The lifecycle of stars

In a nutshell

Stars go through many different stages from birth to death, which are determined by the mass of their core. Stars spend the majority of their life in the main sequence stage where they are stable and hydrogen burning, this is the stage our sun is in currently.

Equations

DESCRIPTION	EQUATION
Schwarzschild radius	$r_s=\dfrac{2GM}{c^2}$

Constants

CONSTANT	SYMBOL	VALUE
$solar \, mass$	$M_{\odot}$	$1.99 × 10^{30}\, kg$
$solar \, luminosity$	$L_{\odot}$	$3.8 \times 10^{26}\, W$
$solar \, surface \,temperature$	$T_{\odot}$	$6000\,K$
$gravitational\, constant$	$G$	$6.67 \times 10^{-11}\,m^3kg^{-1}s^{-2}$
$speed \, of \, light \, in \, vacuum$	$c$	$3 \times 10^8 \, ms^{-1}$

Variables

QUANTITY NAME	SYMBOL	DERIVED UNIT	SI UNIT
$Schwarzschild\, radius$	$r_s$	$m$	$m$
$mass$	$M$	$kg$	$kg$

Definitions

Object	descripTION
Planets	- Objects with a large enough mass to support a spherical shape. - No nuclear fusion occurs. - Has cleared its orbit of other objects.
Dwarf planets	- Orbit has not been cleared of other objects.
Planetary satellites	- Objects that orbit a planet.
Asteroids	- Too small and non spherical to be a planet. - Almost circular orbit around the sun.
Comets	- Small ball of ice, dust and rock. - Orbit the sun in eclectic elliptical orbits.
Solar systems	- Systems comprised of stars and orbiting objects.
Galaxies	- Collections of stars, dust and gas.

Lifecycle of a star

A star has a lifecycle, where each star is 'born' and 'dies' with stages in between, which vary depending on its size.

Physics; Stars; KS5 Year 12; The life cycle of stars

1.	Nebulae
2.	Protostar
3.	Main sequence star
4.	Red giant
4a.	White dwarf
4b.	Black dwarf
5.	Red supergiant
5a.	Supernova
5b.	Black hole
5c.	Neutron star

Formation

Nebulae are the birthplace of stars, and are a cloud of dust and gas. As gravitational attraction pulls them closer together, particles collide more frequently, which causes them to become hot and dense. This hot cloud of dust and gas is called a protostar.

Note: Protostars do not have any nuclear fusion occurring.

For a star to form, the temperature and pressure must be large enough to overcome the electrostatic repulsion and undergo nuclear fusion. The nuclear fusion produces helium nuclei, meaning it is a main sequence star. The star is in equilibrium throughout its main sequence phase, which is a very long period, usually around 10 billion years. At this point, the gravitational force to contract the star equals the radiation pressure emitted from fusion. Eventually, hydrogen runs out, therefore the radiation pressure from fusion will decrease, and the star will no longer be stable. Gravity will then cause the star to collapse, and what happens after this depends on the size of the star's core.

Curiosity: The balance between gravity and the radiation pressure is called hydrostatic equilibrium.

Evolution of a low mass star ( $0.5 \, M_{\odot}$ to $10 \, M_{\odot}$ )

These stars have a smaller, cooler core, so remain in the main sequence for longer. When the hydrogen runs out, the radiation pressure decreases, meaning the force from gravity dominates and the star begins to collapse, and heats up even more. Due to these higher temperatures, helium begins to fuse which releases so much heat that the star begins to expand and appear red. At this stage it is a red giant.

When the helium has run out in the core, the star is not normally big enough for gravity to induce any further nuclear fusion reactions. The core becomes unstable and begins to expel the outer layers of the star leaving behind a planetary nebula.

The inert core then just burns away as a white dwarf. There is no more nuclear fusion happening.

A white dwarf is the hot and dense remnants of the core of the star. Eventually the white dwarf will radiate away all of its energy and cool down over a long period of time. Electron degeneracy pressure stops any further collapse of the core as the mass is below the Chandrasekhar limit ( $1.44\, M_{\odot}$ ).

Evolution of a larger mass star ( $\gt10 \, M_{\odot}$ )

These stars have a larger hotter core, so spend less time in the main sequence. When the hydrogen in the core begins to deplete the radiation pressure decreases, gravity dominates and the core collapse.

As the stars are hotter, the helium nuclei overcome electrostatic repulsion, so helium nuclei fuse to form heavier elements. This forms a red supergiant. It has layers of increasingly heavy elements, all the way up to iron.

Fusing iron absorbs more energy than it releases. This means that the radiation pressure, which originally directed outwards, is now directed inwards as the core absorbs the stars energy. Coupled with gravity also directed inwards, the outer layers of the star crash into the core creating shockwaves with huge amounts of energy. This is the universes largest scale event, a supernova.

The supernova scatters the outer layers of the star across the universe and can also produce gamma ray bursts. The remaining core is too large to withstand the electron degeneracy pressure and collapses into a neutron star.

Curiosity: All elements heavier than iron are formed in supernova explosions.

If the neutron star has a mass of greater than $3\,M_{\odot}$ the star collapses once more into a singularity creating a black hole. A blackhole is a celestial object with a gravitational field so large that even light can't escape it's pull.

Note: A singularity is defined in physics as an infinitely dense point in space.

Schwarzschild radius

The Schwarzschild radius is the distance of the event horizon from the centre of a black hole. The event horizon is the boundary of a black hole from which nothing can escape. It can be described mathematically as:

$r_s= \dfrac{2GM}{c^2}$

where $r_s$ is the Schwarzschild radius, $G$ is the gravitational constant, $M$ is the mass of the object and $c$ is the speed of light.

Example

Calculate the Schwarzschild radius of a star with a mass of $1.8M_{\odot}$

State variables:

$G =6.67 \times 10^{-11}\,m^3kg^{-1}s^{-2} \newline M = 1.8 \times 1.99 × 10^{30}\, kg \newline c = 3 \times 10^8 \, ms^{-1}$

State equation:

$r_s= \dfrac{2GM}{c^2}$

Substitute and solve:

$r_s= \dfrac{2 \times 6.67 \times 10^{-11} \times 1.8 \times 1.99 × 10^{30}}{(3\times 10^8)^2}$

The Schwarzschild radius is $\underline{5300\, m}$ .

Hertzsprung-Russell (H_R) diagram

The Hertzsprung-Russell diagram is a graph showing the relationship between a star's temperature and luminosity. The temperature $x$ axis is inverse, meaning it increases from right to left.

It is useful for helping us understand the differences between types of stars, and how a star changes throughout its lifetime.

There is a specific pattern in the H-R diagram, the hottest most luminous stars are in the top left, and coldest least luminous on the bottom right. Most stars in the main sequence form a curved line between these points. Hot white dwarfs appear on the bottom left of the diagram, and cool red supergiants on the top right.

Physics; Stars; KS5 Year 12; The life cycle of stars

Note: Sometimes the axis for H-R diagrams are logs, due to the large numbers.