Given the average orbital radius of the Sun around the galactic core combined with its angular velocity, I was to find the acceleration.
No problem! That's just "v-squared over r"! Given a 2.472x10^20m radius and velocity of 220000m/s, I get 1.958x10^-10m/s^2. This seems incredible, but our acceleration to the core really is that slow - I suppose the scale of our galaxy is so vast that the continuing change in velocity (at any given moment, velocity in a circular orbit is tangential - an object requires acceleration to the core to keep it moving in a circle) doesn't seem like very much.
Now I need the period. Well, it takes 2π radians to go around a circle, and that distance over the period equals the velocity. A little algebraic switcheroo, and I have 2π radians over the velocity. m over m/s leaves me with a bunch of seconds, which divide into about 223 million years. The published value is around 220 - so far so good.
I'm making some approximations here... I'm calling the orbit circular, and simply letting the centripetal "force" equal gravitational force. F=ma where m is the mass of the solar system (1.9911x10^30kg) and I get 3.898x10^20 Newtons.
Now I can try for the mass of the galaxy.
Fg = ((G)(M1)(M2)/(R^2))
Here you can see that more mass = stronger force of gravity, but larger radius = way way weaker force of gravity (since we square the distance). G is the gravitational constant, and I'm one of those sorry people who uses it more than they understand it (at least for now). It's a very small number, 6.67x10^-11 N m^2 / kg^2. Gravity is a very weak force. You need a lot of mass before it really adds up to anything. But it's pervasive: in an imaginary, non-expanding universe of all empty space where there are two mosquitoes set 10,000 light-years apart, eventually they will come together! That would actually be fun to calculate, but I need to finish this assignment before the start of tomorrow's class. I'll come back to this later.
A little siwtcheroo, and I isolate "M2" - the mass of the galaxy:
((Fg)(R^2))/((G)(M1)) = M2
When performing this, the items you're pulling over switch sides, so radius squared goes on top while G and the mass of the solar system go to the bottom.
Click click click and I get... 1.794x10^41kg. Basically that's 1 followed by a 794 and 38 more zeros. If that doesn't seem very heavy, take a piece of paper and start writing out all the zeros. I don't know how you'd calibrate a scale big enough to weigh it, though. Keep in mind we're no longer in the Earth's particular gravitational field, so your bathroom scales, even though they display results in kilograms (strictly speaking, they're "converting" to kilograms based on the pull you experience towards the center of the Earth), won't work here.
Now this all sounds well and good right? Except that when I go to check the accepted values for the mass of the galaxy, I'm off by a whole order of magnitude! Wolfram Alpha gives me 6x10^42kg, which is about 33 times more massive than my estimation. Ouch.
I triple-checked what I did, and I welcome people to challenge my numbers. I really wonder about my estimations of acceleration and the centripetal force. My units check out, which is always a good sign.
So I'm digging around, and I find this: Galaxy Rotation Curve
Contrary to what is observed in the solar system, stars revolve around the center of galaxies at a constant speed over a large range of distances from the center of the galaxy. (In the solar system, the velocities slow as you march outward from the Sun (yes, and there's a greater path to sweep).) Well, if the speed stays the same, then anything I do relating the centripetal force to radius, and by extension any calculations of mass, all comes into question.
The best part is that I've "independently" stumbled upon the reality that real astrophysicists and cosmologists face - the galaxy and the universe are more massive than can be explained with what is presently known. In other words, their behaviour is not the same as what you would expect if you took everything you could see and added it all up.
My present goal is to see if I can come to understand some of this stuff.