Example 2: NGC 3766 analysis, part b

In this next example, you are going to learn the steps to actually fit the isochrone model to the NGC 3766 cluster HR diagram you began creating in Example 1. This model, known as an isochrone model, plots the positions of stars that are all at a given distance, whose light is scattered by the same amount of interstellar dust, that are all the same age and have the same chemical composition, but which all formed with different masses. Therefore, isochrone models essentially show the positions of stars of different masses that are all part of the same cluster, so by adjusting parameters to fit the isochrone model to a cluster’s color-magnitude data, we can actually estimate the common physical properties of all stars in the cluster—i.e. their distance, age, chemical composition, and the amount of light from them that’s been obscured by dust.

In essence, we can learn a lot of information about a star cluster by plotting its stars’ positions on a color-magnitude diagram and fitting an isochrone model, all because we understand the physical causes of brightness and temperature differences among stars of different masses. And because science is super cool, and astronomy in particular is the coolest!

If you have not yet worked through Example 1, please go back and follow the steps to create your graph, as these instructions will pick up from where we left off.

1. To proceed, start by skipping past Archive Fetching and select “Isochrone Matching” from the stepper menu in Clustermancer.
2. At the bottom-left, in the filter selection area, select the Blue filter to be B, the Red filter to be V, and the Lum filter to be V, then click Add, and at the bottom ensure “CM” is selected in the CM/HR toggle. You should now see a colour-magnitude diagram (specifically, V magnitude plotted vs B-V colour index), with an isochrone curve on the graph that by default does not pass through the cluster stars.
3. Before attempting to fit the isochrone model to the data, there are a few properties of color-magnitude diagrams and HR diagrams, a few model parameters that should be explained, and a few general terms that we should define, to ensure you understand how to properly fit these models:
   • The two axes of the diagram are plotted in magnitude units, and you can toggle between displaying the data in a color-magnitude diagram or in an HR diagram.
     • The apparent magnitude measured from the flux of starlight in each image is calculated as
       (2.1)  $m_x=-2.5\log_{10}\frac{F_x}{F_{x,0}},$
       where $x$ is the photometric filter (in this case, the filters used were $B$ and $V$), $F_x$ is the flux of light measured in the $x$-filter (essentially, the number of photons observed from a particular star), and $F_{x,0}$ is the image zero-point, a reference flux for the filter determined by calibrating to a reference catalogue to determine the dimmest flux measurable in your image.
     • Since the stars in a cluster are all at the same distance, it is possible to calculate the stars’ absolute magnitudes (analogous to luminosity) using the distance-modulus equation,
        (2.2). $M_x=m_x-5\log_{10}\left(\frac{d}{1~\mathrm{kpc}}\right)-10-A_x,$
       where $A_x$ is the galactic dust extinction in the $x$ band.
     • If the amount of galactic dust extinction $A_x$ is known, it can be used to calculate the stars’ intrinsic colours $(B-V)_0$ from their apparent colours $(B-V)$ which have been reddened by interstellar dust.
     • Thus, Clustermancer can take calibrated apparent magnitudes $m_x$ from your afterglow photometry file and use the distance $d$ and galactic dust extinction $A_x = 3.1\cdot E(B-V)$ parameters you set with the sliders on the left-hand panel, and calculate each star’s absolute magnitude $M_x$ and intrinsic colour $(B-V)_0$.
     • Whereas a plot of magnitude vs apparent color is called a color-magnitude diagram, a plot of absolute magnitude/luminosity vs intrinsic color/surface temperature is an HR diagram. You can toggle your graph between the color-magnitude diagram and the HR diagram using the CM/HR toggle at the bottom-left.
   • The horizontal axis plots colour as photometric colour index, which is defined as the difference in magnitude values measured for the same star using different filters; e.g. you should currently have plotted the $B-V$ colour index. Since the magnitude scale is inverted (so lower magnitude values correspond to brighter fluxes; see the sign in Equation (2.1)), this means $B-V$ will be negative if a star is brighter when observed through the $B$ filter (which transmits blue light) than when observed through the $V$ filter (which transmits green light). Conversely, the star will have a positive $B-V$ colour index if the star is brighter in $V$ than it is in $B$.
