make your own color magnitude diagram ! – prepare to astronomize

willb/ January 13, 2020/ research log/ 0 comments

Luritja aboriginal astronomers have a classification system for by-eye observation of stars: Tjilkera (white) stars, Tataka Tjilkera (red/white), Tataka (red) and Tataka Indora (very red) [1, 2]. Astronomers today, thanks to Williamina Flemming and Annie Jump Cannon (among other Harvard Computers), use a rather opaque system of stellar classification called the Harvard spectral classification: OBAFGKM(LTY), ranging from white/blue to red/brown. As abstract as these classification methods seem, binning data can help us more readily construct patterns, theories, and narratives to help further our understanding of the stellar population.

At the turn of the century, western astronomers were concerned with the ultraviolet catastrophe (a dramatic divergence between observed results and analytical models of spectral energy distributions) and Max Plank, in attempting to address that concern, fell head-first into the world of quantum physics [3]. As the particles within an object move (interacting according to gravity, electrodynamic laws, quantum mandates, all in response to other particles) their interactions excite (increase the energy of) particles within the object. Like people, generalized objects don’t like to be excited for too long (it can get tiring) and the easiest way to shed energy is by emitting it as light (a photon). That light, emitted somewhere within the body, will take a while to escape entirely, bouncing between particles and being absorbed and re-emitted in different ways. This process occurs on a large enough scale that the tiny details of each photon’s color, energy, and source become unimportant. The object writ large emits all kinds of light all the time, the amount of any given color/energy of photon being dictated by how excited the object is [4].

Plank’s law is a formula that describes a family of curves that represent this; it describes the amount of a given wavelength (or frequency, or energy, or color – for light, these are all related and interchangeable properties) of light will be emitted by something with a given temperature. For some historical reasons [5] we call these Blackbody Functions. So Plank’s law is shaped differently for every temperature, but comes from the same equation.

This image shows Plank’s law for various temperatures, as well as the “ultraviolet catastrophe” encountered by previous models (which imply the existence of infinite energy, yikes!) Image credit: Darth Kule [Public domain]

Try thinking through what this graph means: the y-axis is measured in kW (a measure of power, or energy per second) per steradian per meters squared per nanometers, while the x-axis is measured in micrometers. You can imagine steradian meters squared as an area, so a quantity with units power per area is describing the amount of energy from some object that passes through a given area every second (we call this the Irradiance of an object). The meat of the graph is that it shows you how much of the irradiance of an object of a given temperature comes from any particular wavelength (color) of light. This implies that the temperature of an object determines the color of that object, because the temperature dictates what wavelength of light the object will emit the most energy at.

Stars are fairly simple things, in comparison to emitters that humans interact with in their day to day lives. They don’t have many (relatively) weird compounds, reflective substances, odd structures and surfaces. Let’s be clear, stars have really interesting things going on in all those regards, but they aren’t as complex as a whole strictly speaking as, say, a tree, or your mobile device. This is partly because they’re so big: stars are massive enough to be defined easily only by very fundamental processes like gravity, pressure, and radiation. So, unlike plenty of colors in our day to day life, the colors of stars are defined really well by Plank’s law with a few addenda. You could imagine Plank’s law as the basis, or progenitor, for the color in your day to day life, with lots and lots of addenda, like reflection, scattering, emissivity, etc (much more than is involved in our understanding of stars) that occurs between the initial production of light and color, and our eyes receiving it.

The Sun’s spectral energy distribution can be modeled fairly well by Plank’s law, with only a handful of addenda to explain the observable differences. Image credit: Sch [CC BY-SA]

Another fun property to explore is how bright a star is over all; that is, adding up all the energy you get from all the kinds of light the star emits, how much is there? Thinking observationally, how bright is the star? The brightness of a star is determined by its luminosity, which is dependent on the star’s mass (by the mass-luminosity relation).

