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:
- 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
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 ~
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%.
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*220/4.43x10-3 = 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 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 "mV [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 in 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.