# Gravity and notes about Kittinger's jump

## Contents

# Circular Earth orbits

Velocity required for orbit is independent of mass and size. A bullet orbits at the same velocity as a feather.

The force of the Moon's gravity is 83.3% (or 5/6) less than Earth's force of g. It is less dense, so its difference in size does not correspond to its difference in g. If you weigh 175 pounds on Earth you would weigh about 30 pounds on the Moon.

The Moon is 1/4 of the diameter of the Earth. mass of the Moon = 7.36 * 10^22 kilograms mass of Earth = 5.97 * 10^24 kilograms Radius of Earth = 6378 kilometers, 3963 miles

The force of the Earth's gravity (g) at various places:

g at San Francisco: 9.800 m / s^2 g at Denver: 9.796 m / s^2 g at North Pole: 9.832 m / s^2

## Edge of space -- “Kármán Line”

The **United States Department of Defense** uses 50 miles (~80 km) as their arbitrary definition for being an **astronaut** -- fly above this altitude and they award you an **astronaut badge**. They picked this number because it's a nice high, round number; it has nothing to do with leaving the atmosphere and entering **space**. The definition of "space" is arbitrary. The atmosphere doesn't just end and space begins. It's all rather fuzzy. Is there a more satisfying way to decide when you have reached space? Theodore von Kármán (or Karman) came up with a definition in the 1950's. His idea was to ask, "At what altitude is the air so thin that an aircraft would have to fly **faster** than orbital velocity to gain any aerodynamic lift?" He calculated that this altitude is roughly 100 kilometers (62 mi). Basically, this is the altitude at which a plane would have to fly so fast to stay up that it would basically be in orbit anyway, so you might as well stop worrying about aerodynamics at this point. At this altitude, the atmospheric pressure is 3.2×10^{−2} Pa and the density is 5.6×10^{−7} kg/m^{3}. So the **Kármán Line** is a point where the air is so thin that aerodynamic lift is not useful. This line is the edge between **flying** and **orbiting**. Obviously, the design of an aircraft will effect this altitude limit, so even this definition is still a fuzzy line (each aircraft should have it's own **Kármán Line**), but the idea of a line between **flying** and **orbiting** is a good starting place for defining what it means to be **in space** and 100 kilometers is the line for any aircraft so far. This definition is used by NASA and the Fédération Aéronautique Internationale (FAI) for the **edge of space**.

In practice, air drag and aerodynamic forces are still significant higher than the **Kármán Line**, so it's not practical to actually orbit at the **Kármán Line** because your orbit would decay so quickly that you wouldn't make a single complete orbit. You still have some aerodynamic control above the **Kármán Line** -- control surfaces can effectively alter the orientation of your craft, but you don't get much useful lift above 100 kilometers. NASA uses 122 kilometers (75.8 miles) as their definition of **re-entry** altitude for the Space Shuttle. This is the altitude where it's time for the pilots to start paying attention to aerodynamics and start using aerodynamic control surfaces for maneuvering. Above this altitude the pilots rely on the **reaction control system** (RCS -- small rocket thrusters) to control their attitude and orientation.

The **GOCE** satellite orbited as low as 229 km, but required frequent use of its ion engine to maintain altitude. Its orbit was low enough that it actually made sense to design it with a streamlined profile to minimize atmospheric drag, and to add stabilizing fins. After running out of propellant it was able to stay in orbit only two weeks before falling low enough to burn up in the lower atmosphere.

Karman Line altitude: 100 kilometers, 62 miles radius: 6478 kilometers, 4025 miles g at this altitude: 9.49 m / s^2

## Sputnik-1 altitude

The first artificial satellite, Sputnik-1, had a perigee (minimum altitude) of 215 kilometers (133.6 miles). It had an eccentric orbit with an apogee (maximum altitude) of 939 km (583.5 miles), so it presumably suffered less air drag than a satellite in a nearly circular orbit close to 215 km. It's orbit lasted only 3 months before it reentered the atmosphere. Yuri Gagarin, the first human to orbit the Earth, had an orbit with a perigee of 169 km (105 miles).

## Edge of space -- gravitational influence of the Earth

If going by nearest gravitational source then you have to go about 21 million kilometers away from the Earth for it to no longer be the most significant gravitational body. This is the distance where the strongest gravitational source is no longer the Earth. For reference, the average distance from the Earth to the Sun is 150 million kilometers (also called 1 Astronomical Unit).

