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.

Show Your Work

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

This goes hand-in-hand with the previous comment. A scientific writeup is an explanation, not a calculation. Don't just hand me a piece of paper with numbers and plots. I expect your writeups to be well written, using verbal explanations, complete sentences, good grammar, etc -- no different from the standards and expectations we scientists have in writing up our research in scientific journals. I don't demand that you typeset or use a word processor to write up your HW (writing out mathematical expressions on word processors is a much bigger pain than it's worth), but your writeups should be neat and readable -- if I can't read your handwriting (or if I can but it gives me a splitting headache), or if I can't follow your logical steps, it will be hard to give full credit on your answers.

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 (using these approximations for course calculations is absolutely fine):

1 year ~ $\pi$ x 10⁷ seconds (to about 0.5% accuracy)
1 km/s ~ 1 pc/Myr ~ 1 kpc/Gyr (to about 2% accuracy)
If you are working with distances in AU, time in years, and masses in solar masses, G=4 $\pi^2$ AU³ Msun^-1 yr^-2 (exactly).
If you measure distances in parsecs (pc), time in millions of years (Myr), masses in solar masses (Msun), and speeds in km/s, then G ~ 4.30x10^-3 pc (km/s)² Msun^-1. 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.
More generally for unit conversions, it may help you to get to know the astropy units package.

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%.

Unit analysis can help you track down a problem

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*220/4.3x10^-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 on the top
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 "m_V [mags]", and bright blue stars would be at the upper left of the plot!

And while we are on the subject of plots, if you are making an x-y plot and y spans more than an order of magnitude, don't plot it on a linear plot! Either plot log(y), or use a plot with a logarithmic y-axis. (The same goes for x, of course!)

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.

Comment on Wrong Answers

If you work out an answer that you know to be wrong, please say so! For example, if you work out the distance to a star and get a number like 3.31234x10⁶ meters, please tell me you know that's a wrong answer, otherwise I'll worry that you think that star really is located somewhere near Los Angeles! (Plus, shouldn't you have worked it out in parsecs, anyway?). Tell me that it's wrong, give me some idea of where you think your error might have happened. Don't just write down an answer that you know to be wrong and leave it there without comment.

Think about the Results

Arriving at a quantitative answer or making a plot is never the end of an exercise. After you've done the analysis, you need to comment on what it means! Again, just about every scientific paper has a "Discussion" section after the "Results" section (even if they are not formally labelled like that), and you should take the same approach. Don't just attach a plot and walk off stage -- instead, talk about how your result fits in with the bigger picture of whatever the problem is talking about.

Also, often times I will ask you to comment on sources of uncertainty or error. Don't just say "the data could be bad" -- that's kind of a "duh" comment. Of course, the data can always be bad. Instead, I'm asking you to comment on sources of systematic uncertainty -- problems with any assumptions that were made, or with how the data might have been collected, etc -- and how that uncertainty affects your answer/result. For example, if I gave you a color-magnitude diagram and asked you why the spread in the main sequnce got worse for low mass stars, saying "the apparent magnitudes of the stars might be wrong" is a duh answer, but saying "low mass stars are intrinsically low luminosity, and measuring accurate colors and magnitudes becomes harder as you go fainter, so there is just more measurement error for low mass stars" is a much more thoughtful answer.

Cite your Sources!

If you are comparing your answers to studies in the scientific literature, you must always cite your sources! And remember, the source is the journal article you found the result in, not the webtool you used to find the information. So, for example, if you used ADS to search for papers on M101 and found a paper by some guy named Mihos who said that M101's disk scale length was 2.4 arcminutes, you would say:

M101 has a disc scale length of 2.4 arcmin (Mihos et al 2013)...

and then at the end of your write-up you would have the reference:

Mihos, J.C. et al 2013, ApJ, 762, 82.

You shouldn't say "I found a value of 2.4 arcminutes by searching online", or "ADS says M101 has a scale length of 2.4 arrcminutes", since neither of those statements can be connected back to the source you found.