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--- courses:ast301:3 [2024/03/15 06:42] – [1.2 Secondary equations] asad
+++ courses:ast301:3 [2024/03/15 13:19] (current) – [3. HR diagram] asad
@@ Line 127: / Line 127: @@
 The seven equations given in Section 1, five primary and two secondary, are used to model a star. Ideally a mathematical model can be constructed using only two parameters: mass and chemical composition. Given these two, the model will automatically give the radius, luminosity and temperature. The radiative and convective zones will also be identified using the temperature gradients.
-For modelling a spherical nonrotating star, $\kappa$ and $\epsilon$ are, first, determined independently from observation or theory. Then if the composition is considered uniform throughout the star, Eqn. \ref{6} can be used to eliminate $\rho$ from Eqns. \ref{1}--\ref{5}. Then the first five equations become single-variable resulting in $P(r)$, $M(r)$, $L(r)$ and $T(r)$. The boundary conditions $P(R)=0$, $M(0)=0$, $M(R)=\mathcal{M}$, $L(0)=0$ and $T(R)=0$ are specified. The solution of the first five differential equations are then carried out numerically using, for example, the Runge-Kutta methods.
+For modelling a spherical nonrotating star, $\kappa$ and $\epsilon$ are, first, determined independently from observation or theory. Then if the composition is considered uniform throughout the star, Eqn. \ref{6} can be used to eliminate $\rho$ from Eqns. \ref{1}--\ref{5}. Then the first five equations become single-variable resulting in $P(r)$, $M(r)$, $L(r)$ and $T(r)$. The boundary conditions $P(R)=0$, $M(0)=0$, $M(R)=\mathcal{M}$, $L(0)=0$ and $T(R)=0$ are specified. The solution of the first five differential equations are then carried out numerically using, for example, the [[wp>Runge-Kutta methods]].
+One such modelling has been described in Appendix L of //Introduction to Modern Astrophysics// (edition 2) by Carroll and Ostlie and [[http://spiff.rit.edu/classes/phys370/lectures/statstar/statstar.html|in this website]] by Michael Richmond. You have to use a python code to make a model of a star and postdict and predict its life. You can chose any type of star.
+Stars are classified based on their //effective temperature// $T_e$ on the surface which is estimated from the [[uv:spectral-line|absorption lines]] found in their spectra. The hottest to the coldest stars are assigned the **spectral types** O, B, A, F, G, K and M. O-type stars are divided into ten more types called by the numbers 0 to 9; B0 stars would be hotter than B9 stars. Stars also have a **luminosity class** given the Roman numerals I to VII. Class V stars are in the **main sequence**, meaning they are in hydrostatic equilibrium. Stellar classification is summarized below.
+^ Type ^ $M_V$ ^ $T_e$ [kK] ^ $M/M_\odot$ ^ $R/R_\odot$ ^
+| Main sequence stars (Class V) |||||
+| O3 |  |  | 120 | 15 |
+| O5 | -5.7 | 42 | 60 | 12 |
+| B0 | -4.0 | 30 | 17.5 | 7.4 |
+| A0 | +0.65 | 9.79 | 2.9 | 2.4 |
+| F0 | +2.7 | 7.3 | 1.6 | 1.5 |
+| G0 | +4.4 | 5.94 | 1.05 | 1.1 |
+| K0 | +5.9 | 5.15 | 0.79 | 0.85 |
+| M0 | +8.8 | 3.84 | 0.51 | 0.60 |
+| M8 |  |  | 0.06 | 0.10 |
+| Giant stars (Class II) |||||
+| B0 |  |  | 20 | 15 |
+| A0 |  |  | 4 | 5 |
+| G5 | +0.9 | 5.05 | 1.1 | 10 |
+| K5 | -0.2 | 4.05 | 1.2 | 25 |
+| M0 | -0.4 | 3.69 | 1.2 | 40 |
+| Supergiant stars (Class I) |||||
+| O5 |  |  | 70 | 30 |
+| B0 | -6.5 | 28 | 25 | 30 |
+| A0 | -6.3 | 9.98 | 16 | 60 |
+| G0 | -6.4 | 5.37 | 10 | 120 |
+| M0 | -5.6 | 3.62 | 13 | 500 |
+Here $M_V$ is the absolute [[uv:magnitude]] in visible wavelengths which is a dimensionless measure of luminosity. As you can see, the mass of main sequence stars can vary from 120 to 0.06 solar mass, but their radius varies from 15 to 0.10 solar radius. The variation in luminosity can be from 0.011 solar value to almost a million times more than the sun.
+{{:courses:ast301:starzones.webp?nolink&700|}}
+The models also provide the information about the radiative and convective zones inside a star. The lowest mass stars are completely convective and the size of central radiative zone increases with mass. In a solar-mass star only the outer envelope (30%) is convective, but a star becomes completely radiative if its mass is 1.5 solar mass. Stars more than two times heavier than the sun develop a convective zone at the core and the central convective zone keeps increasing in size with mass. So lightweight stars produce nuclear energy at the radiative core whereas heavyweight stars do that in their convective core.
-One such modelling has been described in Appendix L of //Introduction to Modern Astrophysics// (edition 2) by Carroll and Ostlie and [[http://spiff.rit.edu/classes/phys370/lectures/statstar/statstar.html|in this website]] by Michael Richmond.
 ===== - HR diagram =====
+HR ([[wp>Ejnar_Hertzsprung|Hertzsprung]]-[[wp>Henry_Norris_Russell|Russell]]) diagram related the luminosity and temperature of stars. These two quantities are linked to the radius via the equation of flux we introduced before: $\mathscr{F}=\sigma T_e^4=L/(4\pi R^2)$ where $T_e$ is the **effective temperature**.
+$$ R = \frac{\sqrt{L}}{\sqrt{4\pi\sigma}T^2} = \frac{\sqrt{L/L_\odot}}{(T/T_\odot)^2} \ R_\odot $$
+which means there are diagonal constant-radius lines in the HR diagram. HR diagram can be shown using either $T$-$L$ or the corresponding observable quantities color index and absolute magnitude. Both temperature-luminosity and color-magnitude versions are shown below.
+[[https://astro.unl.edu/naap/hr/animations/hr.html|{{:courses:ast301:hrd.webp?nolink|}}]]
+B-V color index is the difference between apparent magnitudes in the blue and visual or yellow bands, i. e. $m_B-m_V$. The absolute magnitude on the $y$-axis is the yellow magnitude $M_V$. A negative B-V index means the $m_B<m_V$ and, hence, the brightness in blue wavelengths is greater than in the visual band. Note that $M_V\propto - \log L$ and $(m_B-m_V)\propto -\log T$ which will be evident if you compare the axes of the two panels.
+The green patch in the HR diagram is the main sequence where a star stays during its stable period. Before the birth a star is to the lower right hand side of the main sequence. When the equilibrium of the star is broken, it moves to the upper right hand side of the main sequence toward the giant (light red) or supergiant (blue) branches. After death stars move to the lower left hand side of the main sequence near the white dwarf patch (grey). Some massive stars oscillate periodically and they are found in the instability strip (dark red). On the main sequence the heavier stars are found in the upper part, and vice versa.
 ===== - Main sequence and giant branch =====