Hohmann transfer

Leafnode, CC BY-SA 2.5, via Wikimedia Commons

The first type of transfer orbit is Hohmann transfer. As shown above the Hohmann transfer is basically an elliptical orbit that is tangent to both the inner and outer circle at its apse line. Let us denote radius of inner circle by R and outer circle by R’. Then the periapse (closest point from focus) and apoapse (farthest point from focus) of the transfer ellipse will be the radii of the inner and outer circles, i.e. R and R’ respectively.

We can see from the diagram above that the space-craft fly through only one-half of the transfer ellipse. It can be either from the inner circle to outer circle or vice versa. One important thing to note here is that the Hohmann transfer is the most energy efficient two-impulse maneuver for transferring the spacecraft or satellite between two coplanar circular orbits having a common focus.

Now let us have a look at how the orbit will be transferred. If the spacecraft is needed to transfer from inner circle to outer circle, then some extra energy will be required to boost the spacecraft or the satellite. So at the starting of transfer ellipse a velocity increment of ∆v is required in the direction of

flight to boost the satellite onto the higher-energy elliptical trajectory.

After travelling half the transfer ellipse, another forward velocity increment of ∆v’ is required to transfer the satellite in the outer circular orbit. If the satellite was not provided the velocity increment of ∆v’, then it would have travelled on the Hohmann transfer ellipse and eventually returned to the same initial point again. The total energy required will be equal to the total ∆v requirement or in mathematical form we can write ∆v total = ∆v + ∆v’.

What if we want the satellite to transfer orbit from outer circle to inner circle? Nothing different, in this case the thrust will be applied in the reverse direction to decrease the energy of the spacecraft, so that it can transfer to lower energy orbit. Note that the magnitude of will still remain the same as before.

Bi-elliptic Hohmann transfer

AndrewBuck, CC BY-SA 4.0, via Wikimedia Commons

In bi-elliptical Hohmann transfer a spacecraft or satellite uses two coaxial semi-ellipses, as shown in green and orange color in the above figure. One of these two ellipses is tangent to the inner circular orbit, while the other one is tangent to the outer circular orbit, and they both are tangent to each other at point 2, which is basically the apoapse (farthest point from focus) of both ellipses.

The actual logic behind bi-elliptic Hohmann transfer is to place the point 2 far away from the focus so that at 2 became very small. Interestingly if the radius of outer circle r3 is less than about 11.9 times that of the radius of inner circle (r1), then the Hohmann transfer will be more energy efficient.

And if the radius of outer circle r3 is more than about 15 times that of the radius of inner circle (r1), then the bi-elliptical hohmann transfer will be more energy efficient. In between these two values of radius, if the value of the r2 is larger, then the bi-elliptical transfer will be more favourable, and if r2 is smaller, then the Hohmann transfer will be more favourable.

Phasing maneuver

Esburgos, CC BY-SA 4.0, via Wikimedia Commons

From the figure shown above we can say that phasing maneuver is basically a two-impulse Hohmann transfer initiating from the orbit with a common periapse. Phasing maneuvers are mainly used to change the position of a spacecraft in its orbit.

If two spacecraft or satellite are at different locations in the same orbit, then one of them may perform a phasing maneuver in order to reach the other one. Phasing maneuver is mainly performed by communication and weather satellites in geostationary earth orbit to move to some new location.

If the target satellite is ahead of the chasing satellite, then the above shown phasing orbit can be used to return to periapse point in less period of time. In this case a thrust in opposite direction will be required to slow down the spacecraft in order to speed it up, relative to the main orbit. The reverse will be done if the chaser satellite is ahead of the target.

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