Physics of Stars

Luminosity versus Temperature diagram
in this diagram is indicated that there are physical limits to a star whose energy is generated by nuclear fusion. Most H-R diagrams limit themselves to the area between the red boundaries.

The yellow ribbon denotes the socalled main sequence stars. These are the stars in which hydrogen is fused into helium. Stars will spend most of their time in fusing hydrogen. That is why most stars can be found on that ribbon in the diagram.

There is a lower mass limit of stars. Below that limit of about 0.07 solar masses there is not enough mass to exert sufficient gravitational pressure in the core to start a fusion process.

The upper mass limit is determined by the amount of energy produced by the star. A star produces radiation pressure which exerts outward pressure to its outer layers. This pressure is counteracted by gravity. For medium size stars the gravity and radiation pressure are in equilibrium. But the heavier the star the more energy and therefore radiation pressure is produced. This equilibrium between gravity and readiation pressure can be maintained up to about 300 solar masses. Stars which are heavier will will simply blow away their outer layers for which gravity is not strong enough until they have reached their upper mass limit.

Spectral classes
The classification of the spectral classes (marked in yellow tan) is based on the Morgan-Keenan classification system.
The absence of an upper temperature limit for spectral class O is one of its main features.

Wolf-Rayet stars

White Dwarfs
The high luminosity of Z Andromedae B is due to the fact that it is capturing hydrogen from its red giant companion, which results in hydrogen fusion on the surface of this white dwarf.

Brown Dwarfs

Black body
The photosphere of a star is its outer shell which is responsible for all the radiation its emits. The photosphere can be considered as a black body. It is a physical term to characterize an object which is in thermal equilibrium. The photon energy distribution of such an object is directly related to its absolute temperature T.
As a consequence we can write down how the luminosity of a star, which is directly related to the total emitted energy which depends on the temperature of the photosphere and its surface area: Stefan-Boltzmann law.
We can also write down how the peak of the photon energy distribution depends on the temperature: Wien's displacement law.

The Luminosity of a star is proportional to its surface area (4πR2star) and its surface temperature (T) to the power of four (Stefan-Boltzmann law).

L = σ·4πR2·T4 (Watt (Joule/sec))

σ is the Stefan-Boltzmann constant: 5.670 x 10-8 Wm-2K-4

Relative Luminosity of a star:  
Lstar  =  R2star  ·  T4star
LSun R2Sun T4Sun

Wien's displacement law
The higher the temperature of a star the more bluish its appearance (the shorter the wavelenght).

λmax b

b = 2.898 x 10-3 m K

Magnitude of a star
The magnitude of a star is a measure of its brightness.
This brightness (b) is the amount of energy received per unit area and is inversely proportional to the square of the star distance (d). b Is the measured brightness and is therefore also called apparent brightness
The luminosity (L) of a star is actually an intrinsic property. It is the total amount of energy emitted at its surface

The relationship between apparent brightness (b) and luminosity (L): 
b =  L

The magnitude is a logarithmic value to accomodate the huge dynamic range of the brightness or luminosity.

Absolute magnitude of a star:  
Mstar = MSun + 2.5 · log ( LSun )

The absolute magnitude of a star is by definition related to its apparent brightness as measured from a distance of 10 parsec, 32.6 light-years.
The magnitude scale has been chosen in such a way that a luminosity ratio of 100 results in a magnitude difference of 5.

The luminosity relates to the total emitted energy over the whole electromagnetic spectrum.

Visual and bolometric magnitude
With regard to the absolute magnitude, a distinction is made between the visual magnitude Mv and the bolometric magnitude Mbol. Mv expresses how bright a star is to our eyes, it relates to only that part of the energy which is emitted in the visual part of the electromagnetic spectrum . Mbol expresses how bright a star is to a measuring device integrated over the whole electromagnetic spectrum.

The difference between Mbol and Mv is called the bolometric correction BC and depends on the surface temperature of the star.

The absolute visual magnitude of the Sun is:MSun,v = 4.83
The absolute bolometric magnitude of the Sun is: MSun,bol = 4.75
Its bolometric correction:BCSun = Mbol - Mv = -0.080

Color index
The scale of the color index is based on observational data of stars with surface temperatures between 2 860 and 42 000 Kelvin.
A trend analysis has been used to extrapolate these data to cover a temperature range between 102 and 106 Kelvin.
More information about the analysis can be found on this page.

Personal reminder:
I am not sure about the relationship between Luminosity and Mass for stars less than 0.43 solar mass. The formula seems to agree badly with the observations. I need to check on that. I also need to check to which extent the formulas are phenomenologcal in nature or have a theoretical foundation.
Lifetime of a star
Lifetime (τ) of a star =  amount of fuel (mass)
fuel consumption rate
The fuel (hydrogen) consumption rate of a star is proportional to its luminosity L, so:
τstar  =  Mstar  ·  LSun
τSun Lstar MSun

Extended Luminosity-Temperature diagram
In order to depict the life cycle of super and hyper giants we have to extend range of the Luminosity - Temperature diagram to depict the next phases of these giant stars.

Life cycle phases of super giant stars
All giant stars will eventually go supernova: a gigantic explosion during which so much energy is released that its luminosity is comparable with that of an entire galaxy containing several hundred billions of sunlike stars. During that explosion much gas is released which still contain an abundant amount of hydrogen to give birth to a next generation of stars.

Neutron star or black hole
Also a very compact object is formed at the very heart of the explosion. It is a neutron star when its mass is less than 2 solar mass. A neutron star is the product of the compressive force as a result of the massive explosion acting on the core of the giant star. The mass density will become so high that normal hydrostatic equilibrium in which the gas pressure is in balance with the gravity cannot be achieved. The core will collapse until its mass density is comparable to that of an atomic nucleus in which nuclear forces are counteracting gravity to reach a new equilibrium. Electron and protons have merged into neutrons.

If the core is havier than 2 solar mass then even nuclear froces are not enough to counteract gravity. It will go through a gravitational collapse to become a black hole.

Two cooling phases of a neutron star
A newly formed neutron star is believed to have a surface temperature of several hundred million Kelvin. In that hot phase it will rapidly loose its thermal energy by radiating high energetic neutrinos. In about 10 000 years the surface temperature will drop to about one million Kelvin. From that moment on the cooling process will increasingly be dominated by radiating photons.

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