   • Since stars emit thermal radiation with nearly blackbody spectra, color is analogous to temperature; e.g. with blue stars being the hottest ones, which emit more light at the blue end of the spectrum than the red.
   • Most stars in NGC 3766 fall along the main sequence, the line connecting the hottest, most luminous (bright blue) stars at the top-left to the coolest, least luminous (dim red) stars at the bottom-right. These stars are all fusing hydrogen in their cores, and they mainly differ from one another by mass, with the most massive stars at the top-left and least massive at the bottom-right.
   • The main sequence is not a thin line, as you might expect since all the stars are the same age, they have the same chemical abundance and they are located at the same distance. In fact, the differences in these quantities are so small that they are actually negligible. You might then think that the scatter is due to data quality, but the magnitude measurements in your data set, particularly of the brightest stars, are far more accurate than the amount of scatter you see. The source of scatter in your main sequence actually primarily comes from unresolved binary star systems.
   • Typically, roughly half the stars in a cluster are members of binary star systems, and since the clusters are so far away, we don’t resolve the individual stars but see their combined light. For instance, if two stars have the same color and luminosity, they appear in the HR diagram with the same color but with brighter magnitude. If one star is lower mass than the other, the combined light of the two will be redder and brighter than that of the more massive star. As a result, the points in your HR diagram representing unresolved binary star systems create roughly a magnitude of scatter above the curve of isolated main sequence stars. Therefore, when we fit isochrone models to cluster data we must fit the curve to the densest part in the bottom of the main sequence.
   • The top-left stars are the most massive main sequence stars in the cluster. Massive stars consume their core hydrogen fuel more rapidly than less massive stars, and the stars at the very top of the main sequence have nearly exhausted their fuel supply. When they do, their luminosity increases while their surface temperature cools and they become red giants, moving to the upper-right of the HR Diagram.
     • In Clustermancer, select the HR toggle, then Standard View (top-right). Then, move the log(Age (yrs)) slider to larger values while holding the other parameters constant. You should see that the curve describing the main sequence remains roughly constant, while the point where stars leave the main sequence and migrate to the upper-right makes its way down the main sequence with increasing age. Therefore, this turn-off point at the top-left of the main sequence is what we use to determine a cluster’s age.
       Note: More massive stars have larger cores with more fuel to burn than lower mass stars, so it’s somewhat counter-intuitive that they have shorter main sequence lifetimes. But the reason is actually fairly straightforward: a star’s main sequence lifetime is proportional to its mass $M$, but it’s inversely proportional to luminosity $L$ (since energy generation is proportional to the rate of fuel consumption); and the for main sequence stars, luminosities increase exponentially with mass—i.e. $L\sim M^{3.5}$; therefore, a main sequence star’s lifetime goes like $t\sim M/L\sim M/M^{3.5}=M^{-2.5}$. Stars therefore run out of fuel for core thermonuclear fusion reactions more quickly the more massive they are.
   • Astronomers are funny. We call all elements heavier than hydrogen and helium “metals”, and therefore denote our measure of the abundance of all elements heavier than hydrogen and helium as a star’s metallicity.
     • In Clustermancer, select the HR toggle then move the metallicity slider to larger values while holding the other parameters constant. You should see that by increasing the metallicity, the isochrone model’s main sequence burns brighter and cooler (redder), and the giant branch becomes more evolved as the more “metal rich” clusters burn with higher luminosities to compensate for the extra shielding the metals add to core fusion reactions.
   • When light travels from a cluster towards Earth, it must pass through interstellar dust. This galactic dust scatters some of the light so it never reaches us, and in fact preferentially scatters blue light, so clusters eventually appear both dimmer and redder when viewed from Earth. The parameter that measures interstellar reddening is denoted E(B-V) and is measured in magnitudes. Interstellar reddening is typically proportional to the amount of interstellar extinction $A_x$ that occurs in a given band $x$. For example, in the Milky Way, we find on average that $A_V=3.1E(B-V)$, whereas the proportionality constant $3.1$ is higher when relating reddening to extinction in bluer wavelength bands and lower for redder bands. Therefore, we require only one parameter, i.e. E(B-V), to describe both interstellar extinction and reddening.
