## General Homework Tips, Suggestions, and Strategies

### Chris Mihos, CWRU Astronomy

Start Early, Work Regularly
Many of the problems involve multiple steps that build off one another. If you start the HW set late, find you're stuck on step 1, you'll be in deep trouble for the rest of the problem. Also, I don't always define every single thing you need to solve the problem -- that's by design, to get you used to the idea that sometimes you need to do a bit of your own digging for information. Again, if you can't find that information the night before the due date, you'll be in trouble. I'm always happy to help you find information or work through sticking points, but you need to know where those sticking points are well ahead of the deadline so that you can ask for help. So start the problem sets early, and work a little bit on them every few days so that you know where you might need help well in advance of the due date.

Explain each step of your calculation of analysis, included a description of any assumptions or data/numbers used. Show intermediate steps. Don't just write down an answer with no other explanation -- that won't get full credit, even if it's correct. Two reasons for showing your work: First, a good answer is much more than a calculation, its an explanation. A truly excellent A-level homework writeup should double as the solution set that I can just hand to other students so that they can understand the concepts behind the solution, as well as the quantitative answer. Second, if you show your work, I can tell the difference between a simple calculation error (which don't get penalized much) and a completely wrong approach. If all I have to grade is a final answer, and it's wrong, it's a zero, whereas if I can see steps showing that you did 90% of problem correctly, but stumbled on the last step, you'll get much more credit.

Use Words

Use the units of astronomy; try and limit conversions.
We're doing astronomy, so in general you should use units natural to astronomy-- parsecs, solar masses, years, and variations of those units. Certainly there are times when SI/cgs units are appropriate -- if you were calculating the mass of a comet, for example, expressing it in solar masses would be kind of silly. So you'll have to make decisions about the best units to use. But you're used to mixed units that are situationally dependent. In your physics class, if asked to do a calculation of time, you'll probably work out an answer with units of seconds, but if asked your age, you'll give it in years. Or if asked "how far is Toledo", you might even answer in units of time ("its a two hour drive"), not distance. These are "situational units". And in this class, the situation is astronomy, and more specifically, galactic and extragalactic astronomy. So use the units of astronomy. Converting back and forth from astronomy units to SI/cgs is bound to lead to silly mistakes, and is one of the biggest sources of error I see on HW sets.

And finally, yes, we use magnitudes. We're astronomers. Learn to be comfortable with them.

Here are some tips and shortcuts to make your life easier:
• if you measure distances in parsecs (pc), time in millions of years (Myr), masses in solar masses (Msun), and speeds in km/s, G=4.43x10-3 pc3 Msun-1 Myr2. Don't convert everything to SI, plug in G=6.67x10-11 m3 kg-1 s-2, then convert back -- you're apt to make a silly conversion error.
• Similarly if you are a planetary scientist working with distances in AU, time in years, and masses in solar masses, G=4$\pi^2$ AU3 Msun-1 yr2.
• 1 km/s ~ 1 pc/Myr (which means I could just as easily have said  G=4.43x10-3 pc (km/s)2 Msun-1)
• 1 year ~ $\pi$ x 107 seconds
• For small magnitude errors (< few tenths), the relative flux uncertainty is roughly equal to the magnitude uncertainty. So a magnitude uncertainty of 0.1 mag is roughly a 10% uncertainty in flux.
• For small errors in distance modulus, the relative distance uncertainty is about half the distance modulus uncertainty. So a distance modulus uncertainty of 0.1 mag is a distance uncertainty of 5%.

Remember that you can do math on units as well. Let's say you were doing a problem using the speed of the Sun's orbit around the Galaxy to work out the Galaxy's mass. OK, so the Sun is about 8 kpc from the galactic center, and the orbital speed is about 220 km/s. So you say

M = rv/G = 8000*220/4.43x10-3 = 4x107 Msun. Hmm that seems really small if the Galaxy really has billions of stars in it, what went wrong?

So let's check the units:

M = rv/G = [pc] * [km/s] / [pc (km/s)2 Msun-1]
• the pc on top and bottom cancel out
• Msun-1 on the bottom becomes Msun
• one power of km/s on top divided by two powers of km/s on the bottom leaves a km/s on the bottom
• so my final unit on my answer is Msun / (km/s) -- that's not a mass! So my answer can't be right, I've messed up the velocity part!
and yes, that's the problem -- it's not M = rv/G, it's M=rv2/G. If you plug the numbers into that correct expression, you get a much more reasonable mass: 8.7x109 Msun. (Although remember that's only the mass inside the Sun's orbital radius!)

So if you're numbers aren't working out, use unit analysis to help track down a problem.

Answers have units, plots have labels

Numerical answers always have units -- make sure you give them. Working out a distance to a globular cluster of "7600" is not correct, it ought to be "7600 pc" or, better yet, "7.6 kpc". When you make a plot, axes should be labeled both with what they are showing and what the units are. For example, a color magnitude diagram would have an x-axis label that says "B-V [mags]" and a y-axis that says "mV [mags]", and bright blue stars would be at the upper left of the plot!

Give quantitative estimates of uncertainties whenever possible

On numerical data analysis problems, you should always try to give an quantitative estimate of an uncertainty. If you calculate a value, calculate or estimate uncertainty using propogation of errors, if possible. When fitting a line to data, report not only your fitted values, but also their uncertainty based on the fit. If you are unsure how to do this, or the level of detail you need to provide on the error estimate, ask me or the TA for clarification and advice!

Don't write out every digit your calculator displays!

Think about significant digits, not necessarily in the strict sense, but in terms of common sense. If I said to you that Columbus was 140 miles away and you were driving 75 mph, I hope you wouldn't tell me it would take 1.86666666667 hours to get there, right? (Plus, why would you want to go to Clodumbus?) Stop quoting digits where they stop being meaningful. Sometimes "meaningful" will have a quantitative definition -- for example, how the answer compares to the uncertainty -- other times it will have a common sense answer based on the quality of the assumptions.