How do I calculate ISP at a given pressure?

[Laie]

When looking at an engine's config, I see this:

		atmosphereCurve
		{
			key = 0 300
			key = 1 270
			key = 7 0.001
		}

I take this to be the ISPs at zero, one, and seven atmospheres, respectively.

How do I calculate the ISP this engine would have at (say) 0.35 or 2.6 atmospheres?

Edited May 28, 2015 by Laie

[kerbiloid]

http://forum.kerbalspaceprogram.com/threads/93332

[OhioBob]

The equation is a cubic Hermite spline, which has the basic form

p(t) = (2t³ - 3t² + 1) p_o + (t³ - 2t² + t) m_o + (-2t³ + 3t²) p₁ + (t³ - t²) m₁

Between the first and second keys we have

p_o = 300

p₁= 270

m_o = 270 - 300 = -30

m₁= ((270-300)+(0.001-270)/(7-1))/2*(1-0) = -37.4999

and, t = (pressure-0)/(1-0)

For example, if the pressure is 0.35 atm, we have

t = (0.35-0)/(1-0) = 0.35

p(0.35) = (2*0.35³-3*0.35²+1)*300 + (0.35³-2*0.35²+0.35)*-30 + (-2*0.35³+3*0.35²)*270+(0.35³-0.35²)*-37.4999 = 290.0972

Between the second and third keys we have

p_o = 270

p₁= 0.001

m_o = ((270-300)+(0.001-270)/(7-1))/2*(7-1) = -224.9995

m₁= 0.001 - 270 = -269.999

and, t = (pressure-1)/(7-1)

For example, if the pressure is 2.6 atm, we have

t = (2.6-1)/(7-1) = 0.26666667

p(0.26666667) = (2*0.26666667³-3*0.26666667²+1)*270 + (0.26666667³-2*0.26666667²+0.26666667)*-224.9995 + (-2*0.26666667³+3*0.26666667²)*0.001+(0.26666667³-0.26666667²)*-269.999 = 204.4535

Edited May 28, 2015 by OhioBob