The Main Sequence

If we plot the properties of bright stars observed, the distribution on the plot is not random.

We can plot things like colour vs magnitude. Or temperature vs luminosity.

When we do we see a track going from red and dim to blue and bright, following a clear sequence. This is the main sequence.

The main sequence is where normal stars of different mass settle down.

The rate of nuclear burning in a star is determined by the temperature and pressure in the centre of the star. The more massive the star, the higher the pressure and temperature in the core and the hotter and faster the hydrogen fuses.

Even though more massive stars have more hydrogen fuel, they also burn it proportionally more quickly, so the more massive a star is, the more quickly it runs out of fuel.

The luminosity of a star is directly reflecting the rate of burning in the core.

So higher mass stars have higher luminosities.

Higher mass stars are also bigger, and have more surface area to radiate their energy away, but their luminosity goes up faster than their area, as the mass increases. So, the more massive stars, on the main sequence, are also hotter.

The reason stars are hot is the Stefan-Boltzmann law. They have a lot of power generated in their cores which must be radiated away, and increasing the temperature is a very efficient way to let the energy flow away.

The Hertzsprung-Russell diagram

The plot of colour vs luminosity is know as the H-R diagram.

A typical H-R diagram shows the main sequence running from upper left to lower right.

Above the main sequence is the giant branch and above that lie the supergiants.

In the lower left corner are the white dwarfs.

Giants are bigger than the main sequence stars, typically 10-100 times bigger in radius (not mass! Giants have different density than main sequence stars).

Supergiants have radii up to 1000 times bigger than the Sun.

Giants have a different structure from main sequence stars and are undergoing different nuclear reactions, which causes their high luminosity and size.

Mass-luminosity relation

Stellar masses range from about one tenth the mass of the Sun to about 100 times the mass of the Sun.

As the main sequence mass increases, the higher the pressure and temperature in the core of the star, and the higher the rate at which hydrogen is fused.

It turns out that luminosity, L, is proportional to mass, M, to the 3.5 power.

L* = Lsun*(M*/MSun)3.5

What this means is that a star that is one seventh the mass of the Sun - close to the minimum mass found for stars - has a luminosity one thousandth that of the Sun.

A star 10 times the mass of the Sun has a luminosity of about 3000 times that of the Sun, and a star twice the mass of the Sun has a luminosity of about 11 times that of the Sun.

But, the amount of fuel available is just the mass (of the core) of the star!

So, more massive stars run out of fuel faster.

The lifetime of a star, t*, goes like M-2.5

So, a star that is 10 times more massive than the Sun lives only one three hundreth as long. - We think the Sun will last about 9000 million years, so a 10 solar mass star will live about 30 million years.

Similarly a 2 solar mass star will live about 5 times shorter than the Sun or about 1.8 billion years.

For very massive (> 10 solar masses) and very low mass (< 0.8 solar mass) stars, the simple relationship above doesn't hold. Very massive stars all seem to last a few million years at least.

Low mass stars do live much longer than the Sun, but not as long as the simple relationship would imply - so a star with 70% of the mass of the Sun will last about 15 billion years.

Stellar Evolution

As stars grow old they move away from the main sequence, becoming sub-giants and then giants.

Eventually stars run out of fuel in their core completely and fusion stops.

The stars then collapse under their own weight and die.

Some stars become white dwarfs, others leave more exotic remnants.

White Dwarfs

White dwarfs are the left over, burnt out cores of dead stars.

White dwarfs are typically a few thousand km across - Earth size! But have half to one solar masses.

White dwarfs are extremely dense, and usually very hot. The heat is remnant heat and the white dwarfs are passively cooling very slowly, they do not generate energy internally.

White dwarfs are hot, but have small surface areas, so they are cannot radiate their energy very quickly.

A typical white dwarf has luminosity of less than one hundreth that of the Sun, but the amount of heat energy stored in it is comparable to that stored in the core of the Sun.

It would take the Sun about 20 million years to cool passively if the core fusion stopped, so a cooling white dwarf with the same heat energy but 0.001 times the luminosity can radiate for 20 billion years!

In practise white dwarfs start off very hot, and more luminous than the Sun, and cool down - initially they cool down very rapidly, and fade correspondingly - a white dwarf may start off at 100,000 Kelvin, so by the time it has cooled to 10,000 Kelvin - still hotter than the Sun - it has faded by a factor of 104 = 10,000!

Sirius has a white dwarf companion, gravitationally bound to it.

Sirius with Sirius B.

Binary Stars

The Sun is a loner. The nearest star is several light years away and its biggest planet is puny compared to a star.

Most stars (maybe ½ - 2/3) have companions. They are in binaries (or in some cases triples, quadruples, quintets etc).

Binary stars are bound - that is to say the two stars are tied to each other through their mutual gravitational attraction and they will orbit each other indefinitely, much as a planet orbits a star.

Binary stars come with all possible ranges of orbital separation (and eccentricity). Some binaries are very widely separated, the stars barely bound by the weakest possible gravitational attraction. Other binaries are so close the two stars practically touch.

Observing binaries

Stars close to each other are hard to separate visually.

If stars in a binary are widely separated, then we can resolve them in telescopes.

Basically when we look at what we think is a single star in a telescope, it separates out into two different stars.

Another way to see if a star is a binary is to look for eclipses.

If the binary is near edge on, then as one star passes in front of the other it blocks out the light from the star at the back, and the combined light we see from the stars is less.

If the two stars are a different colour, then we also see a colour change as well as a brightness change. A famous example of such an eclipsing binary is Algol.

Algol demo

By analysing exactly how the brightness and colour of eclipsing binaries changes, we can measure the sizes of stars and the structure of the outer layers of the stars.

We can also see binaries spectroscopically. As the stars move in orbit around each other, their speed changes. So there is a Doppler shift of the absorption lines of the stars, which changes systematically as each star comes towards us and then goes away from us.

The size of the Doppler shift, combined with the orbital period can tell us what mass the stars are. So spectroscopic binaries are extremely valuable in allowing us to quantify theories of how stars of different masses should behave.

One of the ways of detecting white dwarfs is by finding them in binaries, we can see the main sequence star in the binary moving through the doppler shift of its spectrum, even if its white dwarf companion is too faint to see directly.

Mass transfer

If binaries are close, then stars can transfer mass.

In particular, if one star goes inside the Roche lobe of the other, then the tides from the one star can rip off the outer layers of the other star. The mass ripped off can either accrete onto the star, or it can ejected into space (or bit of both).

Mass transfer can rejuvenate stars low on fuel, and alters the position of a star on the H-R diagram.

Some of the most interesting and energetic phenomena we observe comes from mass transfer binaries.