Introduction
The previous post showed how simplified calculations of a spacecraft trajectory between Earth and Mars orbits could be based on little more than high-school math and physics. In the process it revealed that Mars doesn’t orbit fast enough for the spacecraft to rendezvous with it if Mars and the spacecraft start out in angular alignment. So in this post we’ll make the same calculation as before except that instead of starting Mars out in angular alignment with the spacecraft (and Earth) we’ll have it begin about 44° ahead. As a result the spacecraft will fly by Mars and thereby obtain a speed boost.
The Flyby
The plot above illustrates the resultant flyby. It shows that although with respect to the sun the spacecraft is still traveling counterclockwise (which at the time of the flyby would be right to left in our reference frame) its motion with respect to Mars at that time is clockwise, i.e., left to right. This is because although the spacecraft starts out faster than Mars the sun’s gravity slows it down enough as it travels away from the sun that it’s slower by the time it reaches Mars’ orbit.
The particular timing and thrust parameters we’ve chosen (by trial and error) so separate the spacecraft from Mars as to prevent Mars’ gravity from capturing it. (Actually, we’ve allowed it to approach so close that in real life drag from Mars’ atmosphere would probably slow it enough to cause a collision, but for the sake of simplicity we’re ignoring Mars’ atmosphere.) Fig. 5 below shows that Mars’ gravity nonetheless affects the spacecraft’s speed. The solid curve represents the speed at which the spacecraft approaches Mars. Mars’ gravity makes that approach speed increase slightly at first and then reverse and thereafter decrease in magnitude once the craft has passed Mars.
But that’s only the rate at which the distance to Mars changes. Since the spacecraft isn’t directed purely toward or away from Mars its velocity also has a tangential component. As the dashed curve shows, Mars’ gravitational pull causes the spacecraft speed’s tangential component to increase and then decrease as the craft approaches and then withdraws from the Red Planet. The dotted curve represents the result of combining those components to obtain the craft’s overall speed with respect to Mars.
The Slingshot Effect
Note that in Fig. 5 the overall speed is symmetrical with respect to the point of closest approach: no slingshot effect is apparent. That’s because Fig 5 depicts only speeds with respect to Mars. To observe the slingshot effect the speed we need to plot is instead the speed with respect to the sun:
Fig. 6 shows that as Mars is catching up to the spacecraft Mars’ gravitational pull initially slows the craft down but that once Mars has passed the craft it speeds the craft up. The spacecraft speeds up more than it slows down, because Mars tends to pull it toward Mars’ speed.
We’re calling the net speed increase about 2.4 km/sec. We arrived at that value in a manner we’ll now explain by reference to Fig. 7.
Fig. 7 shows that the spacecraft starts out behind Mars. It’s initially fast enough to pass Mars. But its outward travel against the sun’s gravity slows it down enough that Mars eventually overtakes it. In passing the spacecraft Mars bends the craft’s trajectory into a wider orbit, whose aphelion exceeds Mars’ orbital radius. The basis for Fig. 6’s “2.4 km/sec” legend is that when the spacecraft passes back through Mars’ orbit, at about the 7 o’clock position, it’s traveling about 2.4 km/sec faster that it traveled at that distance from the sun in the previous post’s calculation, in which Mars didn’t give it a boost.
So our calculations have demonstrated the slingshot effect.
They can also give us a sense of how significant the resultant 2.4-km/sec boost could be. Back when the rocket burn was injecting the spacecraft into its trans-Mars trajectory at the start of the trip the craft’s speed increase reached 2.4 km/sec at about 200 seconds into the 300-second burn. Given that the 300-second burn reduced the spacecraft mass by 62% and the rate of mass reduction during the burn was constant we might surmise that enough propellant to add 2.4 km/sec would take up something like 40% the pre-burn mass.
And it turns out that because gravitational force and thrust direction affected our calculations this somewhat understates the boost’s significance. A relationship we’ll encounter in due course tells us that without gravitational effects nearly 49% of the pre-burn mass would need to be propellant in order to obtain such a speed increase from rocket thrust.
Conclusion
By using nothing more than a little high-school math and physics we’ve demonstrated that a planet flyby can give a spacecraft a significant speed boost and thereby save considerable fuel. As we stated before, moreover, this basic approach of calculating acceleration from position, updating velocity from acceleration, and updating position from velocity would still apply if instead of assuming predetermined orbits for Earth and Mars and a stationary position for the sun we calculated those bodies’ changing positions, along with the spacecraft trajectory, from mutual gravitational attractions. Indeed, such a numerical approach is the only way to take all those factors into account.
But even without that complication numerical solutions such as we’ve employed here can involve calculating tens or hundreds of thousands of time intervals’ acceleration, velocity, and position values. If we’re willing to sacrifice accuracy we can avoid that computational cost—and perhaps gain more insight—by reducing the question to a two-body problem, which can be solved analytically, i.e., without the computation-intensive numerical approach we’ve been using. The next post will use that expedient to explore how much a change in mission parameters might increase the slingshot boost beyond the 2.4-km/sec value we obtained above.