I looked at today’s APOD and wondered “how big is that galaxy?”

The article said “the arcing structures form tenuous loops extending more than 150,000 light-years from the narrow, edge-on spiral,” so a thumbnail measurement told me that the disc was around 150,000 light-years across… assuming the loops and the disc were all the same distance from the camera. Clicking on the links in the article gave me a bunch of information that did not include helpful things like “this galaxy has an estimated diameter of…”

But they DID tell me that it was 40,000,000 light-years away*, and was 12.8 arc-seconds in diameter as observed from Earth. It’s been a long time since I took any trigonometry, but I remembered enough to know that I had all the information I needed to calculate diameter. I just needed to google the formula.

The formula, so you know, is D (diameter) = (2pi*(distance)(angular diameter))/360, where angular diameter and “360” are both expressed in degrees. Thus “degrees/degrees” drops that unit from the result, leaving only the unit of measure used for “distance” behind to be applied to the diameter (or in this case “length” since the object being observed is does not present a disc)

Rounding to the comma, the galaxy in question is 146,000 light-years in diameter.

(*Note: I didn’t ask how astronomers measured the distance to the galaxy in question, but I assume it has to do with luminosity, red-shift, and other stuff. Obviously if the distance figure changes as a result of new measurements, the calculated diameter of the object will also change.)

Maybe they went at it from the reverse direction.

Maybe they determined the diameter of the galaxy in question to be 150K ly based on “other factors” and calculated the distance based on that and the arc-seconds data.

Probably redshift, though there’s a chance it’s Cepheid Variables (I forget how far away we can resolve those) or Type Ia supernovae.

In case you’re curious, the formula you quoted is essentially another version of the formula for the circumference of a circle: Circumf = 2pi*radius. 2pi is the number of radians in a full circle, which yields the full circumference of the circle. If you want an arc of a circle, instead of using 2pi you use the appropriate number of radians. In this case the circle is centered around the Earth, the radius is the distance from the Earth to the galaxy, the “arc” is the diameter of the galaxy, and to get the number of radians you create a ratio of measured degrees to the total degrees as you mentioned.

FWIW, when I check your arithmetic, I get 2.5*10^3 ly diameter. Your number (1.46*10^5 ly) seems more probable than mine (and therefore I have an error somewhere) for two reasons though. (1) Both NGC 5907 and our Milky Way are spirals. It’s reasonable to assume we have the same order of magnitude diameters, and the Milky Way is 1*10^5 ly diameter. If you want a better guesstimate w/o doing the math, I’d look up what type of spiral NGC 5907 is and what their typical diameters are. (2) The loops mentioned in the article are quoted as at most 1.5*10^5 ly extended from the galaxy’s center. In the picture these loops are only a little larger than the rest of the galaxy, so I would’ve likely just looked at the picture and said “let’s call the galaxy 1.3*10^5 ly across” and skipped all the math in the first place.

These days, most astronomical distances are calculated by redshift adjusted by using Type 1a supernovae as a “standard candle”, so yeah, that supposition is substantially correct.

If there’s an error in my math it’s my conversion of arc-minutes to decimal degrees. I took divided 1 by 60 (60 arc minutes in a degree) and multiplied that by 12.8 arc minutes.

1/60 = .016666~

12.3*.016666~= .205

Then, using calc.exe, I did 2*3.14159*40,000,000*.205, and then divided the result by 360.

143,116.87~ light years.

Aaand I just found the error. It’s 12.3 arc-minutes, not the 12.8 I listed in my blog post. HAH!

(But that only increases its size to 148,000 ly — that’s not the right error.)

Oh duh, I mistakenly thought that it said 12.8 arc

sec, not arcmin. I still maintain that doing the calculation was unnecessary – perhaps I’m fond of such tricks precisely because I make arithmetic errors so frequently, but why do calculations if you don’t need to? 🙂ooo. Math is sexy. Brains that can do math are fascinating to me. I don’t think in math. I looked at the photo and thought – after “ooo. galaxies are sexy” – we’ll never know what it looks like face-on, but if we could see it face-on, we might not see its beautiful stardust trails. What else is out there that’s amazing, that we’ll never see?

Sorry. Feeling philosophical today. Didn’t mean to get any on you. 😉

You lost me at D=diameter and found me again at red shift. I understand that their is a shift in light the farther away you go, but I can’t do algebra to save my life.

Ona

AHHHHHHH!

Math, my old nemesis, we meet again.