IspExample

This example is a real-world calculation I performed at work recently. I've substituted fake numbers to protect the innocent.

> import qualified Prelude
> import Numeric.Units.Dimensional.Prelude
> import Numeric.Units.Dimensional.NonSI (poundMass, gee)

When deorbiting (retiring) a geostationary spacecraft you normally want to raise its altitude (above the geostationary orbit) by a certain amount. A typical target amount would be 300 kilometers. In the satellites propellant budget the satellite manufacturer would have reserved the necessary propellant to reach this altitude.

> deorbitHeight     = 300   *~ kilo meter
> deorbitPropellant =  20.3 *~ poundMass  -- American manufacturer.

The dry mass of the spacecraft is the weight of the spacecraft without any expendables (i.e. propellant).

> dryMass = 1534.2 *~ kilo gram

However, when performing the deorbit the spacecraft will not be completly dry. Some propellant (in liquid and gas form) will be trapped inside the spacecraft. In addition there will be some uncertainty in how much propellant is actually available onboard the spacecraft. We must assume these masses are still onboard but unusable (a conservative assumption).

> withheldPropellant = m_trapped + m_2sigma
>   where
>       m_trapped = 14.0 *~ poundMass
>       m_2sigma  = 23.2 *~ poundMass  -- 2 sigmas is acceptable here.

The total mass we will be deorbiting is the dry mass, the withheld masses and the propellant reserved for the deorbit (which will be expended during the deorbit, but we'll conservatively assume it remains on board).

> massAtDeorbit = dryMass + withheldPropellant + deorbitPropellant

In order to change the orbit (mean) altitude by a certain amount the velocity of the spacecraft must be changed. In satellite operations the term delta-V or DV is often used for a velocity change. In the vicinity of the goestationary orbit the following formula can be used.

> deorbitDV :: Velocity Double
> deorbitDV = phi / _2 * deorbitHeight
>   where
>       phi = 360.985647 *~ (degree / day) -- Earth's rate of rotation.

Based on the mass of the spacecraft we will be deorbiting and the propellant required to reach the deorbit altitude we can extract the specific impulse the satellite manufacturer expected for the deorbit "burns".

> i_sp :: Time Double
> i_sp = massAtDeorbit * deorbitDV / (g * deorbitPropellant)
>   where
>       g = 1 *~ gee -- Earth's gravitational acceleration at surface.

> main = print i_sp