Difference between revisions of "Orbit darkness time"

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(*use vectorized image; -specific size; +description what those variables mean; -random case; *use mass instead of µ, because either mass or µ is given, but µ can be calculated if it isn't given; -specific units; +length of shadow (in text and table);)
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[[File:Orbit darkness.svg|thumb|Orbital darkness schematic]]
 
[[File:Orbit darkness.svg|thumb|Orbital darkness schematic]]
The '''orbit darkness time''' determines is the time a [[craft]] stays in the shadow of orbited [[Celestial body|object]]. To calculate the exact time depends on the current configuration of the orbit and the moons.
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The '''orbit darkness time''' determines the time a [[craft]] is staying in the shadow of an orbited [[Celestial body|object]]. The exact time depends on the current configuration of the orbit and the moons.
  
 
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 [[Battery|batteries]] are needed for a craft to remain powered during the dark portion of orbit.
 
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 [[Battery|batteries]] are needed for a craft to remain powered during the dark portion of orbit.

Revision as of 22:15, 22 September 2013

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Orbital darkness schematic

The orbit darkness time determines the time a craft is staying in the shadow of an orbited object. The exact time depends on the current configuration of the orbit and the moons.

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.

Result

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

where is the semi-major axis, the semi-minor axis, the specific angular momentum, the eccentricity, and the radius of the planet or moon. For reference these terms can be calculated by knowing the apoapsis (Ap), periapsis (Pe) and body to orbit:

  • , the apoapsis measured from the center of the body
  • , the periapsis measured from the center of the body
  • , the semi-major axis
  • , the semi-minor axis
  • , the eccentricity
  • , the gravitational parameter
  • , the specific angular momentum

The following parameters are the minimum requirements to calculate the values from above:

  • , planned apoapsis of the craft's orbit from the surface of the body
  • : planned periapsis of the craft's orbit from the surface of the body
  • : radius (not the diameter!) of body to orbit
  • : mass of body to orbit

The radius and mass of a body can be obtained by visiting the page of the body (e.g. Kerbin) or by visiting Kerbol System/Table which contain the mass and radius of all celestial bodies. The actual page of a body also shows the gravitational parameter directly so it doesn't need to be multiplied by G.

When using kilometer for the orbital parameters the gravitational parameter needs to be divided by 1000³. For example Kerbin has equatorial radius 600 000 m or 600 km. Its gravitational parameter is 3.5316000×1012 m3/s2 or 3.5316000×103 km3/s2.

For circular orbits this equation gets rather simple, which can act as an approximation for almost circular orbits.

Limitations

This method assumes the orbit is stable and elliptic. It also assumes the sun's rays are parallel across the orbiting planet, although all planets' shadow end within the sphere of influence. The method does not take into account darkness caused by eclipses of a different body than the orbited body, like orbiting Laythe but Jool blocks the sun. When orbiting a moon it is impossible to travel outside the moon's shadow.

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.

Application

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

where is the battery storage required in the electricity unit e, is the rate of energy use in units of e/sec, and is the darkness time in seconds.

The darkness time can also be used to calculate how many solar panels are needed to recharge the batteries before the next cycle of darkness. However, the time spent in sunlight is quite long, and typical recharge rates are much smaller than the power provided by even a single solar panel.

Examples

Orbiting Kerbin with 100 km circular orbit

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:

So at least 57.8 e of electricity is 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 0.21.1[outdated]) is the Z-100 with 100 e of storage.

Orbiting Jool with apoapsis 700 km and periapsis 200 km

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:

So at least 505 e of electricity is needed in storage to make it through the darkness for this orbit around Jool, using 0.21 e per second.

Darkness times for planets and moons

The following table contain the darkness times for the planets and moons. The orbit is circular and has an altitude of 20% of the body's radius.

Orbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/RowOrbit Darkness Time/Row
Body Darkness time Orbital period Altitude above sea level Length of shadow