Difference between revisions of "Atmosphere"

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m (I realised a note that says the area is always taken to be 1. Therefore, my previous tests weren't quite right. Therefore, redoing the same tests, I found that the 0.008 actually needs to be 0.00696 to give the right evolution. Finally, I got rid of al...)
m (Edit for consistency)
Line 89: Line 89:
 
: <math>v_T = \sqrt{\frac{2\, GM}{r^2\, \rho \cdot 35.07}}</math>
 
: <math>v_T = \sqrt{\frac{2\, GM}{r^2\, \rho \cdot 35.07}}</math>
  
: <math>\rho = 1 \cdot e^{-0/5000} \cdot 1.2230948554874 \cdot 0.008</math>
+
: <math>\rho = 1 \cdot e^{-0/5000} \cdot 1.2002 \cdot 0.00696</math>
  
: <math>v_T = \sqrt{\frac{2 \cdot 6.674e \cdot 10^{-11} \cdot 5.2915793 \cdot 10^{22}}{600000^2 \cdot 1.2230948554874 \cdot 0.008 \cdot 35.07}} = 7.56 \operatorname{m/s}</math>
+
: <math>v_T = \sqrt{\frac{2 \cdot 6.674 \cdot 10^{-11} \cdot 5.2915793 \cdot 10^{22}}{600000^2 \cdot 1.2002 \cdot 0.00696 \cdot 35.07}} = 8.18 \operatorname{m/s}</math>
  
 
=== Examples ===
 
=== Examples ===

Revision as of 14:14, 26 July 2013

Planets
TinyEve.png Eve
TinyKerbin.png Kerbin
TinyDuna.png Duna
TinyJool.png Jool
Moons
TinyLaythe.png Laythe

The atmosphere of a celestial body slows the movement of any object passing through it, a force known as atmospheric drag (or simply drag). An atmosphere also allows for aerodynamic lift. The celestial bodies with atmospheres are the planets Eve, Kerbin, Duna and Jool, as well as Laythe, a moon of Jool. Only Kerbin and Laythe have atmospheres that contain oxygen.

Atmospheric pressure diminishes exponentially with increasing altitude. An atmosphere's scale height is the distance over which atmospheric pressure changes as a factor of e, or 2.718. For example, Kerbin's atmosphere has a scale height of 5000 m, meaning the atmospheric pressure at altitude n is 2.718 times greater than the pressure at altitude n + 5000.

Atmospheres vary in temperature, though this has no bearing on gameplay.

Atmosphere allows aerobraking and easier landing. When atmosphere contains oxygen, it allows jet engines to work. However, atmosphere makes taking off from a planet more difficult and increases a stable orbit altitude.

Drag

File:Ml16-XL parachute.JPG
A Mk1-2 pod with a Mk16-XL parachute being slowed by drag in Kerbin's atmosphere.

In the game, the force of atmospheric drag (FD) is modeled as follows:[1]

F_{D}=0.5\,\rho \,v^{2}\,m\,d\,A

where ρ is the atmospheric density (kg/m3), v is the ship's velocity (m/s), m is the ship's mass (kg), d is the coefficient of drag (dimensionless), and A is the cross-sectional area (m2).

Note that the m term is not present in the real-word drag equation. In the game, this causes acceleration due to drag (a = FD / m) to be unaffected by a ship's mass. (It also causes the units of the drag equation to have an extra "kg" term.)

ρ can be derived from atmospheric pressure (p of unit atm), which is a function of the atmosphere's pressure at altitude 0 (p0) and scale height (H):

p=p_{0}\cdot e^{{-altitude/H}}
\rho =p\cdot 1.2002\cdot 0.00696

The 1.2002 value can be derived from the ideal gas law (assuming that T is always 293.15K as it is inconclusive if Temperature is important) and the factor of 0.00696 is just the way the Kerbal universe works. This yields

\rho ~=p\cdot {\frac  {101{\text{kPa}}}{1{\text{atm}}\cdot 287.058{\text{J/(kg·K)}}\cdot 293.15{\text{K}}}}\cdot 0.00696


The coefficient of drag (d) is calculated as the mass-weighted average of the max_drag values of all parts on the ship. For most ships without deployed parachutes, d will be very near 0.2, since this is the max_drag value of the vast majority of parts.

