Welcome Guest [Log In] [Register]
Sky Dragon is up and running. We need people to sign up and add content. Introduce yourself at the General discussion forum. Speak up to welcome a few others, start a thread, or contribute to someone else's thread.
Viewing Single Post From: The giant space ship example
Chris Ho-Stuart
Member Avatar
Administrator
At Climate Etc, gbaikie provided some comparisons and calculations to get started with the starship. I'll repeat some of them here, in tabular form.

The following numbers define what is given for the ship, and can be a basis for comparison with Earth, or the moon, or other bodies.

  • Surface gravity. (Starship has a surface gravity 1/10 of Earth.)
  • Radius. (Starship has a radius of 1200 km.)
  • Surface air pressure. (Starship has surface air pressure 30% of what is on Earth.)
  • Surface temperature. (Starship surface temperature is 10 C.)

Values for bodies in the solar system are available from NASA at the Planetary Fact Sheet.

Calculating values invariably works better with SI units, so that's what I will be using. Some points to bear in mind on units:

Temperature. Many thermodynamic calculations need to use an absolute temperature scale, where the coldest possible temperature is given as 0. Absolute temperature is measured in Kelvins. The temperature in degrees Celsius is equal to the temperature in Kelvin, plus 273.16; adding 273 is adequate for rough conversions.

Pressure. The SI unit for pressure is the Pascal (Pa), which is the same as one Newton per square meter. The force on one kilogram in Earth's gravity is 9.8 Newtons. Earth's atmospheric pressure at sea level is very close to 100,000 Pa. The unit "hectoPascal" (hPa) is sometimes used; a hectoPascal is 100 Pascals, and this is equivalent to 1 millibar, and a "bar" is an unit representing an early estimate of pressure at Earth's surface. The "standard atmosphere" pressure is now defined to be 101325 Pa. One psi is equal to 6895 Pa.

The pressure at Earth's surface in Pascals is the mass of the atmospheric column above a square meter of surface, times the gravitational acceleration.

The gravitational acceleration g at the surface of a uniform sphere of mass M and radius R is equal to GM/R2, where G = 6.673 e-11 is the gravitational constant. Hence M = gR2/G. The volume V of a sphere of radius R is (4/3).pi.R3, so the density will be M/V = 0.75*g/G.R.pi.

The dry adiabatic lapse rate (DALR) in an atmosphere is -g/Cp (unit K/km), where Cp is the specific heat at constant pressure (in units J/g/K). For SI purists, note that these are not quite SI units; Cp is typically given as Joule per gram per Kelvin (not per kilogram), and the lapse rate in Kelvin per kilometer (not per meter). I'll use these units also, where it shows up.

On Earth, where the atmosphere is mainly Nitrogen and Oxygen, Cp is very close to 1. On Mars, the atmosphere is mainly Carbon Dioxide, with Cp being about 0.83, so there the DALR is about -3.71/0.83, or the DALR is thus about 9.8.

For an atmosphere consisting mainly of Nitrogen and Oxygen, the specific heat (at constant pressure) Cp is pretty close to 1 (unit J/g/K); and the DALR is about 4.5. In the starship, Cp should be similar to Earth, since you want an atmosphere that is mostly Nitrogen and Oxygen, both of which have Cp close to 1. Hence the DALR will be about 1 as well.

Using these formulae, we can find some basic properties for Earth, Mars, our Moon, and the starship.
Properties
EarthMoonMarsStarship

Radius, in km
6378173833961200

Surface gravity, in m s-2
9.801.623.711

Surface temperature (average), in C (Celsius)
15-20-6510

Surface air pressure, in bars (1 bar is 1 atmosphere)
100.006360.3

Surface air pressure, in Pa
10000063630000

Total mass, in kg
5.97e247.33e226.41e232.16e22

Density, in kg m-3
5.53.33.93.0

Total mass atmosphere in kg
5.1e182.5e165.4e17

Atmospheric specific heat (constant pressure) in J/g/K
10.831

Dry Adiabatic Lapse Rate (DALR) in K/km
-9.8-4.5-1


     (late edit to fix the DALR on Mars; I previously had -3.7 which failed to account for the reduced Cp value.)

Gbaikie, this matches your estimates so far, apart from lapse rate. You were correct to suppose a weaker lapse rate on the starship than on Earth or Mars; but it's weaker even than you had guessed.
Edited by Chris Ho-Stuart, Dec 4 2011, 09:59 PM.
Offline Profile Quote Post
The giant space ship example · Physical theory for climate