Skip to main content

Order of Magnitude Astrophysics


From the Ay20 blog, here's a solution to one of the week 1 worksheet problems.


Estimating The Luminosity of a Sun-like Star


by: John A. Johnson, Jackie Villadsen


Abstract


We present the solution to Worksheet problem #2, from week 1, estimating the power output of a Sun-like star. Each group should submit one to two of these per week. Decide amongst your group members who will be first author, second author, etc. Acknowledge people and resources used in your solution. Cite ancillary information. State your assumptions clearly. Write your solution such that a frosh could duplicate your steps and arrive at the same solution.


Introduction


The oldest astronomical instrument is the human eye. A marvel of evolution, the eye has both high sensitivity and a large dynamic range. A classic study of the eye's response to light conducted in 1942 showed that of order 10 photons need to impinge on the eye in order for the brain to register detection (Hecht, Schlaer & Pirenne 1942). In other words, the eye has a gain of 10 photons/DN. In this contribution we use this fact as a starting point for estimating the luminosity (power output) of a Sun-like star. As additional input for our calculation we note that a Sun-like star at 100 light years is just barely visible to the naked eye if the star is viewed from a dark site. (As a side note, this corresponds to a G2V star with an apparent magnitude of V=6).

Order of Magnitude (OoM) Calculation

We start with a rough estimate of the aperture area of the eye. Fully dilated, an eye has an entrance diameter of roughly Reye = 0.5 cm, corresponding to an area of 0.25 cm^2. From here on we consider only a single eye since it is unclear how two eyes would combine for the detection of a faint star, and since we will only incur a factor-of-two error at most, which is insignificant for our OoM calculation. As an additional assumption we ignore absorption by the Earth's atmosphere and set interstellar reddening to zero.

The star is at a distance of 100 light years. Light travels at 3x10^10 cm/s, and there are (π x 10^7) seconds in a year. A light year is therefore D ~ (10 x 10^10 x 10^7) = 10^18 cm. The star emits some number Nemit photons isotropically, and the eye subtends a tiny fraction of the area of a sphere with a radius of D = 10^20 cm and receives 10 photons. This fractional area is (AD/Aeye), where Aeye is the area of the eye and Ais the area of the sphere surrounding the star. Thus



We are interested in the power output of the star, which is the energy emitted per second. We can get the energy corresponding to Nemit photons with 
where \lambda is the wavelength of light. We can assume that the eye's spectral response is well-tuned to the peak of the Sun's spectral energy distribution, which corresponds to about 550 nm (we'll learn more about this after we estimate the Sun's temperature and learn about black body radiation). Thus


where I have used cgs throughout (note that 550 nm = 550 x 10^-7 cm). Now we need to figure out the time interval. The eye detects the 10 photons at a certain "readout rate." This can be estimated by noting that movies are typically shot at 24 frames per second. At a slower rate the eye would notice a distinct slowing of the movie scenes (imagine watching a movie that shows one frame every second, i.e. a slide show), and at a faster rate the movie studio would be wasting film. So the time the brain takes to "read out" the eye is about 10 milliseconds or 0.01 seconds, to an OoM. Thus, the power output of the Sun-like star is


This compares well to the actual luminosity of the Sun, which is 3.862 x 10^33 ergs/s. 

Summary and Discussion

We have performed an OoM calculation of the Sun's luminosity by noting that a Sun-like star at 100 pc is barely visible to the naked eye. Our final answer is correct to within a factor of 4, demonstrating the usefulness of OoM calculations. By not worrying about the exact numbers, but instead focusing on the problem-solving process, we are free to concentrate on the physics of the problem knowing that we can perform the exact calculation using the same reasoning and a bit more time/effort. 

Acknowledgements


We thank Owen and Marcus Johnson for playing nicely with each other for the 45 minutes it took Daddy to write this. We made use of WolframAlpha when our initial estimate of the photon energy was off by two orders of magnitude, and when we couldn't remember Planck's constant in cgs. WolframAlpha helped us realize that we needed the wavelength of a green photon in cm rather than meters. Duh. The equations were generated using CodeCogs online LaTeX editor.

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 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…