How To Reach Other Planets - A Picture Guide! (aka How to Calculate Hohmann Transfers)

[EtherDragon]

Here it is - the step by step - point by point - bit by detailed bit picture guide to calculating your own Hohmann Transfers.

Part 1: What is a Hohmann Transfer?

Simply put, a Hohmann Minimal Energy Transfer Orbit, commonly referred to as a Hohmann Transfer, is a special orbit used to transfer from one planetary orbit, like Kerbin, to another planetary orbit, like Duna. This math tutorial will make it dirt simple for you to calculate your own Hohmann Transfers from any planet to any other planet. Have a look at this:

The solid lines are the orbits of the planets in question, in this case, Kerbin and Duna.

The dashed line is the proposed Hohmann Transfer.

Looking at this we see that we're starting at Kerbin at about 3 o-clock and we want to meet up with Duna at 9 o-clock. The trick is, we need to figure out how much of Duna's orbit will Duna traverse, during the same time our ship is traversing the Hohmann Transfer. Time for some math!

Part 2: The Math, Part One - The Big Equation

The equation below is derived simply from Kepler's Third Law of Planetary Motion which says, "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit." In mathematical terms:

r = p²/a³

where:

r is the ratio, which is constant for any system

p is the orbital period

and a is the semi-major axis

This means that for any two planets the following equation must be true:

p₁²/a₁²=p₂²/a₂³

On other words the ratio for Duna's period squared to Duna's semi-major axis cubed, must the the same as the ratio for Kerbin.

This is extrapolated to any two orbits, not matter their eccentricity! What this means is that the same ratio holds true for a Hohmann Transfer.

Our goal is to solve for the unknown, which is which portion of Duna's Orbit will be completed during our transfer.

With that in mind, we do some algebra, to derive the final equation:

It looks more complicated than it really is, so don't sweat it too much. At this point, all we need to do is plug in some numbers.

Part 3: Gathering the Data

We need just two pieces if information to plug into the equations above. First, we need the semi-major axis of our starting point. In this example, we'll use Kerbin at 13.6 million-km. Second we need the semi-major axis if our destination, which is Duna at 20.7 million-km.

Now, we need to calculate our Hohmann Semi-Major Axis for use in the larger equation.

Part 4: The Math Part 2 - Hohmann Semi-Major Axis

The Hohmann Semi Major Axis is equal to the average of the Apoapsis and Periapsis. In math terms:

a_h=(h_a+h_p)/2

So we plug in our numbers:

(h_a+h_p)/2 = (20.7+13.6)/2 = 17.15

That's it! Now we can plug that into the final equation!

Part 5: The Math Part 3 - The Transfer Solution

In the final equation, we have some terms to fill in.

a_h is what we determined above in Part 4.

a_t is the semi-major axis of the target planet, in this case Duna at 20.7 million-km.

The rest is just plugging in numbers to work through them:

a_t³/a_h³ = 20.7³/17.15³ = ~1.7585

2*√1.7585 = ~2.6521

1/2.6521 = ~0.377

thus

r_t = ~0.377

Duna will traverse about 37.7% of its orbit during our Hohmann Transfer. Converting that to a clock-face basically says that we should depart when Duna is at about 1:30!

Now I leave it to you to determine the return trip from Duna back to Kerbin:

(I get 70.8% of Kerbin's Orbit, or a departure from Duna when Kerbin is at 5:30 on the clock...)

Coming soon - the Video version of this tutorial

[EtherDragon]

A follow up!

How did I come up with the equation above?

Well - let's start with Kepler's Third Law:

r=p²/a³

This defines that the ratio between the Period Squared and the Semi-Major Axis Cubed is always the same for all orbiting objects with the same center of mass. We can use that fact to find out some things:

Let's define the orbital period of Kerbin to "1 year" and plug in Kerbin's semi-major of 13.6 million-km. If we wanted to know how many orbits Duna completed during a single Kerbin orbit, we can set up the following equation:

p_k²/a_k³=p_d²/a_d³

Where

p_k is the period of Kerbin's orbit

a_k is the semi-major axis of Kerbin's orbit

p_d is the period of Duna's orbit

and a_d is the semi-major axis of Duna's orbit

Since we defined "1" as the Period for Kerbin we get:

1²/13.6³ = p_d²/a_d³

Simplify to:

1/13.6³ = p_d²/a_d³

We're looking for the Period of Duna compared to Kerbin - as that's the unknown, but to get there we need to plug in what we do know: Duna's semi-major axis is 20.7 million-km. Thus:

1/13.6³=p_d²/20.7³

20.7³/13.6³=p_d²

~3.526=p_d²

√~3.526=p

p=~1.878 Kerbin Years for each complete orbit of Duna.

Now, we can really do the same thing for a Hohmann Transfer since the following equation must be true:

p_h²/a_h³=p_t²/a_t³

Where

p_h is the period of our full transfer orbit

a_h is the semi-major axis of the transfer

p_t is the period of the target orbit, in this case Duna

and a_t is the semi-major axis of the target orbit

To get the general form that can be used for any two planets we start by defining our Hohmann Transfer period as "1":

1²/a_h³ = p_t²/a_t³

Solve for P:

a_t³/a_h³ = p_t²

√(a_t³/a_h³) = p_t

So we have how many of our orbits we will complete compared to Duna. Now we need to convert that into Duna orbits. Again, by setting our period to "1" we get this equation as the ratio between our Hohmann orbit and Duna's:

r_h = 1/p_t

Now our Hohmann Transfer is not a full orbit, it's a half of one:

r_t = 1/(2p_t)

Substituting in the full equation for P_t, above:

r_h = 1/(2√(a_t³/a_h³))

Fin.

Notes:

*1 - On Earth an AU is defined as the Semi-Major Axis for the Earth around the Sun.

Edited July 19, 2015 by EtherDragon
Math clarifications

[EtherDragon]

Hi! I added a Video Version of the first Tutorial:

[be77solo]

Thanks for this! Will give it a try this evening.