8.3.1 Satellites

Suppose, however, that we launched the projectile from a tower of height $h=230\text{mi}\approx 3.8\times {10}^5\text{m,}$ just above the earth’s atmosphere. This is approximately the height of the International Space Station and other low-earth-orbit satellites. Note that $h\ll R_{\text{e}}$, so the radius of the orbit $r=R_{\text{e}}+h=6.75\times {10}^6\text{m}$ is only 5% greater than the earth’s radius. Many people have a mental image that satellites orbit far above the earth, but in fact many satellites come pretty close to skimming the surface. Our calculation of $v_{\text{orbit}}$ thus turns out to be quite a good estimate of the speed of a satellite in low earth orbit.

We can use $v_{\text{orbit}}$ to calculate the period of a satellite orbit:

For a low earth orbit, with $r=R_{\text{e}}+230\text{miles,}$ we find $T=5210\text{s}=87\text{min}$. The period of the International Space Station at an altitude of 230 mi is, indeed, close to 87 minutes. (The actual period is 93 min. The difference, you’ll learn in Chapter 13, arises because g is slightly less at a satellite’s altitude.)

When we discussed weightlessness in Chapter 6, we discovered that it occurs during free fall. We asked the question, at the end of 《 SECTION 6.3, whether astronauts and their spacecraft were in free fall. We can now give an affirmative answer: They are, indeed, in free fall. They are falling continuously around the earth, under the influence of only the gravitational force, but never getting any closer to the ground because the earth’s surface curves beneath them. Weightlessness in space is no different from the weightlessness in a free-falling elevator. It does not occur from an absence of gravity. Instead, the astronaut, the spacecraft, and everything in it are weightless because they are all falling together.