Lecture 4: On to the Stars

The most obvious difference between stars is their brightness. The Sun is 10¹⁰ x (ten thousand million times) brighter than the next brightest star. This is why we can only see the other stars at night - their light is completely drowned out by the Sun during the day. If they are intrinsically the same as the Sun then they must be immensely far away in order to appear so much dimmer.

Luminosity is the amount of energy emitted per second (power). So this is plainly constant, depending on the star not on where we observe it from. The apparent brightness that we see depends on our distance as the light gets spread out as it travels out in all directions. The area of a sphere increases as the square of the distance, so the apparent brightness we see decreases with the square of the distance as long as there is no fuzzy stuff in the way to decrease the brightness still further. This is called the inverse square law. So if we know the luminosity of a source, and its brightness, we can find its distance. Take the brightest star (Sirius) which is about 10 ¹⁰ x dimmer than the Sun. Assuming it is intrinsically the same as the Sun (i.e. has the same luminosity) then it must be 10⁵x further away than the Sun. The Sun is 8 light minutes away from the Earth, so Sirius would be 8 x 10 ⁵ light minutes away = 1.5 light years (its actually rather further than this - see below!).

But not all stars are like the Sun. They have different temperatures. For a few stars we can see this directly - e.g. Betelgeuse in Orion does look reddish, while Sirius looks white. Rather than just looking we can plot the spectrum of the star. The wavelength of maximum intensity is related to the temperature - long wavelength is low temperature, short wavelength is high temperature. But when you look, the spectrum of a star is not simply a blackbody - a close look at the spectrum of the sun shows that its has lots of dark absorption lines (don't bother about clicking on anything, just look at it) superimposed on the blackbody continuum emission. What are these ?

To answer this we need to digress into some really strange physics, into the quantum world.... Why do atoms exist ? If you think of electrons going around the nucleus in an orbit like the earth going around the Sun, the all you have to do is look at it from the side and you see the electron going up and down. So it should radiate electromagnetic energy and so spiral in closer to the nucleus - and calculations show that it would fall into the nucleus in about 10^-16 seconds. So it seems that atoms shouldn't exist. But they do! How ? In desperation the physicists (most notably one called Niels Bohr) finally said, suppose that electrons CAN exist stably in atoms, but only with certain energy states. Then tranistions between these states have specific energy associated with them - if an electron absorbs a photon of exactly the right energy then it can jump up to a higher level (giving a dark absorption line) but it tries to be in the lowest energy state, so it will jump back down again, emitting the excess energy as a photon of a specific energy (and emission line). Have a browse through the quantumzone for more on this. Other nice sites are spectral lines, and the Bohr atom. For those with more science background there are more detailed notes on formation of spectral lines and the Bohr model atom and how atoms produce their spectra. As for the reason WHY such strange behaviour might exist, we get back to the strange world of quantum mechanics. On very small scales, when we look at subatomic particles like electrons, they behave not so much like billiard ball point particles, but more like a wave. This is called wave-particle duality, and is plainly very odd. But if an electron can behave a bit like a wave, then it has a wavelength. There will be some special radii in an atom where an electron wave can fit exactly, folding back on itself, while at other radii it doesn't exactly fit, so eventually a peak lines up with a trough, and the wave cancels itself out! See this nice animation of electron waves in atoms. The energy (ie the wavelength) of the electron wave depends on how tightly the electron is held by the electrostatic attraction of the nucleus. So every chemical element has its own set of lines. But the pattern of lines also depends on temperature - for example, if a star is very hot, then the atoms are moving very fast, and collisions can detach the electron from the proton in hydrogen (ionise it). Since the electron is no longer bound to the proton then there are no lines! For more details see the temperature dependence of hydrogen lines. The result is that the pattern of absorption lines gives us another way to measure the temperature of a star, which is less ambiguous (for reasons we'll talk about in a few lectures time) than simply measuring the peak of the spectrum to get the blackbody temperature.

So, now we have ways to measure the temperature, both from the wavelength of the peak intensity of the blackbody, and/or more accurately from the spectral lines. And stars do indeed have a wide range in temperature, some hotter and some cooler than the Sun. First classified as spectral type by the line emission - the sequence from hottest to coolest is O-B-A-F-G-K-M going from ~50,000 K to 3,000 K (the Sun is spectral type G). But if stars have different temperatures from the Sun then there is no reason for them to have the same luminosity as the Sun! So how can we measure distance ?

Go to a cluster of stars - these are bound together so they are all at the same distance, so their apparent brightness relative to each other is determined only by their absolute luminosity. There are open clusters and globular clusters shown on this star clusters page. For either type of cluster we can plot temperature and apparent brightness: This is called a Hertzsprung-Russell (HR) diagram. Most stars lie on the 'Main Sequence', where temperature and luminosity are tightly linked: high temperature implies high luminosity and vica versa. We can find a star with a spectrum like the sun, and then assume that it has the same luminosity, and so get its distance. This then tells us the distance to the whole cluster so we can convert the apparent brightness of other types of stars in the cluster into luminosities. Then we can calibrate the whole HR diagram and get it in terms of temperature and luminosity. Then for any main sequence star, all we have to do is use its spectrum to determine its spectral type (so we know what its luminosity should be), and measure its apparent brightness, and then we can use the inverse square law to find its distance (assuming there is no fuzz in the way). This method of finding distance is called spectroscopic parallax