Sorry for the really bad picture

Sorry for the really bad picture

I am looking at some data from the Parker Solar Probe mission and I was looking at the speeds of PSP at perihelion and aphelion at each new, different, orbit (the orbits caused by a gravity assist of which there are 8) and I found something which I can't explain. Firstly the speeds at perihelion increase after each orbit which makes sense because PSP is getting closer and closer to the Sun (shown in the picture v3>v2>v1). Then I noticed that the speeds at aphelion actually decrease after each new orbit (v6>v5>v4) which I don't understand because the aphelion positions are also getting closer and closer to the Sun. I was thinking that the only way for these speeds to be decreasing would be because of the gravity assist. If this is the case I still don't understand why these speeds would be decreasing since the Venus assist should get less effective as PSP gets faster (not more effective which is what is suggested by the fact that the speeds are decreasing each new orbit). In fact the assist wouldn't just have to be getting slightly more effective but quite a bit more effective in order to not only decrease the speed more than the previous orbit, but do so with PSP moving much faster than before (each new orbit is closer to the Sun). As far as I know the factors that effect the magnitude of the delta-v from a gravity assist are the hyperbolic excess speed (where, generally, lower speed is more desirable), periapsis altitude (which stays fairly constant for each gravity assist - there are a couple outliers but I see no trend with those outliers and the speed of aphelion data), and gravitational parameter (constant). On top of this, even if the gravity assist was capable of getting more effective and decreasing the speeds more and more, this would mean that Venus would always have to be positioned before aphelion in order to decrease its speed by the time it reaches aphelion which is also something I think is unlikely. Does anyone know what may be happening here? *** all my knowledge on this subject is from the internet so I might be missing something really simple ***

This is the velocity at perihelion and aphelion data where the first row is orbit 1, the second is orbit 2, third is orbit 3, ..., and the last is orbit 8. There appears to be similarities with the 2nd and 3rd orbits and the 5th and 6th which is also possibly important. I know that the pairs of gravity assists 1st and 2nd, 3rd and 4th, 5th and 6th occur at roughly the same position of Venus in its orbit but even this doesn't seem to be what is reflected by the similarities between orbits 2 and 3, 5 and 6 in the data below.

v_peri = 84.362, 95.293, 109.0, 129.348, 148.007, 162.669, 176.786, 190.492
v_apo  = 17.239, 16.865, 16.213, 15.04, 13.988, 13.253, 12.577, 11.987

This is the velocity at perihelion and aphelion data where the first row is orbit 1, the second is orbit 2, third is orbit 3, ..., and the last is orbit 8. There appears to be similarities with the 2nd and 3rd orbits and the 5th and 6th which is also possibly important. I know that the pairs of gravity assists 1st and 2nd, 3rd and 4th, 5th and 6th occur at roughly the same position of Venus in its orbit but even this doesn't seem to be what is reflected by the similarities between orbits 2 and 3, 5 and 6 in the data below.

This is a table of all the data I calculated (the perihelion and aphelion distances I found from NASA).

enter image description here

I also found a graph of PSP's speed over time and it displays the same thing that I found: despite the speed at perihelion increasing, the aphelion speed is slightly decreasing.

I have the idea that this decreasing in speed might be the reason why PSP is able to keep its velocity going into the gravity assist fairly constant (I also calculated this) despite its speed at perihelion increasing significantly. I calculated the incoming v_inf 's of PSP for each gravity assist and they are all approximately 22.9 km/s despite the fact that PSP would be getting a lot more speed from the Sun after each assist (I have a weird exception in the speeds for the 6th flyby which I found is 43.7 km/s but I suspect something is wrong with this data since it also has an eccentricity of 140 000). I was wondering how PSP is able to keep such a constant v_inf going into the assist and now I guess it has something to do with it decreasing speed at aphelion.

