# Color-Magnitude Diagrams (in general)

The original version of this text copied from this site

There is a relationship between the luminosity & surface temperature based upon

• the initial mass of a star
• its age
• its composition (usually a small effect)

Note that the rest of this page assumes you know what photometry and magnitudes are.

The initial studies of the relation between the absolute magnitudes (equivalent to luminosities) and spectral types (broadly equivalent to temperature) were performed (independently) by Ejnar Hertzsprung (Danish, 1905) and Henry Norris Russell (USA, 1913). Thus the plots of luminosity vs. temperature (or absolute magnitudes vs. spectral types) are often referred to as Hertzsprung-Russell (or HR) diagrams.

Stars of different initial masses follow different evolutionary paths within the luminosity-temperature plane (ex: see this). However, stars spend most of their hydrogen-burning lives within a relatively narrow band within the luminosity-temperature plane known as the Main Sequence. You can use this fact to get relative ages for clusters based on the shape of the main sequence ("main sequence fitting").

However, instead of plotting stars directly on the actual luminosity-temperature plane, an equivalent plot, known as a color-magnitude (CM) diagram is often used instead (such as the figure on the right). The reasons are as follows:

To first order, the light emitted by a star is a black body. Thus, rather than actually measure & plot (in Kelvin) the temperature of every star, it is MUCH easier and quicker to simply measure & plot the ratio of the intensity of the star in two spectral bands. This ratio is then directly related to the black body function and hence temperature. For primarily historical reasons, the ratio is usually expressed as the difference (in magnitudes) between two standard (optical/IR) spectral bands and is known as the color. Traditionally, the most commonly used color is the difference between the B and V bands (centered at 440 & 550 nm, respectively) and usually written as simply B - V.

Similarly, rather than actually measure & plot (in W/m^2) the total flux of every star, it is MUCH easier to simply measure & plot the flux in a standard (optical/IR) spectral band. Again since the emitted spectra are black bodies, this is directly related to the total flux. Traditionally, the most commonly used band is the V band (550 nm) and usually written as simply V.

Hence, typically CM diagrams are used rather than HR diagrams. For example

• if the distances to all the stars have been determined, then this might be a plot of B - V versus absolute V band magnitude.
• if the distances to all the stars have NOT been determined (say for a star cluster), then this might be a plot of B - V versus apparent V band magnitude.

These are both equivalent to plots of luminosity vs. temperature.

Near- and mid-infrared measurements (such as those taken with Spitzer) can be combined with optical data or used on their own to make CMDs.

# Color-Color Diagrams (in general)

In order to compare CM diagrams of two different clusters at two different distances, you need to know the distance to each cluster in order to calculate the absolute magnitude from the apparent magnitude. But what if you don't know the distances to the clusters? Wouldn't it be nice to still be able to learn something about the clusters, even if you don't know the distances? (Because distances are hard to get!)

The magnitude of a star is related to the log of the flux. Therefore, a color (or the difference of two magnitudes) is related to the ratio of the fluxes. When you take the ratio of the fluxes of the same star, the distance cancels out. (Go get the math from the Photometry page and work that out if you don't believe me!) The point is that colors are independent of distances! So a color-color plot is also independent of distance.

By studying main-sequence clusters, we can determine the locations of "normal" stars (or other objects) in nearly any color-color space. Then, stars (or other objects) that have colors different than these normal objects stand out. For example, in the plot on the left, the relationship between these two colors (U-V and V-I) for normal stars is indicated by the line marked "ZAMS Relation." Normal stars are clumped along this line. Stars significantly above this line are brighter than expected in U-V given their observed color in V-I. This plot was used in Rebull et al. (2000) to argue that those stars, those that are brighter than expected in U-V, are likely cluster members.

Near- and mid-infrared measurements (such as those taken with Spitzer) can be combined with optical data or used on their own to make CC diagrams.

Color-color plots can be used to separate objects of different types, such as distinguishing galaxies from stars. The plot on the right comes from Fig.2 from Fan et al. 2001, AJ, 122, 2833, and was used to show how the quasars (lower right) are different than brown dwarfs (top) and more boring objects (cluster of points).

# Color-Color plots and Young Stars

A color-color diagram in general is a useful tool in (a) finding the young stars, and (b) making a guess at the age of young stars. Using Spitzer colors, we can do this more easily than in many other color-color spaces.

Here is an example of a color-color plot using IRAC colors. By making models of stars, astronomers have determined that a star's position on this graph is related to its age since the starâ€™s position on the graph is related to how warm or cool it is. Stars start out as cool (or red) and become warmer (bluer), therefore they shift positions on the diagram as they age.

In the diagram, "photospheres" means plain stars, e.g., stars without circumstellar dust. (They could be older brethren of the stars elsewhere in the diagram, or they could be unrelated foreground or background objects.) There is a sample set of error bars in the lower right, so that you can see how much measurement error might move the points around. The red arrows show an indication of the direction the points could move when subject to interstellar reddening - e.g., from clouds of gas and dust often found in star-forming regions (larger scale -- and at a different temperature and density -- than the circumstellar gas and dust in the disk).

--Class 0: these are very young stars, still buried deep inside their cocoon of gas. These do not appear on the diagram at left, because they are so buried, they do not appear even at 3.6 or 4.5 microns.

--Class I: these are protostars surrounded by an infalling envelope of gas.

--Class II: these protostars, now in the T-Tauri stage, have developed protoplanetary disks.

