of luminosity and radius for models X2, X3, and X4 corresponding to the same central pressure of model X1 that matches Iben’s table, namely 1.617 x 1017 dyne/cm2. Note that the row with central temperature of 1.39 x 107 K and central mass-density of 85 g/cm3 has a central pressure of 1.58x1017 dyne/cm2 which is considered close enough to the pressure in Iben’s table. The central pressure is one parameter on a long time scale that must be the same if the star is to maintain itself in hydrostatic equilibrium. [Please note that the generated table took about 30 hours of computer time. This was done only after all of the bugs were taken out of the programs and also only after suitable techniques for various computations were discovered.]
The following figures represent the average evolutionary path from model X1 to X2, X3, and X4 respectively.
6. Results
As the star consumes hydrogen the central temperature and mass density both increase. The luminosity increases by several times as does the effective temperature which was calculated from the following equation:
The effective temperature of the zero-age main sequence sun is 5740 K. The evolved effective temperatures are based on the zero-age main sequence sun. The values of r that need to be used in the equation should be the effective radius of the star, however; since the program only goes out to 0.5 MSUN, we must use the corresponding value of r at that location. It is hoped that in this comparison there is not too much error made with that assumption. The effective temperature of the zero-age main sequence star may be written as follows:
The next table illustrates the values for the central temperature, central mass-density, luminosity, radius (at 0.5 MSUN), ratio of the luminosity to the luminosity of mopdel X1, and effective temperature for the star for all models. Note that the luminosity and the effective temperature both increase for the evolved models (X2, X3, and X4) relative to the ZAMS model, X1.
Lsun = 2.98x1033 erg/s (at 0.5 Msun)
Model / Tc / ρc / L / r(at 0.5Msun) / L/LSUN / Teff (K)(x107 K) / (g/cm3) / (X1033 erg/s) / (x1010 cm)
X1 / 1.39 / 85 / 2.98 / 1.77 / 1 / 5740
(ZAMS)
X2 / 1.72 / 95 / 9.09 / 1.75 / 3.05 / 7630
(Evolved)
X3 / 1.72 / 95 / 8.14 / 1.85 / 2.73 / 7230
(Evolved)
X4 / 1.8 / 115 / 1.74 / 1.74 / 3.19 / 7750
(Evolved)
7. Discussion
The main objective of this research is to study the dependence of the core properties of the sun-like main sequence stars on hydrogen mass fraction. Many precise models have been done before by other researchers but it is still unclear what the exact path of a solar-like star might be as it evolves from the zero-age main sequence to the point where most of the hydrogen is depleted in the core. In addition, what I have done here is to learn how physicists and astronomers resolved and understood the evolution of main sequence stars like the sun. The first model or “test.for” is the program using to check all equations to compare the results referred to in Iben’s table (Novotny, p. 375). It gives very accurate results. The average of the results is close up to 97%. Once the tested model has been created, other models have been created as the hydrogen contents have been exhausted. From the above models, we find that:
From the above table at the same central temperature and density, we assume that models with X4 are older than models with composition of X3, X2 or X1. It is obviously seen that the pressure changes as the hydrogen contents changes. For example, at ρc = 75.0 g/cm3 and Tc = 1.20 107 K, the pressure is 1.112763 x 1016 dyne/cm2 for the model with X1; while it is 0.507763 x 1016 dyne/cm2 for models with X2; and 0.129671 x 1016 dyne/cm2 for models with X3. The pressure drops as hydrogen contents convert to helium by the nuclear fusion reaction. Look up to the other models with the same hydrogen composition and the same density, models with higher temperature tend to have the higher pressure, luminosity, effective temperature and density, but lower radius. For models with the same central temperature but different central density, models with higher central density tend to have lower pressure, effective temperature and density, but higher luminosity and radius.
For models with X2 or X3 to get the same pressure as the model with X1 at ρc = 75.0 g/cm3 and Tc = 1.20 107 K, they have to have higher central density and temperature to get the pressure close to the pressure for X1.
This can be explained by the fact that Pc must remain the same because of hydrostatic equilibrium. The material nears the core of the star must become hotter and more densely to keep the star extended, support its weight and keep it to be held outwards. By solving the equation for different ρc and Tc leads to the conclusion that they must increase to keepPc the same.