PSP velocity vs time (orange)

  • $\begingroup$ I wonder if it would be possible to calculate the change in orbital energy and figure out whether there is a discrepancy larger than the change rendered by the gravitational assists. $\endgroup$ – BMF Jul 11 '20 at 0:27
  • $\begingroup$ @BMF So your suggesting to calculate the difference in orbital energies (E = -GM/2a) of 2 orbits, say the 1st and 2nd, and then the difference in orbital energies of the two corresponding hyperbolic orbits cause by the gravity assists? (which should be the same formula except without the negative sign I think). I don't really understand why this would be useful to do. Could you clarify how these energies might help explain the decrease in aphelion speeds? $\endgroup$ – Alexander Ivanov Jul 11 '20 at 1:18
  • $\begingroup$ You may find the following useful (unfortunate paywall): sciencedirect.com/science/article/pii/S0094576520306512 $\endgroup$ – honeste_vivere Jan 19 at 22:44

Another way to express the results of the vis-viva equation at apoapsis is $${v_a}^2 = \frac{2\mu}{r_p+r_a}\frac{r_p}{r_a} = \frac{2\mu}{r_a}\frac{r_p}{r_p+r_a}\tag{1}$$ where $v_a$ is the velocity at apoapsis, $\mu$ is the standard gravitational parameter $\mu\equiv GM$, and $r_a$ and $r_p$ are the apoapsis and periapsis distances.

On holding the apoapsis distance $r_a$ constant, the first term on the right hand side of equation (1), $\frac{2\mu}{r_a}$, is constant, making the apoapsis velocity solely vary according to latter term on the right hand side of equation (1), $\frac{r_p}{r_p+r_a}$. This latter term monotonically increases as the periapsis distance $r_p$ increases from zero to $r_a$.

Alternatively, this latter term monotonically decreases as $r_p$ decreases from $r_a$ to zero. In other words, decreasing periapsis distance while holding apoapsis distance constant results in a decrease in apoapsis velocity.

  • $\begingroup$ Yet another way to look at it is to solve for $r_p$ given $v_a$, $r_a$, and $\mu$:$$r_p = \frac{{v_a}^2 r_a}{\frac{2\mu}{r_a}-{v_a}^2}$$Reducing the apoapsis velocity reduces the periapsis distance. Yet another way to look at it: The delta v needed to make an object dive into the Sun is a good deal more than is the delta v needed to make an object escape the solar system. $\endgroup$ – David Hammen Jul 11 '20 at 7:25
  • $\begingroup$ Oh wow this is a nice way of understanding it. The data actually shows that the apoapsis distance is also decreasing (I added in a table of all my data). However, the apoapsis distances is decreasing at a much slower rate than the periapsis distance and so it makes sense that the overall apoapsis velocity should be decreasing (and not by much, which is exactly what is seen). Thank you! $\endgroup$ – Alexander Ivanov Jul 11 '20 at 7:51

Just a short supplementary to try and add a more "intuitive" understanding to the two excellent "equation-based" answers.

For me the easiest way to think of this is that you have cause and effect reversed in your description of the problem. Consider PSP at aphelion, which is always more or less at Venus distance from the Sun, so that it can use Venus' gravity to shape its orbit. It is moving "around" the Sun (since it's at aphelion, it is momentarily getting neither closer or further away). Where it goes next is determined by how fast it is moving. If it was moving fast enough it would stay in a circular orbit, with the Sun's gravity pulling it around just enough to stop it getting further away, but not enough to bring it closer. (this is what Venus does, near enough). If it is going less fast than that, then the Sun's gravity bends its trajectory in more sharply, so it ends up curving "in" towards the Sun. As it moves closer, gravity both bends and accelerates its trajectory, until in the end it speeds up enough that it stops getting closer to the Sun (perihelion) and then starts to move away again. The slower it was moving at aphelion, the more "falling in" it needs to do to reach perihelion, so the closer to the Sun it gets, but, it turns out, the faster it will be going at perihelion. So this is what the equations are showing -- for a fixed aphelion distance, the slower you are going at aphelion, the lower you perihelion will be, but, in fact, the faster you will be going at perihelion.