If we flip this reasoning, then, observing the color and brightness of a given star will tell us something about its temperature, which can clue us into some of the fundamental processes that dictate the star’s existence. There are lots of fun astrophysics concepts that you can derive from these kinds of observations: how long a star will live and how it compares to other stars for example. In making these comparisons you can then attempt to explain how stars evolve over time, how they form, what their neighborhoods look like, what they do when they retire, and so on. I won’t cover much of that in this article, but hopefully I can spark your curiosity and invite you to dig even deeper – an incredibly accessable and comprehensive source is Searching for the Oldest Stars by Anna Frebel, particularly chapter 4.

In addition to a color classification system, aboriginal astronomers made detailed observations of the brightness of variable red-giant stars recorded in oral tradition [6]. Astronomers quantify the brightness of a star by measuring it’s magnitude (a measurement of the star’s brightness relative to a standard star). Hipparchus (c. 150 B.C.E) is suspected to have popularized ranking stellar magnitudes from 1 – 6; in the 19th century N. R. Pogson placed these on a logarithmic scale, where a magnitude 1 star is 100 times brighter than a magnitude 6 star. Generally, we fix the zeropoint for the apparent magnitudes of stars with the star Vega (the 5th brightest star in our night sky, and the once and future North Pole star).

We then apply various filters to their telescopes and measure the star’s magnitude in different regimes of light (taking a blue filtered image, a green filtered image, and a red filtered image, and then combining them, is how your digital camera creates color images). We quantify “color” by subtracting the magnitude in one filter from the magnitude in another; in this way, a numerical color represents how much more of one color of light a star emits than another. Remarkably similar to the Luritja color scale, which categorizes a star based on how much more red a star is than white, common measurements of color include B-V (blue magnitude minus visual magnitude), and V-R (visual magnitude minus red magnitude). Be careful when interpreting these numbers, because the logarithmic magnitude scale is also backward (yeesh): positive color implies a star is brighter in the subtrahend filter (the filter being subtracted) while negative numbers mean an object is brighter in the minuend filter (the filter that is the object of the subtraction).

There’s a-lot that goes into taking images in various filters, cleaning them, determining their magnitudes, and then plotting them; I had to take a whole class about it (I got an A- ). If you want to try making your own, you can feel free to use the images I collected and the reduction/analysis pipeline I wrote (it’s certainly not perfect, but then again no one’s code is). I put it all in a github repository my awesome lab-mate Karina and I share – the link is here.

A 2Mass infrared image of M39. Image credit: 2Mass Atlas

I took observations of Messier 39, a cluster of young stars in the constellation Cygnus, in Johnson B(lue) V(isual) R(ed) filters. I extracted photometry (measurements of brightness, in this case in units of magnitude) from each of the images, and plotted the color of M39 versus its magnitude.

My BVR image of Messier 39, taken on the Amherst College Observatory’s Beta telescope (Oct. 2019)
Color Magnitude Diagram of M39, constructed in python. Notice the main sequence turn off that’s very apparent in the upper-left of the figure. This feature occurs as more massive, bluer stars will begin burning helium and evolve off the main-sequence (the monotonic function running from top left to bottom right) where hydrogen burning stars spend their lives.

I also was taught how to fit an isochrone to my CMD. Lots of astrophysics go into understanding why and how you might do this, but the purpose is to be able to determine the age and distance to the cluster by fitting a model to the data. I determined the stars within M39 are roughly 300 Myr old, and at a distance of ~320 parsecs from the Earth from this model fit. Surprisingly, these values aren’t too far off the mark! Zhen-Yu Wu (2009) report an age for M39 of 278.6 Myr, so our age estimate has a percent error of 7.7%. Data from the new GAIA telescope, published in the paper Gaia Collaboration; Babusiaux, C. et al (2018) finds a parallax of 3.33 mas, which implies a distance of 299.643 ~ 300 pc, for a percent error of 6.7%. 

A 300 Myr isochrone fit to V vs. B-V CMD

Hopefully this article has piqued your interest in measuring stellar colors, and I’ve been able to convince you how much you can determine about stars and their stories from these kinds of measurements. If you have any questions, feel free to reach out to me. Until next time, happy astronomizing 🙂

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About willb

I'm Will, an undergraduate astronomer studying transition disks, direct imaging, and planet accretion and formation at the Follette Lab at Amherst College. I use they/them/theirs pronouns.

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