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

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…

The Bright Line is not Monotonic

The anthology of myths commonly known as America rests upon the notion that history is linear. In the past people in this country ignorantly did bad things to other people. But thanks to the passage of time, we can now "let the past to be the past," because today we live in a time when things have gotten much better. Furthermore, any problem that our society faces in the present will inevitably be solved as "the old guard" dies off and a new generation of better people takes their place. 
Of course this story isn't told so simply or explicitly. But the assumption lurks beneath the other stories we, as Americans, tell ourselves and each other. The myth certainly undergirds the notion that racism is a thing of the past, and that today we inhabit a "post-racial" world in which all people, regardless of race have equal access to betterment, dignity and happiness. We are lulled into beliving that at some point in the mid to late 1960's, a wise reveren…