  • $\begingroup$ Thank you for this explanation! It seems so obvious now. In order to keep bending the orbit enough to get PSP to be closer and closer to the Sun, it must be moving slower at aphelion as otherwise it would simply have enough speed to make a more circular orbit than before (and the goal is to be more eccentric to get closer to the Sun). $\endgroup$ – Alexander Ivanov Jul 11 '20 at 15:21

When thinking about the speeds and distances in a Keplerian orbit we turn to our friend the vis-viva equation:

$$v^2 = GM \left( \frac{2}{r} - \frac{1}{a} \right)$$

where $v$ is the speed at distance $r$ for an object with a semi-major axis $a$ and $GM$ is the gravitational constant $G$ times the Sun's mass M. We can call that product the standard gravitational parameter of the Sun and it's about 1.327E+20 m^3/s^2.

The distances at periapsis and apoapsis $r_p, r_a$ are given by

$$r_p = a(1-\epsilon)$$ $$r_a = a(1+\epsilon)$$

where $\epsilon$ is the eccentricity of the orbit. Put these together and the speeds at periapsis and apoapsis $v_p, v_a$ are

$$v_p^2 = \frac{GM}{a} \frac{1+\epsilon}{1-\epsilon}$$

$$v_a^2 = \frac{GM}{a} \frac{1-\epsilon}{1+\epsilon}$$

Since you have two equations and two unknowns for each orbit, in principle you can solve for the semimajor axis and the eccentricity for each row in your table. You can imagine that as long as both parameters can vary, you can make the speed at aphelion either increase or decrease even though the speed at perhihelion decreases.

You can get a big hint from the fact that the orbit starts circular and ends very elliptical and suppose that eccentricity generally increases, and remember that Parker needs to regularly access Venus's orbit each time it moves closer to the Sun since it uses Venus' gravity rather than propulsion to lower its perihelion, so it's probably a good first guess to assume that the orbit keeps a roughly constant aphelion and its perihelion and eccentricity both decrease over time.

An orbit with a fixed apoapsis and decreasing periapsis will have increasingly slower speed at apoapsis so that it "falls closer" to the Sun.

  • $\begingroup$ Yes I actually used that formula to get the speeds at perihelion and aphelion. I only had the values for the distance to perihelion and aphelion for each orbit so I used that to get a and e and solve for the velocities. And when you mentioned that the apoapsis would remain at the same position, the data I have (from NASA) shows that it doesn't. The aphelion distance definitely decreases for each orbit and so does the velocity at aphelion, I will put in the whole table of data that I made into the question. $\endgroup$ – Alexander Ivanov Jul 11 '20 at 6:20
  • $\begingroup$ So I added in the table and you can see that r_a decreases while v_a also decreases which is what I am confused about. Since this point is getting closer to the Sun how could it be possible that PSP is moving slower at it. $\endgroup$ – Alexander Ivanov Jul 11 '20 at 6:24
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    $\begingroup$ Right, I suppose my question is not specifically about PSP. Thank you for thinking about it. Another idea I had (I'm kinda just guessing though) is that maybe it is to do with the fact that the closer PSP is to the Sun, the farther down it is in the Sun's gravitational well. And in that case I guess it could be possible that the farther down you go, the "steeper" the travel back up is and so the harder it is to go back up. So I was thinking maybe thats why it would be slower for an aphelion that is actually closer (it has to climb out of a steeper part of the well). $\endgroup$ – Alexander Ivanov Jul 11 '20 at 7:25
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    $\begingroup$ That idea came when I was thinking about those pictures of how relativity describes gravity (same principle as a trampoline). But again I have no idea how accurate that model is or really anything about it, it's just a thought. $\endgroup$ – Alexander Ivanov Jul 11 '20 at 7:27
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    $\begingroup$ @AlexanderIvanov If you keep apoapsis near Venus for gravity assists for orbit lowering and the potential energy at apoapsis therefore stays the same, then the only way to get closer to the Sun is to reduce total energy, which means kinetic energy at apoapsis must be lower which means velocity at apoapsis must be lower. $\endgroup$ – uhoh Jul 11 '20 at 8:24

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