Astronomy 10
HW#1 Solution set.
1) These problems, sometimes called Fermi Questions, are exercises in estimation. The essence of the problem is to make a small number of reasonable assumptions and make a rough calculation, not to get an exact answer. The appropriate precision in these problems would be an answer in the form 
, since typical answers will be off by no more than, perhaps, 10 times.
Example problem: How many cells in the human body?
Assumptions:
The volume of the human body is composed entirely of cells.
The human body has dimensions 2m 
0.5 m 0.25 m.
A cell is a cube 
m on a side.
These assumptions are very rough, but none of them is wrong by a factor of more than a few, except the third. The size of a cell varies greatly, so any guess between 
and 
m is probably OK.
Now the number of cells in the body would be:
Therefore 
.
There are about 
cells in the human body by this calculation.
2) In Meridio, the Son appears to be on the zenith (directly overhead) at noon, on the eastern horizon at sunrise, and on the western horizon at sunset. The Son always makes this exact pattern every day, all year long.
In Dimlee, the Son appears on the horizon all of the time (all horizons are the northern horizon). During the day, the Son will appear to move along the horizon, making a complete circuit every day. The Son always makes this exact pattern every day, all year long.
3) Since the Astra's orbital plane (it's ecliptic plane) and Astra's equator (defining it's Celestial Equator) are the same, the Son appears on the Celestial Equator all year long.
4) There are basically three parts to this problem. The first is determining how far you've gone in your hovercar:
distance = velocity * time = 200 km/hr * 5 hr = 1000km
The next is determining how far that is on the planet:
During the journey along the surface of Astra, the North Celestial Pole appears to get 10 degrees closer to the horizon. Since the altitude of the North Celestial Pole is equal to your latitude, this means that you have moved 10 degrees around the planet.
To prove this to yourself, consider that if you start at the North Pole, the North Celestial Pole appears to be at the zenith (directly overhead). As you move toward the equator, the North Celestial Pole will get lower and lower until it appears on the horizon when you get to the equator. This is a 90 degree change in the location of the pole, and you have gone 90 degrees around the planet.
Finally, putting the two pieces of information together, we know that the circumference of the planet is
where R is the planet's radius. We also know that we have gone 10 degrees around the planet, and that a circle has 360 degrees. Therefore, we have gone
10/360 = 1/36 of the way around the planet. Therefore the total circumference is
and the radius is
