7th Week: Stellar Structure and Evolution
As we look at the sky in the night, we can see that the Universe has a very complicated structure: planets, stars, galaxies, clusters etc. To understand how this stellar structure formed we use our knowledge of nuclear physics combined with cosmology.
Stellar properties
Here we make the way to identify how stars are grouped and classified. This classification scheme identify stars by the following:
magnitudes - observational measure of the brightness, grouped
scale extended in both directions
Sun, m = -26.83 , the faintest objects, m = 30
Proxima Centauri: m = 11.05
Vega: m = 0.03
Zeta Reticuli: m = 5.54
Betelgeuse: m = 0.58
colors - grouped
spectral types - grouped from hottest to coolest
O B A F G K M L T
the harvard classification scheme for star spectral
reordered to reflect the surface temperatures of stars
O: blue-white, hottest
B: hot blue-white
A: white
F: yellow-white
G: yellow
K: cool orange
M: cool red
L: dark red, very cool
T: coolest , infrared
Proxima Centauri: M5V
Sun: G
Vega: A0V
Barnard's Star: M4V
Zeta Reticuli: G2.5V
Betelgeuse: M2Iab
luminosity classes - grouped
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=4\pi R^2\sigma T_{e}^{4}}
based on Maxwell's equation pertaining to radiation pressure and the laws of thermodynamics, so for a star of spherical radius R and area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A =4\pi R^2} and whereFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{e}} is the effective temperature of the surface of the star, utilized due to that stars are not perfect blackbodies
Sun: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L= 3.839 x10^{26} W = L_{\bigodot}}
Vega: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = 37L_{\bigodot}}
Proxima Centauri: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = 6x10^{-6} L_{\bigodot}}
Zeta Reticuli: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = 0.79L_{\bigodot}}
Betelgeuse: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = 135,000L_{\bigodot}}
The Hertzsprung-Russel diagram
a diagram chart that allows for the stellar properties of luminosity, surface temperature and magnitude to be related into a stellar classification scheme
Stellar structure
Stars of different mass and age have varying internal structures. Stellar structure models describe the internal structure of a star in detail and make detailed predictions about the luminosity, the color and the future evolution of the star.
Equations of stellar structure
Pressure equilibrium: Pressure and gravity must balance or the star will expand or contract. The balance of the gravity force and the pressure gradient is known as the hydrostatic balance.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dP_r}{dr}=-\rho_r \frac {GM_r}{r^2} }
Conservation of mass
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {dM_r}{dr}= 4\pi r^2 \rho _r }
Energy generation: To keep the temperature constant everywhere luminosity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L } must be generated.Considering the energy leaving the spherical shell yields the energy equation
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac {dL_r}{dr}=4\pi r^2 \rho _r \epsilon }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } is the energy generation rate (sum of all energy sources and losses) per g and s
Energy transport: The temperature inside a star must increase progressively towards its center to make energy flow from the center to the surface. The energy transport equation takes differing forms depending upon the mode of energy transport.
For the case of convective luminosity transport we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dT_r}{dr}=\left( 1- \frac{1}{\gamma} \right) \frac{T_r}{P_r}\frac{dP_r}{dr} }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = c_p / c_v } is the adiabatic index, the ratio of specific heats in the gas
For radiative energy transport we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dT}{dr}=-\frac{3 \kappa \rho}{4acT^3} \frac{L(r)}{4 \pi r^2} }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa } is the opacity of the matter and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } is the radiation density constant equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7.56591 \times 10^{-15} erg \ cm^{-3} K^{-4} }
For conductive luminosity transport (appropriate for a white dwarf), the energy equation is
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dT}{dr}= - \frac {1}{k} \frac {1}{4 \pi r^2} }
where k is the thermal conductivity.
Stellar evolution
The evolution of stars is due to their sustained emission of radiation originating from an internal source. The energy source that supplies the luminosity of stars during most of their lifetimes is nuclear fusion, which turns a small fraction of the rest mass into energy.
Hydrogen burning I: the p-p chain
The most abundant element in newly born stars is hydrogen. Fusion of hydrogen into the next element, helium, would require an encounter of three or four protons - hydrogen nuclei - within the distances of the order fermis (1E-15 m). The probability of such a multiple encounter is small, therefore this process does not happen at once, but gradually through the chain of reactions. The first link of this chain is the fusion of two protons
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p + p \rightarrow ^2D + e^+ + \nu \ , }
that forms a stable isotope of hydrogen. Then deuterium captures a proton to form the lighter isotope of helium
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D + p \rightarrow ^3He + \gamma \ , }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} indicates the emission of an energetic photon that soon will be absorbed by other particles. Now the reaction becomes more complicated: two Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^3He} nuclei form the stable isotope of helium, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^4He}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^3He + ^3He \rightarrow ^4He + 2p \ , }
and later
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^3He + ^4He \rightarrow ^7Be + \gamma \ . }
This is the end of the so called p-p chain I.
The p-p chain II starts with the electron capture by the beryllium nucleus
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^7Be + e^- \rightarrow ^7Li + \nu \ , }
and then a subsequent capture of the proton
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^7Li + p \rightarrow 2 ^4He \ , }
then ends. The p-p chain III results by capturing proton by beryllium nucleus instead of an electron capture
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^7Be + p \rightarrow ^8B + \gamma \ . }
Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^8B} is radioactive it decays
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^8B \rightarrow ^8Be + e^+ + \nu \ , }
and the since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^8Be} is highly unstable it goes through the reaction
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^8Be \rightarrow 2 ^4He \ . }
Hydrogen burning II: the CNO bi-cycle
The small percentage of initial composition of stars consist of carbon, nitrogen, and oxygen nuclei. These nuclei may induce a chain of reactions that transform the hydrogen into helium. This procedure is first introduced by Bethe and is known as the CNO cycle. Since there are two such cycles, therefore we call it CNO bi-cycle.
The cycle one is as following:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{12}C + ^1H \rightarrow ^{13}N + \gamma \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{13}N \rightarrow ^{13}C + e^+ + \nu \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{13}C + ^1H \rightarrow ^{14}N + \gamma \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{14}N + ^1H \rightarrow ^{15}O + \gamma \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{15}O \rightarrow ^{15}N + e^+ + \nu \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{15}N + ^1H \rightarrow ^{12}C + ^4He \ . }
And the cycle two is:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{14}N + ^1H \rightarrow ^{15}O + \gamma \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{15}O \rightarrow ^{15}N + e^+ + \nu \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{15}N + ^1H \rightarrow ^{16}O + \gamma \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{16}O + ^1H \rightarrow ^{17}F + \gamma \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{17}F \rightarrow ^{17}O + e^+ + \nu \ , }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{17}O + ^1H \rightarrow ^{14}N + ^4He \ . }