## Lowest practical orbit

A practical minimum altitude is depends on your definition of practical. At 180 kilometers a satellite can orbit only a few hours before atmospheric drag will bring it back to Earth. The International Space Station (ISS) orbits at a minimum of 340 kilometers. At this altitude its orbit decays about 2 km/month. The ISS has about one of the lowest altitudes of any orbiting artificial body. If you want a satellite to stay in orbit more than a few days without the need for boosts to counteract orbital decay then 200 kilometers is a good start.

altitude: 200 kilometers, 124 miles radius: 6578 kilometers, 4087 miles g at this altitude: 9.20677898 m / s^2

## Other interesting orbits and the force of g

About half the satellites in orbit are under 1000 kilometers in altitude.

altitude: 1000 milometers, 621 miles radius: 7378 kilometesr, 4585 miles g at this altitude: 7.31843389 m / s^2 about 75% of Earth at sea level

GPS satellites orbit at a very high altitude.

GPS satellite orbital altitude: 20200 km, 12551 miles orbital radius: 26560 km, 14018 miles g at this altitude: 0.5640 m/s^2

Geostationary orbit (orbital velocity in sync with revolution of the Earth):

geostationary altitude: 42300 km, 26284 miles g at this altitude: 0.1681 m/s^2

Moon's orbit:

384403 kilometres, 238857 miles g at this altitude: 0.0026 m/s^2 (insignificant if you are actually near the Moon)

## Gravity and the force it causes

We use the symbol **g** to represent the **force** of gravity on Earth. To calculate the force of gravity on Earth we need to use the gravitational constant, **G**. This constant is the same throughout the entire universe (probably) and allows us to calculate the force at a given distance.

G = 6.673 × 10^-11 m^3/kg*s^2 or G = 6.673 × 10^-11 m^3 kg^-1 s^-2

Normally the force of gravity between two objects is calculated from the distance between their centers. We don't normally think about how far above the center of the Earth we are. We think about our altitude above the surface. So to get the force, **g** (the force of Earth's gravity) from altitude, **A**, we have to add in the radius of the Earth. We ignore the radius from our own center of mass because it is insignificant compared to the size of the Earth. The force of **G** above the Earth can be calculated as follows:

g = G * MassEarth / RadiusOrbit^2 = 6.673 * 10^-11 m^3 kg^-1 s^-2 * 5.97 * 10^24 kg / (6378 km + ALTITUDE km) ^ 2 = (ALTITUDE km + 6378 km) ^-2 * 6.673 * 10^-11 m^3 kg^-1 s^-2 * 5.97 * 10^24 kg

Note that you can copy that formula directly into Google search and it will calculate the force, **g**, for you. Replace **ALTITUDE** with the altitude above Earth you want to find the force, **g**. You **must not** forget the **km** units for your altitude; in this case "100 km", not just "100":

( 100 km + 6378 km ) ^-2 * 6.673 * 10^-11 m^3 kg^-1 s^-2 * 5.97 * 10^24 kg

Google will give you:

= 9.49322051 m / s^2

## Joseph Kittinger's jump

On August 16, 1960, Captain Joseph Kittinger made a jump from the Excelsior III helium balloon gondola at 31.3 kilometers, 19.4 miles, 102800 feet.

g at 31.3 kilometer altitude: 9.6978 m / s^2

The force of gravity at 31 km is about 97% of what you feel at sea level. Kittinger was about 3% lighter than on the ground. So if he was 180 pounds on the ground he would have been about 175 pounds at 31 kilometers.

So, by any of the albeit fuzzy definitions, Kittinger was not "in space". Granted, the air was so thin that he would have passed out in seconds and be dead in a few minutes if his pressure suit had a major failure. He would not have been able to descend in free-fall to a lower altitude quickly enough if his pressure suit failed. Actually, his suit did have a failure in his glove. The glove did not inflate to provide pressure to his hand, so his hand was exposed to the vacuum of 100k feet. It swelled to twice its size and he was unable to use it until he got back to Earth.

## other interesting orbital metrics

Hill sphere

Roche limit

Schwarzschild radius