Cepheid variable stars

earlhy computers

This photograph of the Harvard Observatory in the 1890's shows some of the earliest computers used for astrophysics. Can you spot them?


The women in the photograph were called "computers." They were examining images of stars on glass plates, looking for stars that varied in brightness. You can see graphs on the wall, indicating the oscillation patterns. The women noticed that the oscillation period varied between stars. Some oscillated faster, some slower.


Among the women in this photograph is Henrietta Leavitt (1868-1921) who is credited with discovering the period-luminosity relationship. She focused on comparing oscillation frequencies for stars in the Large Magellenic Cloud. Focusing on this population allowed her to assume that the stars were all roughly the same distance away. This meant that the brighter stars didn't just look brighter, they had higher luminosity. She noticed that the stars with higher luminosity also had longer oscillation periods.


Finding the relationship between the oscillation period and luminosity allowed astronomers to use the oscillation period of a star to determine its luminosity. They could then use the relationship between luminosity and apparent brightness to calculate how far away the star was.


Cepheid variables are unstable stars, nearing the ends of their lives. As the star gets hotter, helium becomes doubly ionized - that is, it loses both of its electrons - and becomes more opaque. When the gas is the most opaque, the star is the dimmest. But since the opaque gas traps in the light, the star heats up and expands. As it expands, it cools, and some of the electrons are able to become bound to the helium nuclei, making the gas more transparent. This allows light to escape, stopping the expansion, and reversing the process. This process happens over and over, and takes longer for higher luminosity stars, since they are larger and more massive, the reaction takes longer to spread through the star.


For more information on Cepheid variables, see Nick Strobel's astronomy notes here.

Cosmic distance ladder

The period-luminosity relationship for Cepheid variables (and RR Lyra variables), plus the fact that these stars were very bright, allowed them to be used as "standard candles." A standard candle is simply a bright object whose luminosity is known. These variable stars are bright enough that they can be seen in nearby galaxies. This knowledge allowed us to push our cosmic distance ladder out to 25,000,000 parsecs away.

Cepheid variables

  • Unstable post main sequence stars
    • Helium ionizes, gas is opaque
    • Star is dim
    • Opaque gas traps light and heats up
    • Star expands
    • Gas cools and gets more transparent
    • Star gets bright
    • Light escapes and star cools and contracts
    • Cycle starts over
  • Long period stars have higher luminosity
    • Periods are a few days to a few weeks
  • Very luminous – can see them in other galaxies
  • Calculate distance:
    • Measure period and apparent brightness
    • Use period-luminosity relationship to get luminosity
    • Use apparent brightness – luminosity relationship to calculate distance

Rotation Curves

wheel-like rotation curve

We can learn a lot about a galaxy by examining how it rotates. The above object has wheel-like rotation, that is, it rotates as a solid. The points A, B, C and D, lying along a radial line from the center of the wheel, will remain in a straight line along a radius. If you plotted the orbital speed as a function of the distance from the center, the plot would be a straight line.

planet-like rotation curve

Kepler's law dictates that the square of the period of a planet's orbit is proportional to the cube of its radius (assuming a circular orbit). The plot above shows how planets orbit around the sun, following Kepler's law. The plot is not a straight line, but follows a very distinctive curve.

Milky Way rotation curve

The above plot shows the rotation speeds of stars in the Milky Way galaxy as a function of radius. The curved dotted line shows what we would expect for Keplerian rotation, similar to the curve for planet rotation but somewhat more complicated than the planets' orbital speed plot, since the system is more complicated. The solar system has one large mass in the center that dominates the system. In the Milky Way, the center is spread out over a large area. Still, we would expect that the stars at the outer edge should be moving more slowly, if we understand how gravity works. There is clearly something else going on here. The rotation curve we see for the Milky Way galaxy is not Keplerian at all. The curve does not even decrease at the outer edge of the Milky Way.

galaxy rotation curves

The Milky Way is not the only galaxy that behaves this way. When we measure the rotation curves of other galaxies, we see that the matter doesn't behave as we would expect. The stars in the outer region are moving way too fast. This means that the stars should be flung off into space. The gravitational force from the inner part of the galaxy should not be strong enough to hold them.

dark matter halo

We think what is going on here is that we are not seeing the whole galaxy. In essence, the stars, dust and gas that we can see are just the tip of the iceberg. There is a huge halo of matter surrounding the galaxy, so that the stars that look to us as residing on the edge are really way in the middle of the expanse of matter. Since we can't detect this matter, we call it Dark Matter.


Now, we know there are forms of matter that are hard to detect. They fall in two categories: machos and wimps.


MACHOs (MAssive Compact Halo Objects) are objects like black holes and dwarf stars. They have quite a bit of mass but do not shine at all (black holes) or very dimly (dwarf stars). The problem with machos being the main source of dark matter is that we have a good idea of how many there should be, judging from what we see in our neighborhood, and it doesn't seem possible that there are enough to make a massive enough halo.