As an example, the coefficient of drag for a craft consisting simply of a Mk1-2 Command Pod (mass 4, drag 0.2) and a deployed Mk16-XL Parachute (mass 0.3, drag 500) is:

{\frac  {4\cdot 0.2+0.3\cdot 500}{4+0.3}}=35.07

Note that the game does not calculate the ship's cross-sectional area (A) and assumes a value of 1 instead. This means the net force of drag on a spacecraft is unaffected by the width or orientation of a spacecraft, even if the spacecraft has very large wings. (Wings do affect the force of lift, however.)

Terminal velocity

The terminal velocity of an object falling through an atmosphere is the velocity at which the force of gravity is equal to the force of drag. Terminal velocity changes as a function of altitude. Given enough time, an object falling into the atmosphere will slow to terminal velocity and then remain at terminal velocity for the rest of its fall.

Terminal velocity is important because:

  1. It describes the amount of velocity which a spacecraft must burn away when it is close to the ground.
  2. It represents the speed at which a ship should be traveling upward during a fuel-optimal ascent.

The force of gravity (FG) is:

F_{G}=m\,a=m\,{\frac  {GM}{r^{2}}}

where m is still the ship's mass, G is the gravitational constant, M is the mass of the planet, and r is the distance from the center of the planet to the falling object.

To find terminal velocity, we set FG equal to FD:

m\,{\frac  {GM}{r^{2}}}=0.5\,\rho \,v^{2}\,m\,d\,A
{\frac  {GM}{r^{2}}}=0.5\,\rho \,v^{2}\,d\,A
v=v_{T}={\sqrt  {{\frac  {2\,GM}{r^{2}\,\rho \,d\,A}}}}

Assuming d is 0.2 (which is a good approximation, provided parachutes are not in use) and given that A is 1, this simplifies to:

v_{T}={\sqrt  {{\frac  {10\,GM}{r^{2}\,\rho }}}}

For the Mk16 pod and parachute example pictured above, the drag coefficient is 35.07, so its terminal velocity at sea level on Kerbin (which is 600 km from Kerbin's center) is:

v_{T}={\sqrt  {{\frac  {2\,GM}{r^{2}\,\rho \cdot 35.07}}}}
\rho =1\cdot e^{{-0/5000}}\cdot 1.2002\cdot 0.00696
v_{T}={\sqrt  {{\frac  {2\cdot 6.674\cdot 10^{{-11}}\cdot 5.2915793\cdot 10^{{22}}}{600000^{2}\cdot 1.2002\cdot 0.00696\cdot 35.07}}}}=8.18\operatorname {m/s}

Examples

Altitude (m) vT (m/s)
Eve Kerbin Duna Jool Laythe
0 58.385 100.13 212.41 12.665 115.62
100 58.783 101.01 214.21 12.686 116.32
1000 62.494 109.30 231.16 12.876 122.83
10000 115.27 240.52 495.18 14.938 211.77

On-rails physics

If a ship is "on rails" (meaning it's further than 2.25 km from the actively-controlled ship) and its orbit passes through a planet's atmosphere, one of two things will happen based on atmospheric pressure at the ship's altitude:

  • below 0.01 atm: no atmospheric drag will occur — the ship will be completely unaffected
  • 0.01 atm or above: the ship will disappear

The following table gives the altitude of this 0.01 atm threshold for each celestial body with an atmosphere:

Body Altitude (m)
Eve 0
Kerbin 0
Duna 0
Jool 0
Laythe 0

Atmospheric height

The atmospheric height depend on the scale height of the celestial body and is where 0.000001th (0.0001 %) of the surface pressure is remaining so the atmospheric pressure at the border isn't constant. Technically a craft in Jool's orbit can get lower into the atmosphere (or the atmosphere starts from a higher pressure).

alt_{{{\text{atmospheric height}}}}=-ln\left(10^{{-6}}\right)\cdot {\text{scale height}}
p_{{{\text{atmospheric height}}}}=p_{0}\cdot 10^{{-6}}

Kerbin's atmosphere ends at 0.000001 atm and to calculate where the other celestial bodies should have the atmospheric height:

alt_{{{\text{atmospheric height (real)}}}}=-ln\left({\frac  {10^{{-6}}}{p_{0}}}\right)\cdot {\text{scale height}}

Notes

  1. http://forum.kerbalspaceprogram.com/showthread.php/5235-Atmospheric-drag?p=88804&viewfull=1#post88804