     • In Clustermancer, select the HR toggle then move the E(B-V) slider to larger values while holding the other parameters constant. As you increase E(B-V) in the plotting tool, notice that the data points on your graph become both brighter (they move up) and bluer (they move left). This is the opposite of what you might expect — after all, more dust should make stars appear dimmer and redder, not brighter and bluer. The reason is that the HR diagram displays corrected values: Clustermancer uses your E(B-V) parameter to calculate how much dimmer and redder the dust has made each star, and then removes that effect from the plotted values. So increasing E(B-V) applies a larger correction, shifting the data points toward their intrinsic, dust-free brightnesses and colours. The actual starlight reaching Earth hasn’t changed — only what the graph is showing you about the stars themselves.
   • All of these physical parameters—the distance, age, metallicity, reddening, and extinction—affect the form/location of the isochrone model within the color-magnitude diagram/HR diagram. And unfortunately they are all somewhat degenerate with one another—a change in one parameter can mimic a possible change in another. By observing the cluster in more than one filter, we can eliminate some of this degeneracy by plotting multiple data sets and fitting the isochrone simultaneously to them all.
     • In Clustermancer, create a second plot of -band magnitude against  color index by setting “Blue” as $R$, setting “Red” as $I$, and setting “Luminosity” as $I$, then clicking Add.
     • Note that Clustermancer supports creating up to four color-magnitude diagrams/HR diagrams at once, and the placement can be adjusted by moving the corresponding boxes up or down at the bottom-left, or graphs can be removed by clicking the X.
   • In the demonstration video below, we quickly review all the key points in this list before moving to the next step where you will see how to determine the physical parameters of NGC 3766 by matching the isochrone model to your stars in Clustermancer. Then, we will return to Afterglow Access to finalize our tri-color image using tools that set color balances based on the calibrated magnitudes of stars in each filter and even remove the effect of reddening to create a natural color image that shows us what the cluster would look like from Earth if there weren’t any dust along our line of sight.
4. Match the isochrone model to your photometry data in Clustermancer:
   • Note that below the distance slider you should see the cluster distance that is indicated by Field Star Removal (FSR). If you step back to the Field Star Removal page, you should see that this distance corresponds with the peak in the distance distribution. Move the slider to this value.
   • At this point, if you have HR diagrams displayed it is recommended to select the Frame on Data option.
   • Increase both age and metallicity to find a shape that is close to the shape of the data set you see. Metallicity should be somewhere approximately close to 0 for open clusters. Then adjust E(B-V) to bring the cluster data closer to your isochrone (HR diagram) or to bring the isochrone closer to the cluster data (color-magnitude diagram). Then modify all three to create a good match, bringing the isochrone up to the bottom of the main sequence by adjusting metallicity, fitting the turnoff by adjusting age, and getting the properly shaped isochrone main sequence to coincide with the data by adjusting the reddening value.
   • When you’ve found a good fit, save your V vs B-V color-magnitude diagram as a PNG by clicking the three horizontal lines at the top-right of the graph..
5. Use the following tri-color image processing sequence in Afterglow Access:
   • (from Example 1, open NGC 3766 BVRI images, group BVR, set colour maps for the three layers)
   • select the top layer of your grouped image “NGC3766.fits”
   • select Afterglow’s Display tool
   • click Color Composite Tools > Link All Layers (Pixel Value)
   • click Color Composite Tools > Photometric Calibration, then in the Photometric Calibration window click “Measure zero points with field calibration”. When zero point calibration completes, click Calibrate Colors.
   • Set Stretch Mode to Midtone, and click the Default Preset. Raise the Background Level Percentile to a value lower than the peak to darken the background. 15 is a good number for these sample images.
   • Click Export Image as JPG to save this image. This is the photometric color calibrated image of NGC 3766, which displays realistic colors for the stars in this cluster as they appear from Earth.
   • Now click Color Composite Tools > Photometric Calibration one more time, and then set the Extinction E(B-V) value to the value you determined in Clustermancer. Click Calibrate Colours, and you will see your graph update colours in the photometric colour calibrated image, removing the effect of reddening so the image now displays the cluster as it would appear from Earth if all the interstellar dust along our line of sight were removed.
   • Click Export Image as JPG to save a copy of this image.