WIMPs (Weakly Interacting Massive Particles) are small particles that may arise in accordance with the theory of supersymmetry. The idea here is that every particle has a supersymmetric counterpart. For protons, there are sprotons; Electrons have selectrons, etc. The math behind this theory is elegant and makes sense. Physicists are busy working on possible ways to detect these particles, if they do exist. The most promising candidate is the neutralino.


If you are interested in learning more about this, here is a good source of information.

interactive graphic showing galaxy rotation curve

Dark Matter interactive

The above PHet interactive allows you to view and make plots of the rotation speed at different locations in a spiral galaxy modeled after the Milky Way. As you can see, in this interactive, you can see the effects of varying the amount of dark matter present.

Gravity lensing

Gravity lensing is a direct effect of general relativity. The theory of general relativity was discovered by Einstein in 1915. It is important to note that we are using the word "theory" here to denote an in-depth collection of interconnected ideas that have been scientifically proven to be fact. Sometimes "theory" is used to denote something we think might be true, that has not been proven. That is not the case for general relativity. It has been experimentally shown to be true in many tests. For more information, please see "More Precisely 22-1" in your text.


In short, the idea behind general relativity is that the density of matter/energy warps spacetime and the curvature of spacetime affects the motion of matter/energy. This is a significant departure from Newton's laws. For example, the force due to gravity is written as a function of the masses of two objects. By this formula, light should not be affected by gravity, since photons have no mass. According to general relativity, light is affected by the curvature of spacetime created my a massive object. The path that light takes is curved around a massive object.


What is spacetime? We know that there are three dimensions to space, you can describe them as x, y and z directions. There is also one dimension of time. Spacetime combines these four dimensions in a very special way, such that the speed of light is the same for any observer, no matter how fast they are going.


For more information on special and general relativity, please read the second half of chapter 22 in your text.

interactive graphic for gravity lensing

Gravity lensing interactive


Light grazing a star curves because of the star's presence. In this very exaggerated diagram, some of the light that would have gone upward actually curves around the star and ends up at your eye. Since we instinctively assume that the light traveled in a straight line, to us, it looks like the star has moved. Please check out this interactive to get a feel for how this works.

light rays refracting through lenses

In effect, the dense matter of the star curving the light acts just like a lens. When light goes through a glass lens, the glass bends the light. Light interacts with the atoms inside the glass as it passes through, effectively slowing it down. Since light follows the path of least time, this causes the ray to bend. For more information, please check out the refraction of light in The Physics Classroom.

We can see the effects of gravity lensing in this image of a galaxy cluster from the Hubble telescope. The smeared lines are actually images of galaxies that are very much farther away, being lensed through the galaxy cluster, as if it was a magnifying glass. The gravity lens allows us to see galaxies much farther away than we could normally view.


The smearing effect takes place because of the "imperfect" shape of the galaxy cluster. We can use the shape of the smeared light coming through the lens to figure out what the density distribution of the gravity lens must be. Analysis of the gravity lens reveals that there must be much more matter present than just the normal matter. This gives us a good idea of how much dark matter there must be, and how it must be distributed.


We can also use gravity lensing to locate smaller objects that are very dense, like neutron stars and black holes which may be hard to see directly.

The above photograph of the Milky Way is somewhat curved via a fisheye lens, to capture it in a single image. The overlaid constellations show that the center of our galaxy is located in the direction of the constellation Sagittarius. We have determined that there is a supermassive black hole lying in the center of our galaxy, and because of the coincidence with the Sagittarius constellation, we have named the supermassive black hole Sag A* (pronounced saj A star).


We believe that similar supermassive black holes lie at the centers of other galaxies as well, and that in general, the mass of this black hole is related to the mass of the parent galaxy. We have used the orbital speeds of nearby stars to estimate the mass of Sag A* to be about four million times the mass of our sun.

The above video illustrates a 16 year study of the center of our galaxy done by the European Southern Observatory commonly called ESO. It begins by peering deeper and deeper into the Milky way, with images taken by their massive telescopes. The video documents the motions of stars in the galactic nucleus. The orbits of these stars indicate what the mass of the central supermassive black hole must be. We have not directly imaged Sag A*, but by deduction, a supermassive black hole is the only object that could contain four million solar masses in such a small volume as is indicated by the surrounding star orbits.

© Kathryn Hadley PhD 2020

earlhy computers
Cosmic distance ladder
wheel-like rotation curve
planet-like rotation curve
Milky Way rotation curve
galaxy rotation curves
dark matter halo
interactive graphic showing galaxy rotation curve
interactive graphic for gravity lensing
light rays refracting through lenses