Back of the Envelope

Why is distant Jupiter so bright but communications satellites in close orbit are only visible when they catch the Sun just right? Satellites are smaller than Jupiter but much closer so they should appear bigger, right?
• Jupiter is between, oh, 380 million and 570 million miles away depending on where we are in our orbit. It's also about 10 times the size of the Earth, or about 80000 miles across. At those distances, the angle covered by Jupiter's disc ranges from 1/2 to 3/4 of an arc-second. (The sky is 180 degrees from horizon to horizon and there are 60 arc-minutes in a degree and 60 arc-seconds in an arc-minute.)
• Let's say we're looking at a Dish Network satellite that's 10 feet across, in Geosynchronous orbit (22,000 miles high). It covers three one-thousandths of an arc-second, or nearly one one-hundredth the size of Jupiter in our sky!
You can do this math yourself.

The tangent of an angle is the ratio of the Y distance covered by the angle to the X distance. Draw an X axis line and a Y axis line on graph paper, then draw a line at the angle you want (any length is fine but longer will be easier to measure), starting at the intersection of the X and Y axes. Measure the distance the line travels in X graph paper squares and Y squares, then calculate y/x. That's the tangent of the angle.

You can see that the tangent will be close to zero if the angle is low and large if the angle is high. Tangent takes the angle and turns it into y/x, but we want the opposite, the arctangent (or arctan or atan), which takes y/x and turns it into an angle.

If you take the radius of something, like Jupiter, and divide by the distance from you, then use the arctan or atan button on a calculator, you get the angle!

(I kind of hacked my answer because I used the diameter for y, but you really need to use the radius and then multiply the angle by two. For angles as small as the ones I've been talking about, multiplying the radius by two and then applying arctangent is really close to multiplying by two after the arctangent.)

My initial numbers are so loose that the final answers are only good for really coarse comparisons. Since the difference is close to one hundred to one, however, it's pretty clear that I could be off by a factor of two in all of my initial numbers and still get the same conclusion: satellites are MUCH smaller in our sky than Jupiter.

Really loose math (like this) to quickly find out relationships between numbers or systems is sometimes called a back of the envelope calculation.