Skip to main content

This Week's Astro Nutshell: How Many Photons?

(Actually, this is from Astro Nutshell two weeks ago)

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.

This week we asked

How many photons per second per cm$^2$ (photon number flux) do you receive from a star as a function of its temperature $T$, radius $R$ and distance away $d$?

Having such an equation would be extremely handy for observation planning. When determining the feasibility of a new project, observers tend to start with a statement of the expected signal-to-noise ratio (SNR) for an observation of an astrophysical object. In the limit of a large expected number of photons, the signal is the number of photons $S = N_\gamma$, and the noise can be approximated as $N{\rm oise} = \sqrt{N_\gamma}$. So SNR$ = \sqrt{N_\gamma}$. So this week's question comes down to, "What is $N_\gamma$ for a star of a given temperature, radius and distance away?"


We start with Wein's Law, which states that the wavelength at which a star's (blackbody's) emission peaks is inversely proportional to the star's temperature
$\lambda_{\rm max} \sim \frac{1}{T}$    (1)
This is Astro 101. The flux level at this peak wavelength can be evaluated using the blackbody function (Planck function), which is given by
$F_\lambda(T) = \frac{2hc^2}{\lambda^5} \frac{1}{\exp{\frac{h c}{\lambda k_{\rm B} T} + 1}}$    (2) 
This gives the energy per unit time (power), per area, per wavelength per unit solid angle, as a function of temperature and wavelength. If we evaluate this at $\lambda_{\rm max}$, and approximate the total flux, which is an integral over all wavelengths, as a box of height $F_{\lambda_{\rm max}}$ and with a 100 nm width (standard observing bandpass). We also need to multiply by the solid angle subtended by the star of radius $R$ at a distance $d$, which is $R^2/d^2$. This leads to
$F_{\rm tot} \sim \frac{T^5 \Delta\lambda}{e^{\rm const} - 1} R^2 d^{-2}$   (3)
Since $\lambda_{\rm max} \sim 1/T$, then $|\Delta \lambda| \sim 1/T^2$ (Wow, check out that calculus slight-of-hand! However, the same scaling falls out of actually doing the integral over $d\lambda$). Finally, the energy per photon near $\lambda_{\rm max}$ is $E_{\lambda_{\rm max}} = h c / \lambda_{\rm max} \sim T$. Dividing Equation 3 by the energy per photon, and replacing $\Delta \lambda$ we get the flux of photons 
$F_\gamma \sim T^5 T^{-2} T^{-1} R^2 d^{-2}$
$F_\gamma \sim T^2 R^2 d^{-2}$ 
Increasing the temperature or radius of the star results in more flux, which should seem fairly intuitive: hotter, bigger stars emit more photons. Also, there's the familiar inverse-square law with distance. Evaluating for the Sun at 10 pc results in
$F_{\gamma,\odot} = [5\times10^{4} {\rm photons}] T^2 R^2 d^{-2}$
(please check my math on this!)

For an M dwarf with 1/5 the Sun's radius and roughly half the temperature, at 10 pc it would emit 100 times less light, but most of these photons will be down near 1 micron in the near infrared. An A-type star like Vega, with twice the Sun's radius and twice the temperature will emit 16 times as many photons, most of them in the ultraviolet.

I hope you find this scaling relationship as handy as I do!

Comments

Popular posts from this blog

back-talk begins

me: "owen, come here. it's time to get a new diaper" him, sprinting down the hall with no pants on: "forget about it!" he's quoting benny the rabbit, a short-lived sesame street character who happens to be in his favorite "count with me" video. i'm turning my head, trying not to let him see me laugh, because his use and tone with the phrase are so spot-on.

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 d

Reader Feedback: Whither Kanake in (white) Astronomy?

Watching the way that the debate about the TMT has come into our field has angered and saddened me so much. Outward blatant racism and then deflecting and defending. I don't want to post this because I am a chicken and fairly vulnerable given my status as a postdoc (Editor's note: How sad is it that our young astronomers feel afraid to speak out on this issue? This should make clear the power dynamics at play in this debate) .  But I thought the number crunching I did might be useful for those on the fence. I wanted to see how badly astronomy itself is failing Native Hawaiians. I'm not trying to get into all of the racist infrastructure that has created an underclass on Hawaii, but if we are going to argue about "well it wasn't astronomers who did it," we should be able to back that assertion with numbers. Having tried to do so, well I think the argument has no standing. At all.  Based on my research, it looks like there are about 1400 jobs in Hawaii r