--Class III: these are nearly fully developed stars, with just a remnant disk (and possibly planets) surrounding them.

# Questions to think about and things to try

Use real data from one of the Spitzer star-forming regions and see if you can construct other color-color diagrams that cause some objects to be in different clumps than most of the rest of the objects. If you do a literature search, you might find real-life examples of Spitzer color-color diagrams being used to separate young stars from other stars. (Hint: try some of the papers mentioned in the IR section of Finding cluster members, but you should feel encouraged to branch out and try other papers, since there are LOTS of color-color combinations, and they all tell you different things.)

# More advanced work on CMDs/CCDs

This YouTube video has a bunch of blackbody curves for a variety of temperatures corresponding to the effective temperatures (Teff) of a variety of spectral types. It has vertical lines indicating the position of various bandpasses, and the idea is that you can see how the shape of the underlying approximate stellar photosphere changes as the Teff changes, and you can then infer how the ratio between measured fluxes in the various bandpasses might change. It may take staring at that for a bit and thinking about it. Look at the peak of the curves, and look at how quickly it changes through the optical bands as a function of temperature. This is why a color such as B-V can be uniquely mapped to a temperature. Look at where the mid-IR Spitzer bands are. This is why the colors of most unadorned (dust-free) main sequence stars are zero color - the slope of that blackbody curve doesn't change through those bands for just about any temperature of star. (In reality, large molecular absorption bands start to 'steal' flux from some of these bands at mid M and later, but that is part of why a blackbody isn't a good approximation to a real star for those cases.)

# Another whack at an explanation

This comes from an email conversation; I am copying it here because I liked the logic, and want to capture it, but I don't have time at just this instant to integrate it or find it a better home.

Think of a red star. What we mean by "red" is that the star is brighter in red than other colors of light. (Usually we restrict this to visible colors, but you now can understand other wavelengths of light.) It is still red if it close or far away. The brightness of red light divided by the brightness of blue light is something that won't change with distance. If the star is farther away it is fainter in blue, and fainter in red. The ratio of intensity doesn't change with distance.

What does change the ratio of the brightness in red to the brightness in blue is the temperature. If you remember the black body curves, as things get hotter they get bluer. These colors give us a measure of temperature. There are also some spectral line things that factor into this. The spectral line effects don't change with distance either. Between all of the effects we can isolate areas of color that indicate different types of stars.

Now we take the difference in magnitude. I have been talking about the ratio of brightness. Magnitudes are a logarithmic measure of brightness. This means that when we take the difference of magnitudes (the difference of logs) we are calculating the log of the ratio. (Log{a}-Log{b}=Log{a/b}) So we are basically taking the log of the ratio. The important part here is that we are calculating the ratio.

A measurement in a single band is a brightness. So when you say it's so many magnitudes at J band, that's a brightness (which can be converted to flux density units). (Technically, the definition of magnitude has in it an implicit comparison to Vega - a magnitude is an implicit flux ratio with Vega.)

It is often useful to take the same object and compare its brightness at two different bands. It's easiest to make such a comparison as a ratio of flux densities, e.g., a difference of magnitudes. Conventionally, you take the bluer band minus the redder band, e.g., B-V, J-K. That way, larger numbers are always redder, and smaller numbers are always bluer. Depending on which bands you pick, exactly, it tells you different things. B-V, for example, can be mapped to temperature. K-[8] is a good way to look for IR excesses, e.g., disks.

Note too that when you compare the brightness of the same object at two different bands, the distance cancels out. Consider a pair of identical objects. The one that is further away will be fainter. But they will both have the same color. Interpreting color-magntiude diagrams requires also worrying about the distances to all the objects in the plot. Color-color diagrams don't involve distances -- distances cancel out.

Remember this video: http://www.youtube.com/watch?v=mjfSy2kHWqQ&list=UUQfN0BTwaSZ6ABsQcieCpdA&index=8&feature=plcp this is a series of blackbodies (stand-ins for stars) with the locations of the bands marked. B-V can be mapped to temperature because those bands are near the peak of the blackbody energy distribution, so B-V changes rapidly as temperature changes. K should mostly still see photosphere in many cases, though it too can be contaminated by circumstellar emission. Longer wavelengths will sample more of the disk.

So - why do we make a plot like this? To compare the ratio of brightnesses at two different broadband filters for a set of many objects at once. How do we pick the bands to use? By thinking about stuff like the movie, and making models of the sorts of sources you expect to care about. How do you know what to expect? By making models, but ultimately by observing things you understand before applying it to things you don't understand. For example, this catalog: http://coolwiki.ipac.caltech.edu/index.php/Taurus_catalog consists of a pile of colors of known young objects. The catalog I sent of "standard main sequence colors" is another sort of catalog, establishing colors for a bunch of stars on the main sequence. You can use catalogs like this to establish where you expect young stars (or main sequence stars) to fall in any given combination, and then, with that knowledge, interpret the diagram of a new field.

In the infrared, it's easier to remember off the top of your head "where the main sequence should be", because all magnitudes are defined to be 0 when they are the same brightness as Vega (or, rather, what Vega would be if it didn't have a disk). Which means that if any *difference* between magnitudes is 0, the SED is the same shape as Vega (or, rather, what Vega would be if it didn't have a disk). So a star with K-x=0 where x is any IR band has no IR excess. K-x > 0 is redder than Vega, e.g., possibly a disk. K-x < 0 means either photometry problems, or it's not a star.