On November 21, 2021, at 08:24 UTC, NASA’s Parker Solar Probe will be at perihelion in another of its orbits around the sun, and it is now diving toward its innermost reaches for another hair-raisingly close and hot rendezvous. If solar plasma is your thing, this is your instrument! Parker’s mission purpose is to measure electric and magnetic fields near the sun and learn more about the solar wind: that elusive stream of particles that plasma physicists love to ponder about. It was named after Eugene Parker, who as a solar physicist at the University of Chicago in the 1950s was curious to find out how stars, including the sun, give off energy. He called the complex mess of plasma, magnetic fields, and energetic particles the “solar wind.” Most of the solar wind is composed of electrons, protons, and helium ions. And he came up with a counterintuitive idea that the part of the solar atmosphere called the corona would be hotter than the surface of the sun itself.

Launched in the summer of 2018, the PSP is itself a technical marvel. Highly engineered to withstand the intense heat of the sun, it will come closer than any other spacecraft to it and is the fastest man-made machine ever to fly. It sports a big four-and-a-half-inch thick white carbon-composite heatshield and instruments that can measure the splats of solar particles and dust, smeared like bugs on a windshield during a desert road trip. Its solar limb sensors keep the heatshield automatically aimed at the sun, acting as a life-saving parasol to its precious water-cooled electronics. Its orbital planning too, was a marvel. Venus lends its momentum on select Parker flybys to slingshot the craft ultimately to within six million kilometers from the sun’s surface, closer than ever before.

I invite you to look at PSP’s impressive website, http://parkersolarprobe.jhuapl.edu/ which gives you a real-time picture of where it is and how fast it is going. As I write this, it is only 21 million kilometers from the sun and closing fast. Its heatshield temperature is 427°C.

Of course, I like to dig into the orbital mathematics a little to check the numbers and see if they make sense to me and put things in context. Galileo said, “nature’s great book is written in mathematical language.” Certainly, the best way to learn about the physical world is to engage with it mathematically. It’s a way of participating in what you’re reading about. All the equations I’m going to use are doable on a pocket calculator or iPhone [1].

Here though, the nuts-and-bolts orbital data from web sources been able to find is somewhat lacking (no doubt because the orbit is being tweaked according to NASA's plan). All we have from those sources is Parker’s perihelion and aphelion distances and its orbital period. That’s about it. (That is, the apsidal distances in its orbit where it is closest to the sun and farthest from the sun, respectively [2].) We do have data from the website on its current speed and distance from the sun, and its anticipated speed distance at perihelion which we can use to check our results [3]. In cases like this it sometimes takes a little bit of sleuthing to learn more, a process I call “forensic mathematics.” Of course, this will be a rough cut: we don’t have exact numbers to work with but hopefully we’ll get a fair idea of what’s going on.

*The Shape of Parker’s Orbit*

Let’s begin with what we have. I learned that Parker’s *aphelion* distance is 109 million kilometers, and its *perihelion* distance is 6.9 million kilometers. (We know that the orbit will be adjusted by NASA in the course of time, but we’ll stick with these numbers for now.) As we’ll see, though, these two numbers can tell us quite a bit. First, we convert them into astronomical units, the astronomer’s standard measuring stick, my dividing each number by 149.6 million kilometers, the mean distance between the earth and the sun, which by definition is one *au*. Thus:

Perihelion distance *P* = 0.046123 au. Aphelion distance *A* = 0.728611 au.

Right away we note something interesting about PSP’s aphelion distance: it’s about the distance of Venus from the sun [4]. This is expected given that the mission planners intended Venus to act as Parker’s occasional catapult, nudging it into ever tighter orbit. And from Parker’s perihelion and aphelion numbers there’s a convenient equation that will tell us the *eccentricity* of its orbit, revealing its shape:

I get an eccentricity for Parker’s orbit of 0.88093. This is far from circular. The most eccentric planet in the solar system is Mercury, with an eccentricity of only a fourth of that, at .2056. Venus’s eccentricity is a boring 0.0067, not all that far from a circle’s zero eccentricity. At the other extreme, kamikaze-like comets from the Oort cloud occasionally dive into the solar system looking for the sun with hairpin orbits around it with eccentricities of .9999 etc., awfully close to the break-open shape of a parabola at 1.0 eccentricity.

