By Mike Johnson
At the January Mensa Forum Dr. Stephen Odewahn of ASU showed some beautiful pictures taken from the Hubble space telescope (HST). One that especially intrigued me was a long term exposure of a small region of the sky only two minutes (of angle) square (for 4 square minutes).
I can't picture four square minutes so consider the following. The sun and moon each subtend about 1/2 degree or 30 minutes of arc. This is about 700 square minutes or about 200 times the area of the HST!
At the Cork 'n' Cleaver post forum debriefing Dr. Odewahn said this slide showed about 1,500 to 2,000 galaxies! I asked how many this translated to over the whole sky. Dr. Odewahn wasn't sure. I often refer to rough figures of 100 BILLION galaxies (with, coincidentally, also about 100 Billion stars per galaxy). I didn't remember where I got these numbers. (I've since confirmed that the figures come from Carl Sagan's "Cosmos" (page 5), copyrighted in 1980, long before the HST.) I wondered how well the two values agreed so I did a rough mental calculation.
I couldn't remember any solid geometry, if in fact I had ever known any, so I took a very sloppy approach but came up with a number surprisingly close to the 100 billion.
Lying in bed that night I realized that I didn't need solid geometry so long as I knew the area of a sphere. I remembered this as 4 x pie x r2 for any radius (r). I could figure the "area of the picture" (at the same radius, r). The ratio of these two areas would give me the multiplication factor to calculate all the total number of HST observable galaxies.
Assume 2,000 galaxies in four square minutes; that's about 500 per square minute. Now we need to know how much bigger the whole sky would be. At any distance r the area contained in the picture would be the square of the LENGTH of its side (L), If we convert the one minute of angle into radians and call it A for angle we get:
A = 1/(60x57) or about 1/3,400 radians,
since 1 degree = 60 minutes and
1 radian = about 57 degrees
L = radius x radians = r/3,400
The AREA of this "little" segment equals L squared or r2/1.2E+7
I remembered the area of a sphere as 4 x pie x r2 so the ratio of the whole sky area to the segment must be
R = (4 x pi x r2)/(r2/1.2E+7)
= 4 x pi x 1.2E+7
= 1.5E+8
If each segment contains 500 galaxies the total should be
T = 5E+2 x 1.5E+8 = 7.5E+10
or about 75 BILLION!
These are just rough mental calculations, (a more precise calculation with a calculator gives 7.43 billion!) but I was really excited that the numbers jive well with the 100 billion estimate.