Orbit darkness time

Orbit Darkness Time

This page will give an expression for the worst-case scenario of how long a craft will be in darkness during an orbit. This information can be used to determine how many batteries are needed for a craft to remain powered during the dark portion of orbit.

Orbital Darkness Schematic

Result

Here is the result for calculating the longest amount of time spent in darkness (in units of seconds):

$\Delta t={\frac {2l^{2}}{h}}\left({\frac {e\,{\text{sin}}(\theta )}{(e^{2}-1)(e\,{\text{cos}}(\theta )+1)}}-{\frac {2\,{\text{tan}}^{-1}\left({\frac {(e-1){\text{tan}}\left({\frac {\theta }{2}}\right)}{\sqrt {1-e^{2}}}}\right)}{\sqrt {(1-e^{2})^{3}}}}\right)$

where you calculate the following intermediate quantities first:

$r_{a}=Ap+d$

$r_{p}=Pe+d$

$a={\frac {r_{a}+r_{p}}{2}}$

$b={\sqrt {r_{a}r_{p}}}$

$c={\sqrt {a^{2}-b^{2}}}$

$e={\frac {r_{a}-r_{p}}{r_{a}+r_{p}}}$

$l={\frac {2r_{a}r_{p}}{r_{a}+r_{p}}}$

$h={\sqrt {l\mu }}$

$x={\sqrt {a^{2}\left(1-{\frac {d^{2}}{b^{2}}}\right)}}$

$\theta ={\text{tan}}^{-1}\left({\frac {d}{x+c}}\right)$

where you need to provide the following parameters:

$Ap$: planned Apoapsis of planet the craft will orbit [km]

$Pe$: planned Periapsis of planet the craft will orbit [km]

$d$: radius of planet to orbit [km] (see wiki page on planet for data, ex: Kerbin)

$\mu$: gravitational parameter of planet to orbit [km^3/s^2] (see wiki page on planet for data, ex: Kerbin)

Limitations

This method assumes the orbit is stable and elliptic. This includes the special case of circular orbits. The method gives the longest amount of time spent in darkness, which for polar orbits, will only be experienced periodically. However, it is a good idea to plan on the worst-case amount of time in darkness. Lastly, the method does not take into account darkness caused by a different body than the orbited body (i.e. if you are orbiting Laythe and Jool blocks the sun, this method does not account for that).

Conclusion

By entering the equations into a program such as Microsoft Excel, the darkness time can be calculated for various orbits around any planet or moon. Knowing the amount of time spent in darkness and the energy drain, the battery storage can be calculated by

$B={\dot {E}}\Delta t$

where $B$ is the battery storage required in the game units of "e", ${\dot {E}}$ is the rate of energy use in units of e/sec, and $\Delta t$ is the darkness time in seconds.

Examples

Orbiting Kerbin with Apoapsis 700 km and Periapsis 100 km

$\Delta t=925{\text{ seconds (4m 43s)}}$

Using 0.05 e/s for the RC-001S and 0.04 e/s for the Illuminator Mk1 for a total drain of 0.09 e/s:

$B={\dot {E}}\Delta t=0.09{\text{ e/s}}*925{\text{ s}}=83.2{\text{ e}}$

So at least 83.2 e of electricity are needed in storage to make it through the darkness for this orbit around Kerbin, using 0.09 e per second. The battery with the next largest amount of storage (as of 0.21) is the Z-100 with 100 e of storage.

Orbiting Jool with Apoapsis 500 km and Periapsis 500 km

$\Delta t=2319{\text{ seconds (38m 39s)}}$

Using 0.05 e/s for the RC-001S and 0.16 e/s for 4x Illuminator Mk1 for a total drain of 0.21 e/s:

$B={\dot {E}}\Delta t=0.21{\text{ e/s}}*2319{\text{ s}}=487{\text{ e}}$

So at least 487 e of electricity are needed in storage to make it through the darkness for this orbit around Jool, using 0.21 e per second. This capacity can be achieved by coupling a Z-400 with a Z-100 for a total of 500 e of electricity.