How to Estimate the Age (and Distance) of an Open Cluster with Amateur Equipment

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by Ken Hose

Star clusters are objects of great interest to astronomers. The stars in a given open or globular cluster are considered to have been created at the same time from material of the same composition. The stars in a cluster are about the same distance from Earth. This means that any differences in brightness are not due to differences in distance but are due to intrinsic differences in luminosity. These characteristics make it possible to get accurate estimates of age and distance. I set out to see if I could get good estimates for M67 with amateur equipment. M67 is an open cluster in the constellation Cancer. Surprisingly, I was able to get pretty close to accepted values.

In the early 1900s astronomers found that by plotting stars by brightness and temperature on a graph, certain groupings became obvious. One obvious feature was a tight grouping of stars along a diagonal representing hot bright stars on one end and cool dim stars on the other. There was a smattering of stars in the upper right quadrant, too. Stars along the diagonal are called Main-Sequence stars. When stars are “born” they start out on the main sequence and eventually evolve away towards the upper right part of the chart. Such a chart gives insight into stellar evolution and is called a Hertzsprung-Russell (HR) diagram.

Color Index is a Proxy for Temperature

Last year I took images of M67 through 2 different photometric filters with the goal of making a type of HR diagram to study the distance and evolution of the cluster. One filter was sensitive to blue light, the B filter, and one filter was sensitive to green light, the V filter. I measured the brightness of about 300 stars in the cluster and plotted them. It turns out that subtracting the V measurement from the B measurement will correlate to surface temperature of the star and is often used as a proxy for temperature. It is called the color index. So these measurements gave me a way to make a type of HR diagram called a color-magnitude (CM) diagram.

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The graphs show the CM diagram for M67. The color index is on the x-axis. Note that smaller numbers mean hotter stars. The vertical axis is the apparent magnitude with brighter stars at the top. So hot bright stars are in the upper left and cool dim stars are in the lower right. The red line on the left graph shows the position of main-sequence stars. Note that stars seem to be evolving off the main sequence around a B-V value of about 0.5. This color-index would correspond to an F-type star which would be about 10% hotter than our Sun. Remember that all the stars in M67 are the same age so the different positions on the graph represent different stages of evolution of stars of different initial mass. All the stars hotter than spectral type F (O, B, A) have already evolved off the main sequence. The faint horizontal line of stars in the middle is the sub-giant branch and the rising part on the right is the giant branch of evolution.

Estimating the Distance

The absolute magnitude of a star is the magnitude it would have at a distance of 10 pc from Earth. If we know the absolute magnitude of an ideal main-sequence star and if we can identify a corresponding main-sequence star in M67, we should be able to calculate the distance using the difference between the absolute magnitude and the apparent magnitude from my plot. To make this work, I found B and V values of ideal main-sequence stars (from a reference source) and plotted them with my data. All I needed to do is add a correction term to the V value and vary the term until the ideal curve matched the main-sequence stars in my plot. This is the red line in the graph. The correction term is the so-called distance modulus which can be used to calculate the distance. In this case the distance modulus was 9.6 so the distance is about 832 pc or about 2700 light years. Wikipedia gives a range of 800 to 900 pc for the distance. The Excel formula for distance is d = 10*10^(DM/5) in parsecs.

Using Isochrones to Estimate Age

Estimating the age of M67 relies on stellar evolution models. The idea is that if we can get an accurate evolution model, we should be able to make a curve that will match the CM diagram since it depicts the evolution of M67. This works because all the stars in the cluster are assumed to be the same age. So if we can generate a series of curves of different ages from the model, we simply find the one that most closely matches my plotted data. Such curves are called isochrones. I used a stellar evolution model based on MESA, Modules for Experiments in Stellar Evolution. A web-based tool, MIST, made it easy to specify the boundary conditions for a run and get a quick turnaround of a new model. Some Python code was required to extract the data. Fellow RCA member Sean Curry helped with Python. The best fit gives an age of about 4 billion years. Wikipedia gives an age range of 3.2 – 5 billion years.

The ideas presented here can be used to explore other open clusters and globular clusters, too. This work was done with a 12.5” scope and a CCD camera with photometric filters. This is fairly modest amateur equipment. This shows how amateur astronomers don’t need professional gear to do real science.