Problem #1:a) The Sun orbits the center of the Milky Way galaxy in roughly a circle of radius 23000 light years, and with a speed of 220 km/s. Calculate the time required for the Sun to complete one circuit around the galaxy.
b) How many times will the Sun orbit the galaxy in its 10 billion year lifetime?
Solution -- Part a): This is a straightforward problem designed to give you an idea how big the Milky Way galaxy really is. We're asked to calculate how long it takes the Sun to travel around the galaxy, and we're given the radius of the Sun's orbit and the Sun's speed. Since we have the speed, all we need to do is calculate the distance the Sun travels in one orbit and we get get the time, since
Now the Sun travels in a circle around the center of the galaxy, so it covers a distance equal to the circumference of a circle with radius 23000 light years. The circumference is
So,
where I've converted from km/s to m/s on the fly. Rearranging terms (be careful here!), we get
where I've figured out the number of seconds in a year by multiplying 60 sec/minute x 60 minutes/hour x 24 hours/day x 365 days/year, and I've converted to years because I have no clue how long 1015 seconds is.
Solution -- Part b): Since I've already figured out how many years it takes for the Sun to make one circuit around the galaxy, it's pretty easy to figure out how many laps it will do in its 10 billion year lifetime. The number of laps is just the total time available divided by the time it takes to do one lap:
Problem #2:a) Rydberg's constant, which we calculated in Lab #3, and which is the energy required to ionize a hydrogen atom from the ground state, is equal to 2.18 x 10-19 Joules. Calculate the wavelength of a photon which has this energy.
b) Determine the surface temperature of a star whose blackbody spectrum peaks at this wavelength.
Solution -- Part a): You're given the energy required to ionize a hydrogen atom, and asked what kind of photon has this energy. Well, we know a relationship between photon energy and wavelength. It's
where h is Planck's constant, c is the speed of light, and lambda is the wavelength of the photon. Rearranging terms and putting in the values of these constants from the table in the back of the book, we get
Mea culpa: This is a correct manipulation of the energy-wavelength relationship, and should be the answer that you got, but unfortunately, it's not the wavelength of a photon which can ionize a hydrogren atom from its ground state. That's because there's a typo in the question. The energy required to ionize a hydrogen atom from its ground state is 2.18 x 10-18 Joules, not 2.18 x 10-19 Joules, and consequently, the wavelegnth of a photon of this energy is 91.7 nm, not 917 nm. This is my error, and I apologize if it caused any confusion. Obviously, you'll receive full credit for the incorrect answer.
Solution -- Part b): To find the surface temperature of a star whose blackbody spectrum peaks at this wavelength, you need Wien's displacement law:
where lambdapeak is the wavelength of the peak of the spectrum, T is the surface temperature of the star, and 0.003 m K is a constant. Rearranging and using the (incorrect) answer from above, we get
which is the surface temperature of a K or M star. This is unfortunate because it implies that virtually any star (most stars have surface temperatures equal to or higher than this value) has a large number of photons with sufficient energy to ionize hydrogen. This is not true.
If you use the wavelength corresponding the the correct ionization energy for hydrogen, you'll find that the surface temperature must be a factor of 10 greater, or 33,000 K. This is a pretty high surface temperature for stars, and only O and some B stars are this hot. Consequently, regions of ionized hydrogen (or HII regions) are found only around O and B stars, and therefore are pretty rare.
I apologize again for creating this extra bit of confusion.
Problem #3:a) The nearest "real" galaxy to us is the Andromeda Galaxy, also known as M 31. This galaxy has an apparent size of 2 degrees (about four times the size of the full moon!). If its linear size is 25 kpc, how far away from us is it?
b) How long does it take for light from that galaxy to reach us?
Solution -- Part a): This is an Observer's Triangle problem. If you don't understand it, go back and read the Special Web Page on this topic. I guarantee you'll be asked to demonstrate your knowledge of this topic within a week.
We're given the angular size of the Andromeda Galaxy (2 degrees) and it's linear size (25 kpc), so we can use the Observer's Triangle relation as follows:
or, solving for the distance
Note that I didn't need to do any converting of the 25 kpc to meters or anything else. That's because the right hand side of the Observer's Triangle is a ratio of the linear size of the object to its distance from us. Since it's a ratio, I'm free to use any units I want, provided I use the same units top and bottom. That means if I quote the linear size in kpc, I get a distance in kpc. If we really need the answer in meters (and we will in a minute), we can convert this answer to meters as follows:
Solution -- Part b): We now know how far away the galaxy is, and we know how fast light travels, so
and rearranging terms,