Anyway, now that we have eccentricity, we can find the length of the *semimajor axis* *a* of Parker’s elliptical orbit. The semimajor axis is half its long axis. It tells us how big the orbit is. It can be found by this little equation:

For this I get *a* = 0.387366 au. While we are at it, let’s find Parker’s *semi-minor axis* dimension *b. *That’s the length of the short axis of the ellipse. Here’s a simple equation for it, using the same values for Parker’s perihelion and aphelion distances that we began with:

Which comes out to *b* = 0.183318 au. We’re starting to get a sense of the shape of the orbit; the long axis is about twice the length of the short axis. Here is sketch of Parker’s orbit in relation to the orbits of Venus and Mercury that I generated with mathematical software using Parker’s orbital parameters that we determined above [5].The sun is the small dot in the center:

*Parker’s Orbital Period *

With just the semimajor axis and Kepler’s Harmonic Law, we can now find the *period* T of Parker’s orbit. Kepler in 1619 found that each of the planets’ orbital periods squared is related to each of their distances from the sun cubed. Also known as Kepler’s Third Law, it was a remarkable discovery with significant consequences in the development of mathematical astronomy. (I talked about it in my June 22, 2021, post about Juno and Ganymede.) The equation for the period is this:

where *T* is Parker’s orbital period in days and *a* is the semimajor axis that we found above. I find that T = 88.05885 days for Parker to go around the sun once. This squares with the website’s value. Interestingly, this is almost exactly the orbital period of Mercury, which takes 87.969 days to round the sun. And this *must* be the case. Why? Because the equation defines the orbital period *solely* by the semimajor axis distance *a*, which for Parker almost exactly matches that of Mercury (whose *a *= 0.3871). So its period must likewise mimic Mercury’s period. We might describe Parker’s orbit as a flattened Mercury-orbit, squashed out to touch Venus’s orbit on one side and to brush the sun on the other.

*Parker’s Amazing Perihelion Velocity*

Now let’s look at Parker’s speed. NASA says that the probe “is the fastest human-made object.” I wanted to see what that means. Do you mean faster than a bullet? Faster than the reentry of a spacecraft from orbit? We have an elegant equation it tells us the velocity of an object in an elliptical orbit at perihelion:

where *e* is eccentricity, *P* is perihelion distance, and *k* = 29.7847, a proportionality constant for heliocentric orbits that converts the answer into kilometers per second. Inserting the appropriate ingredients into this mathematical recipe, we get a perihelion velocity of 190.2 km/s, which is almost 16 times faster than its perihelion speed of 12.04 km/s. (The aphelion equation is the same as the above, except with a negative sign in the numerator and the aphelion distance *A* in the denominator.)

Well, 190 kilometers per *second* is *very* fast, over 425,000 mph! An AR-15 bullet travels at something like 2,200 miles an hour, so the probe is 200 times faster than that. The calculated perihelion velocity is consistent with the PSP website, but I wanted to know how it compares with other fast things in our solar system. Here’s a chart showing Parker’s perihelion velocity in relation to some other well-known celestial visitors:

As you can see, its velocity at perihelion fits comfortably within the ranges of non-man-made objects that have also come screamingly close to the sun.

The increasing velocity of Parker as it falls toward the sun can be neatly graphed using the so-called *vis viva* equation, which I analyzed in detail in my May 2020 post on ‘Oumuamua. It is a famous old equation, and I used it to graph Parker’s velocity against its distance from the sun:

It will be fun to monitor Parker in the days ahead to see the changes in its heatshield temperature as it rounds the sun. There will be many more orbits to go in the coming years, including two more Venus flybys, in August 2023 and November 2024, as Parker edges closer to the boiling furnace!

NOTES

[1] As you will see, the equations are surprisingly simple. Their derivations can be found in my *Newton's Gravity*: *An Introductory Guide to the Mechanics of the Universe* (Springer, 2012)*.*
[2] The endpoints of an orbital ellipse are called its *apsides*.
[3] If you look at the NASA website, http://parkersolarprobe.jhuapl.edu/, you can see that Parker's orbit is and will be being adjusted according to plan by ground engineers, including by additional Venus flybys. Hence, we would not expect a set of fixed, planet-like orbital parameters. But what we know today can give us a good sense of the dynamics of the mission.
[4] Venus’s distance from the sun varies in its nearly circular orbit between .728 au at aphelion and .7185 au at perihelion.
[5] Technical note: A more accurate depiction would have included traditional orbital elements discussed in previous posts, such as the longitude of Parker’s ascending node and of its perihelion. The simple configuration of the orbital shapes in the diagram are in any event adequate for the purposes of this article.

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