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# Stellar characteristics

## Hertzsprung-Russell diagram

A Hertzsprung-Russell (HR) diagram plots stars according to their luminosity (vertical axis) and surface temperature (horizontal axis). Unlike most graphs in Physics, the HR diagram is not used to obtain a perfect correlation but is instead used to explore the evolution of stars.

There are a few points to note about the diagram:

• The temperature axis is absolute (K), decreasing from left to right and has a non-linear scale.
• The luminosity axis is logarithmic. It corresponds to the quantity absolute magnitude (not on the course!), which becomes more negative as stars become more powerful.

The spectral classes are categories of star, inconveniently in the order O B A F G K M from hottest to coolest. One possible mnemonic to help remember this is 'Oh Be A Fine Girl/Guy, Kiss Me'!

Sometimes the HR diagram is shown with diagonal lines representing constant star radius. While unlikely to be reproduced on an exam script, this ESO HR diagram also indicates the colour of stars according to their temperature and, hence, spectral class.

The regions of the HR diagram reveal some groupings of stars with similar surface temperature and luminosity. We now know that each of these represents a portion of a star's evolution. These are the main sequence, red giants, super giants and white dwarfs. Note that there are other stages in the evolution of a star, but these cannot be accommodated within the axes of the diagram.

## Types of star

#### Main sequence

Stars in the main sequence have balanced forces between inward gravitation and outward radiation pressure. This is the longest stage in a star's evolution and continues until all hydrogen in the star has fused to become helium. The duration of the main sequence decreases as the mass of the star increases due to the higher temperatures in the core.

Main sequence stars can be of any spectral class due to the possible luminosities and temperatures. The Sun is an example of a main sequence star with surface temperature 6000 K. This makes it spectral class G.

The luminosity of a main sequence star increases with its mass according to the mass-luminosity relation:

$$L\propto M^{3.5}$$

• $$L$$ is the luminosity of the star (W, or any consistent unit of power due to the proportional relationship)
• $$M$$ is the mass of the star (kg, or any consistent unit of mass due to the proportional relationship)
• $$3.5$$ is the exponent for main sequence stars

Alternatively the relationship could be expressed as:

$${L_1 \over L_2}=({M_1\over M_2})^{3.5}$$

#### Red giants

When no hydrogen remains in the core, the outer layers of the star collapse under gravity, increasing the temperature of the core. Depending on the mass of the star, the temperature rise may be sufficient to recommence fusion of hydrogren from the outer layers or to commence fusion of helium. This process will repeat for as long as the temperature will allow, with the star contracting and expanding as each new type of nucleus, including carbon, is born. The expanded star is a red giant.

With their large radius (the diagram shows the scale that the Sun will reach as a red giant), the surface temperature of a red giant is lower than in the main sequence. The appearance of the red giant is from yellow-orange to red, including the spectral types K and M.

The contractions under gravity retain a radius sufficient to keep the electrons from entering the same quantum mechanical state. This electron degeneracy creates an outward pressure that prevents total collapse.

#### White dwarfs

White dwarf stars can form in one of two ways:

• Directly from the main sequence, for stars of mass under 0.3 times that of the Sun.
• From red giants with mass under 10 times that of the Sun, having cast off a planetary nebula of dust and gas due to the weaker gravitational field at the outer layers of the star.

White dwarfs are hot and small, on a similar scale to the Earth, but with no fusion taking place to produce light. This places white dwarf stars to the bottom right of the HR diagram, in spectral classes O, B and A.

Approximately 97% of stars in the Milky Way will become a white dwarf at the end of their lives. They cool with time.

#### Super giants

Super giant stars are large enough to fuse nuclei to produce elements as large as iron, the nucleus of highest stability according to binding energy. They come from main sequence stars of spectral class O and B with masses over 8 times that of the sun.

Super giant stars are both massive and luminous, placing them at the top of the HR diagram. The temperature range of supergiant stars spans 3000 K to over 20 000 K with any spectral class possible.

The Sun will never become a super giant, because of its limted mass. However, the image shows the scale of a blue supergiant in comparison to Jupiter's orbit in the Solar System.

When fusion of iron ceases in a super giant, the star collapses once and then explodes in a supernova, releasing mass outwards. A supernova's luminosity is too great to be displayed on the HR diagram. A supernova is the only cosmological event that is sufficiently energetic for the fusion of elements heavier than iron.

The outcome of a supernova depends, once again, on the mass of the material remaining. A neutron star or black hole will form. These vary from white dwarfs because their mass exceeds the Chandrasekhar limit, above which the electron degeneracy pressure in the star's core is insufficient to balance the inward force of gravity. The Chandrasekhar limit is 1.4 solar masses.

#### Neutron stars

Neutron stars (after a supernova of 10 to 29 solar masses) have a mass up to two or three times that of the Sun and a radius in the order of 10 km (roughly a city!). This gives neutron stars a density of approximately 1017 kgm-3.

An example of a neutron star is a pulsar with a periodically rotating magentic field. These can be detected using radio telescopes on Earth, provided that the plane of the magnetic field intersects with that of the detector.

The incredibly high temperatures of a neutron star means that they are not visible on the HR diagram. The radius of a neutron star is upheld by neutron degeneracy, an outward pressure that prevents total collapse.

#### Black holes

What happens to masses above three times that of the Sun after a supernova is unclear. So far mankind has not detected stars of between 3 to 5 solar masses. Beyond 10 solar masses, the gravitational field is so strong that the star collapses to produce a black hole, in spite of neutron degeneracy.

The Oppenheimer–Volkoff limit is the maximum mass that a neutron star may have before further collapse into a black hole. This is estimated to be 3 solar masses.

Black holes are given their title as their escape velocity exceeds the speed of light, which means that no electromagnetic radiation is observed from their core. The gravitational field is strong enough to distort the path of light passing nearby.