# How to calculate the maximum heliocentric speed during a flyby?

The following graphs are related to the Voyager 2 / Jupiter flyby occurred on July 1979 (I did the calculation with the SPICE library and NAIF’s data files). We see that the maximum speed relative to Jupiter (Vjup) is when the s/c is at the periapsis (as expected), while the maximum heliocentric speed (Vsun) is about 8.54 h after the periapsis. Why? In other words, suppose that I know only the position of all the bodies (I don’t know the speed), is there any way to calculate the maximum heliocentric speed using only the gravitational accelerations?

The question comes from the fact that if I plot the position of the bodies, I don’t see any particular geometry to justify a peak in the heliocentric speed when the s/c is in that particular position: The points are plotted with a 2-hour constant step size. The red points represent Jupiter and the s/c when the s/c is at the periapsis, while the two magenta points are for the maximum s/c heliocentric speed.

As an additional example, here are the graphs for the Rosetta / Earth flyby in the ecliptic plane of J2000: and in the XZ plane (not to scale to amplify the Z axis): The maximum heliocentric speed is reached about 15 min after the perigee, but, again, I don’t see any particular geometry to justify the fact that the peek is reached in that particular position and not 1 minute before or after.

This question is slightly related to the question Why is the highest speed that Voyager 2 achieved from the Jupiter gravity assist not at perijove?

• What if you try to calculate the vector of the gravitational pull of the planet to the spacecraft? Then take the component of that vector working in the flight direction of the spacecraft. This component should change its sign from positive (acceleration of the spacecraft) to negative at the time of maximum heliocentric speed. The direction of the pull should be vertical to the trajectory of the spacecraft at that time. – Uwe Jul 3 '20 at 20:44
• I'm probably wrong in calculating the component in the flight direction. Now I calculate the projection of the total acceleration vector on the unit velocity vector (relative to Sun): it works! Thank you. – Cristiano Jul 3 '20 at 22:16