Going to Mars? Then you'd better plan ahead and book early, because the best departure times fill up fast.
And you'll have to choose between the first-class express or the slower economy passage. And whatever you do, unless it's an absolute emergency, don't get rooked into taking one of those May-June or Oct-Nov departures, they'll cost you an arm and a leg, on top of being a little on the slow side.

What am I yammering about?

Just the fact that when you leave for Mars has a large impact on the trip.

First of all, lets assume that we don't have delta V to burn (as is the case now) and we want to go to Mars. the most economical way to do it is by using a Hohmann transfer orbit, which is basically an elliptical orbit which has its perihelion at Earth's orbit and its aphelion at Mars'. Of course, such an orbit only works if the timing is correct (It doesn't do you much good to get to Mars' orbit if Mars is on the other side of the Sun at the time of your arrival). This limits your launch window. ( Of course if Delta-V was no problem, launch oppurtunities wouldn't be so limited. The more Dv you have to play with, the greater your options)

To figure out how much delta V you need for such a trip and the time it would take you do the following:

Earth's mean distance from the Sun is 149.6 million km

Mars' mean distance from the Sun is 227.94 million km.

An orbit using these two values as its perihelion and aphelion would have a semi-major axis of

a = (149.6+227.94)/2 or

188.77 million km.

The period of such an orbit would be :

T = 2 * sqrt(pi^2*(1.8877e11)^3/ (6.672e-11)*(1.99e30)) or

4.47e7 sec or

518 days.

(Where the distance is measured in meters and NeX is shorthand for Nx10^X)

To get from Earth to Mars you only have to travel 1/2 of a complete orbit so the trip time would be equal to:
259 days.

The delta V needed to achieve the transfer orbit from Earth orbit is:

dV = 29785*(sqrt(2*2.2794e11/(2.2794e11+1.496e11))-1)

or
2945 m/s

where 29785 is the mean orbital speed of the Earth in meters/sec.

The only problem with these numbers is that they are calculated for a transfer orbit from one circular orbit to another, and neither the Earth nor Mars travel in perfect circles. So while they are good "ballpark" figures, the exact figures have to take this into account.

Earth's distance from the Sun varies by 5 million km, and it orbital velocity by 1000m/s over the course of a year, and Mars' by 43 million km and 4500m/s over its year. This complicates the problem somewhat.

In order to work out the proper values, we first must determine how the orbits of Earth and Mars align. Earth's perihelion occurs around Jan 1, and using Earth's orbit as reference, Mars perihelion aligns with Aug. 25.

Then by working out the Earth's distance from the Sun at launch time and Mar's distance at arrival time, we can plug these values into the formula for determining travel time. One result would be the chart below.

Launch Travel Time.
Date (days)

Jan. 25 241
Feb. 25 239
Mar. 25 243
Apr. 25 251
May. 25 263
Jun. 25 275
Jul. 25 284
Aug. 25 287
Sep. 25 282
Oct. 25 271
Nov. 25 259
Dec. 25 248

As you can see, the trip times vary by as much as 43 days, with the shortest trip being for a launch date of Feb. 25, and the longest at Aug. 25.

The next thing to tackle is the Delta V requirements for each launch date, but before we do, there are a couple of things to discuss first.

The first thing is that the delta V needed to attain the transfer orbit is only part of the delta V needed to get you to Mars, assuming you want to do more than just a fly-by. Once you get to Mars' vicinity you are going to have to match speeds with it in order to go into orbit around it. Mars' own gravity will help some here but can't do it all.

One good way to determine just how closely you have to match Mars' velocity to assure capture is to determine the radius of Mars' gravitational sphere of influence(GSI). This is the distance where Mar's gravitational influence over rides that of the Sun's.

This can be calculated using the formula:

R = Dsm * (Mm/Ms)^(2/5)

where Dsm is the distance from the Sun to Mars,
Mm is the mass of Mars
Ms is the mass o f the Sun

Notice that as Dsm gets larger or smaller, the GSI follows suit. This means that the GSI is smaller at perihelion than at aphelion.

