# How would I calculate celestial body characteristics without a priori knowledge?

Without any a priori knowledge of the mass, speed, distance, and size of local celestial bodies (aside from Earth's size), what can I calculate and how from my "backyard" through observation?

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My goal is to create a high-school level physics project that students can conduct over the course of a year/term as both a physics problem mixed with a bit of astrophysics history. Core concepts would include observation (within reason). This isn't meant to be extremely precise work (within an order of magnitude would be excellent), but to demonstrate the methods.

Given a bunch of knowns, it's trivial to find out one unknown, but I am unsure of how to start from square one like the early physicists and astronomers did in times past.

• Is this just a single observation or are you talking about a series of measurements (so that you get some time-dependent characteristics of body's movement)? Commented Nov 15, 2010 at 15:57
• A series of measurements would be fine; something that a student or class could do over the span of a few or possibly several months. Commented Nov 15, 2010 at 16:05
• Well, with good enough telescopes and good enough knowledge of classical physics, you should be able to calculate anything you want (this is what astronomers do all the time, in fact. Except for satellites we don't have any other means for observing the space). But in practice both measurements and calculations might be very hard to carry out. Commented Nov 15, 2010 at 20:13

1. calculate the size of the earth e.g. through vertical sticks (what Erathostenes did) (or just assume it, this requires travelling)

2. measure the acceleration of gravity at the earth's surface and calculate the earth's mass (this is cheating by assuming newton's law of gravity, which was developed astronomically)

3. assume the moon's mass is small and calculate its distance from its orbital period. confirm the orbit is roughly circular by measuring the moons position at a specific time of day over 1 month (or so).

4. in principle the sun's distance can then be determined by measuring how much of the moon is lit on the night exactly between new and full moon, thus measuring the angles of the sun-moon-earth triangle (greeks did it, but this is probably pretty hard to get right)

5. sun's and moon's diameter can be determined through their viewing angle

6. sun's mass can be calculated through the earth's orbital period

• Point 4, using the angle of the lunar terminator to estimate the astronomical unit (distance to the sun) never really worked for Aristarchus, who first proposed the method. He was only able to develop a wild underestimate as a lower bound. This measurement eventually required the measurement of the transit of Venus. This is the toughest link in the chain to measure. However, given that we can directly measure G at home by the Cavendish experiment, maybe something can be done using e.g. the strength of solar tides. Commented Nov 17, 2010 at 9:11
• Continuing the above, I would try to independently estimate the speed of light in an experiment at home (see physics.stackexchange.com/questions/357/… ), maybe using standing nodes in conjunction with a beat chain to measure frequencies, or a computer to measure small time delays, or some other method, and then reverse the logic of Rømer's 1676 speed of light estimate to get an accurate number for the astronomical unit, solving point 4. Commented Nov 17, 2010 at 15:33

Possibly the easiest one, but often overlooked: earth itself. How would you determine it's size? There are a number of fairly easy ways to measure it within an oder of magnitude.

Next up is the moon. Obviously there's the earth's shadow, and we've just determined that radius. So, how big is the moon in comparison? How heavy would it be? (It's rock, you might assume the same density) What is it's rotation period? Its orbit?

The big advantage of the moon is that it's quite easy to find and observe, even under less than stellar atmospheric conditions.

There's a surprising amount you can get from very basic observations; basically just looking at how bright certain stars are over time, that is, measuring their light curves. I would look into the literature of variable stars. In particular, (if you're in the United States) the AAVSO is an organization of amateur observers that may have resources on their website for people interested in these things.

Let me sketch one prototypical example. There is a famous class of stars called Cepheid variables with a well-known relationship between the frequency of oscillation and their absolute magnitude. By comparing the absolute magnitude to the apparent magnitude, you can actually determine the distance to these stars! People have used these sorts of observations to measure the distances of nearby galaxies, and is an important step in the cosmic distance ladder.

Of course, many of these relationships between period and brightness are purely empirical -- I don't know that we currently understand stellar astrophysics well enough to derive these relationships from scratch. But it's been a while since I've read about these things. If you're more interested in this side of things, I'd look for books on stellar astrophysics; though the background you'll need is fairly broad (nuclear physics, statistical mechanics and thermodynamics, knowledge of radiative processes)...

• This is a nice answer but perhaps to a different question than the one being asked? I thought OP was asking about local bodies, and wanted to calculate properties from trajectories of the bodies on the celestial sphere (e.g. reconstructing ellipses from this). Commented Nov 16, 2010 at 0:55
• @Marek correct, I'm wondering if I can create a HS level project for students to do over a year/term as both a physics problem mixed with a bit of astrophysics history. Commented Nov 16, 2010 at 1:53
• Heh, I guess it depends on what you consider "local". In any case, I first got interested in astrophysics and variable stars as a high schooler, so it's not totally out of reach...
– j.c.
Commented Nov 18, 2010 at 2:33