# Single-Impulse Coplanar Orbit Transfer

This model represents the change in orbit when a Δv is applied. Orbit 0 is the initial orbit some time before the Δv is applied. Orbit 0i is the initial orbit the moment before the Δv is applied. Orbit 1i in the new orbit the moment after the Δv is applied. Orbit 1 is the new orbit some time later.

Press «enter» to see the model. To infer results, try one of the examples below, or your own inputs. To clear results, refresh your browser.

## Circular orbit to GTO (geosynchronous transfer orbit)

- A satellite begins in a 300 km circular orbit. Specify zero eccentricity and 300 km altitude by typing:

0i.e = 0 «enter» 0i.h = 300 «enter»

The model infers the initial orbital characteristics.

- The ‘cheapest’ GTO has periapsis at our initial altitude of 300 km and apoapsis at geosynchronous altitude of 35,786 km:

1i.hp = 300 «enter» 1i.ha = 35786 «enter»

The model infers that this takes a ΔV of 2.4 km/s. - How long after the ΔV does it take to reach geosynchronous altitude? Setting the altitude of orbit 1 to apoapsis and telling the model to assume a fractional orbit:

1.h==ha = true «enter» 1.as = true «enter»

The model infers that the duration from point 1i to point 1 (Δt₁.h), takes 5.3 hours, which is exactly half the period of this GTO (1i.T.h).

## Rocket cutoff to circular orbit

- A microwave thermal rocket, let’s say, reaches 7.3 km/s at 80 km altitude with a flight-path angle of 10°. What is the ΔV needed to circularize the orbit at 500 km? To calculate this, start by specifying the cutoff conditions in orbit 0:

0.v = 7.3 «enter» 0.h = 80 «enter» 0.g = 10 «enter» 0.as = t «enter»

The model infers orbit 0 (type 0 «enter» to see). For example, the eccentricity is 0.2 and periapsis altitude is -1950 km, meaning that the rocket will not complete a full orbit and instead reenter. The apoapsis is 549 km.

- Only a ΔV at 500 km can nudge the payload into a circular orbit at this altitude (the before and after orbits have to intersect). Luckily, the previous step found that the peak altitude (apoapsis) is above 500 km, so circularization is possible. The desired circular orbital altitude of 500 km can be entered by typing:

h = 500 «enter» 1i.ic = t «enter» 0i.io = t «enter»

The latter entry specifies that the ΔV occurs while height is still increasing (as opposed to decreasing when it comes down again later). The model infers that the rocket takes 8 minutes after cutoff to reach 500 km, and that it then takes a ΔV of 1 km/s with the nose of the rocket pointed 26° below horizontal (γ_{ΔV}= -26°) for the orbit to be made circular.