The next thing we do is calculate the escape velocity for Mars at the distance of R. This will gives us the minimum velocity we can have relative to Mars for capture. (Anything that enters the GSI at a speed greater than this can't be captured.) Since escape velocity goes down as you get further from the planet, a smaller GSI means you can have a larger escape velocity and vice versa. This makes it easier to match speeds at Perihelion than Aphelion.

Using our "ballpark" transfer orbit from above we can give an example.

R = 2.2794e11 *(6.4219e23/1.99e30)^(2/5) = 5.77e8 m = 577000 km

Vesc = sqrt(2*6.672e-11*6.4219e23/5.77e8) = 385 m/s

The mean orbital velocity of Mars is 24800 m/s

and the orbital velocity of the transfer orbit when it reaches Mars is:

21485 m/s

for a difference of

3315 m/s

so the total delta V we need to apply for capture is:

3315 - 385 = 2930 m/s

adding this to the 2945 m/s we needed to attain the transfer orbit gives a combined Delta v for the trip of

5875 m/s.

Again, this is only a "ballpark" figure and varies from launch date to launch date.

Another factor to consider is the relative directions of the paths of the ship and the planets.

Since Mars and Earth travel in ellipses around the Sun and not circles, the direction of their orbital velocities are not always at a right angle to the a line joining them with the Sun. (In fact his only happens during aphelion and perihelion.) the shape and nature of the transfer orbit requires that the ship, however does have such a right angle velocity when leaving Earth and again when approaching Mars. (This makes sense when you consider that it's at perihelion when it leaves Earth and aphelion when it reaches Mars.) The variation between the direction vectors at launch and arrival add to the delta V needed for both. If launch or arrival coincides with the respective planet's aphelion or perihelion this correction is 0, but become quite significant if it happens half-way between.

Okay, now to determine the needed Delta v for attaining the tranfer orbit we do the following (I'll leave the actual equations out):

First determine the orbital distance from the Sun for the Earth on the launch date
Then detemine same same for Mars on the arrival date.
This determines the tranfer orbit and allows you to determine The aphelion and perihelion velocites for this orbit.
Determine the orbital velocitity and direction of the Earth at launch date,
Do the same For Mars at the arrival date.
Find the vector difference between the Earth's velocity and the tranfer orbits perihelion velocity
Find the same difernce between Mars' velocity and the aphelion velocity.
Combine the two answers.

Putting it all together, here is a table that shows the travel time and combined Delta V for different launch dates.


Launch Travel Time. Delta V
Date (days) (m/s)

Jan. 25 241 5365
Feb. 25 239 5221
Mar. 25 243 5418
Apr. 25 251 5783
May. 25 263 6001
Jun. 25 275 5887
Jul. 25 284 5427
Aug. 25 287 5134
Sep. 25 282 5435
Oct. 25 271 5847
Nov. 25 259 5943
Dec. 25 248 5708


From this you can see that the shortest trip has a launch date of Feb. 25, but the most economical (by a small amount) in terms of delta V is Aug. 25. The savings for Aug. 25 comes from the fact that Mars is at aphelion upon arrival and thus has its lowest orbital velocity, and the ship's and Mars' paths parallel each other. This makes for a lower velocity difference between the ship and planet. The delta V requirements go up quickly on either side because as Mars moves away from aphelion, its orbital speed increases and you have to start correcting for the differences in the direction of velocities of the ship and planet.

It is the correction for these variations in direction that drive the Delta v cost up so severely for May and Nov. launches, even though the travel times are around the median.

For a manned mission, it would appear that Feb. 25 would be the best launch date the small amount of extra delta-v is more than made up for by the shorter travel time(Which would result in a lower launch weight since fewer consumables would have to be on board.)

Unfortunately, the proper line up for this optimum launch date only occurs every 15 to 17 years, so like I said, book early!

Janus