Skip to main content

This Week's Astro Nutshell: It's full of stars!

Each week I work with first-year grad students Marta and Becky on "order of magnitude" problems at the blackboard. I put that in quotes because we tend to do many more scaling arguments than true OoM. The idea is for them to draw on what they've picked up in class and apply it to common problems that arrise in astronomy.

Several weeks ago we asked

Suppose you have a magnitude-limited survey such that all stars have magnitudes $m < m_{\rm max}$. What will be the most common type (mass) of star in your survey?


This question is pretty much the same as "What types of stars visible in the night sky are most numerous?" This type of problem was first addressed by Swedish astronomer Gunnar Malmquist back in the 20's, which led to what we now refer to as the Malmquist Bias.

Initially, one might thing: well red dwarfs are the most common stars in the Galaxy, so M dwarfs will be the most common in our survey (or sky). However, M dwarfs are very faint (low luminosities). If the Sun is a 1000 Watt lightbulb then a typical M dwarf would be a Christmas tree light (thanks to Dave Charbonneau for the analogy). Since we're magnitude-limited (brightness-limited), we might not see many M dwarfs.

If we denote the number of stars in our survey as a function of mass $N(M)$, then
$N(M) \sim $(density of stars) $\times$ Volume
Where the "Volume" is characterized by the distance $d_{\rm max}$ out to which you can see a star of a given mass ($V_{\rm max} = d_{\rm max}^3$). Let's denote the density of stars by $\phi$, which is the number of stars per unit volume. This results in
$N(M) \sim \phi \times d_{\rm max}^3$   (1)
The density of stars in a given volume is given by the present-day mass function. Note that this is different from the initial mass function (IMF) because the stars in our survey will not be newly born, but  will instead represent a well-mixed sample of stars of all ages. Since massive stars die young, there will be even fewer massive stars than predicted by the IMF. The PDMF has the form $\phi \sim M^\alpha$, where $\alpha = -1.35$ for stars less massive than the Sun (the standard Salpeter IMF), and $\alpha = -5.2$ for stars more massive than the Sun. Plugging into Eqn 1 gives:
$N(M) \sim M^\alpha \times d_{\rm max}^3$   (1)
As for $d_{\rm max}$, we can use the handy equation that we derived a couple weeks ago (I'll blog about it later), which gives the scaling of the flux received from a star at the peak of its spectral energy distribution. The peak shifts to longer wavelengths for cooler stars, and shorter wavelengths for hotter stars. This all encompassed by the simple scaling relationship
$F \sim T^2 R^2 d^{-2}$ (1)
As the temperature $T$ increases, the flux increases. The same as when the star's radius $R$ increases. Move the star further away, the flux drops. Since our survey is sensitive only up to a limiting magnitude, we can only observe stars with $F < F_{\rm min}$. This means
$d_{\rm max} \sim T R F_{\rm min}^{-1/2}$    (2)
 From stellar structure, we recall that $R \sim M$, and $T \sim M^{1/2}$. Subbing into Eqn 2, we get
$d_{\rm max} \sim M^{1/2} M F_{\rm min}^{-1/2}$ 
And since our flux (magnitude) limit is fixed, there is a maximum distance out to which we can see a star of a given mass, given by
$d_{\rm max} \sim M^{3/2}$
We can now evaluate Equation 1 in terms of stellar mass, $M$:
$N(M) \sim M^\alpha \times M^{9/2}$   

For the different mass regimes, the present-day mass function has different values of $\alpha$:
$N(M) \sim M^{3.2}$      for  $M < 1~M_{\rm sun}$
$N(M) \sim M^{-0.7}$  for  $M > 1~M_{\rm sun}$
Given that $N(M)$ has different slopes on either side of 1 $M_{\rm sun}$, then it's clear that stars like our Sun will dominate your stellar sample. Even though M dwarfs are 75% of the stars in the Galaxy, you won't see many of them. This is why so many of the planets found in wide-field transit surveys such as HAT and WASP show up around Sun-like stars. The visible sky is full of G2-F8 stars!

At higher stellar masses, there aren't many stars formed, and those that do form die young because stellar lifetimes scale as $M^{-3}$ or so. But the effect isn't as severe as on the low-mass side. It's a gentle fall-off toward A and B dwarfs.




Comments

Popular posts from this blog

The Long Con

Hiding in Plain Sight

ESPN has a series of sports documentaries called 30 For 30. One of my favorites is called Broke which is about how professional athletes often make tens of millions of dollars in their careers yet retire with nothing. One of the major "leaks" turns out to be con artists, who lure athletes into elaborate real estate schemes or business ventures. This naturally raises the question: In a tightly-knit social structure that is a sports team, how can con artists operate so effectively and extensively? The answer is quite simple: very few people taken in by con artists ever tell anyone what happened. Thus, con artists can operate out in the open with little fear of consequences because they are shielded by the collective silence of their victims.
I can empathize with this. I've lost money in two different con schemes. One was when I was in college, and I received a phone call that I had won an all-expenses-paid trip to the Bahamas. All I needed to do was p…

An annual note to all the (NSF) haters

It's that time of year again: students have recently been notified about whether they received the prestigious NSF Graduate Student Research Fellowship. Known in the STEM community as "The NSF," the fellowship provides a student with three years of graduate school tuition and stipend, with the latter typically 5-10% above the standard institutional support for first- and second-year students. It's a sweet deal, and a real accellerant for young students to get their research career humming along smoothly because they don't need to restrict themselves to only advisors who have funding: the students fund themselves!
This is also the time of year that many a white dude executes what I call the "academic soccer flop." It looks kinda like this:


It typically sounds like this: "Congrats! Of course it's easier for you to win the NSF because you're, you know, the right demographic." Or worse: "She only won because she's Hispanic."…

Culture: Made Fresh Daily

There are two inspirations for this essay worth noting. The first is an impromptu talk I gave to the board of trustees at Thatcher School while I was visiting in October as an Anacapa Fellow. Spending time on this remarkable campus interacting with the students, faculty and staff helped solidify my notions about how culture can be intentionally created. The second source is Beam Times and Lifetimes by Sharon Tarweek, an in-depth exploration of the culture of particle physics told by an anthropologist embedded at SLAC for two decades. It's a fascinating look at the strange practices and norms that scientists take for granted.
One of the stories that scientists tell themselves, whether implicitly or explicitly, is that science exists outside of and independent of society. A corollary of this notion is that if a scientific subfield has a culture, e.g. the culture of astronomy vs. the culture of chemistry, that culture is essential rather than constructed. That is to say, scientific c…