Orbital mechanics

Starlink configurations

Posted on by John

My nephew recently told me about being on a camping trip and seeing a long line of lights in the sky. The lights turned out to be Starlink satellites. It’s fairly common for people report seeing lines of these satellites.

Four lights in the sky in a line

Why would the satellites be in a line? Wouldn’t it be much more efficient to spread them out? They do spread out, but they’re launched in groups. Satellites released into orbit at the same time initially orbit in a line close together.

It would seem the optimal strategy would be to spread communication satellites out evenly in a sphere. There are several reasons why that is neither desirable or possible. It is not desirable because human population is far from evenly distributed. It’s very nice to have some coverage over the least-populated places on earth, such as Antarctica, but there is far more demand for service over the middle latitudes.

It is not possible to evenly distribute more than 20 points on a sphere, and so it would not be possible to spread out thousands of satellites perfectly evenly. However there are ways to arbitrarily many points somewhat evenly, such as in a Fibonacci lattice.

It’s also not possible to distribute satellites in a static configuration. Unless a satellite is in geostationary orbit, it will constantly move relative to the earth. One problem with geostationary orbit is that it is at an altitude of 42,000 km. Starlink satellites are in low earth orbit (LEO) between 300 km and 600 km altitude. It is less expensive to put satellites into LEO and there is less latency bouncing signals off satellites closer to the ground.

Satellites orbit at different altitudes, and altitude and velocity are tightly linked. You want satellites orbiting at different altitudes to avoid collisions, they’re orbiting at different velocities. Even if you wanted all satellites to orbit at the same altitude, this would require constant maintenance due to various real-world departures from ideal Keplerian conditions. Satellites are going to move around relative to each other whether you want them to or not.

Related posts

The vis-viva equation

Posted on by John

The vis-viva equation greatly simplifies some calculations in orbital mechanics. It is reminiscent of how conservation of energy can sometimes trivialize what appears to be a complicated problem. In fact, the vis-viva equation is derived from conservation of energy, but the derivation is not trivial. Which is good: the effort required in the derivation implies the equation is a shortcut to a place that might take longer to arrive at starting from first principles.

The term vis viva is Latin for “life force” and was applied to mechanics by Liebnitz around 350 years ago. The vis-viva equation is also known as the vis-viva law or the vis-viva integral.

The vis-viva equation says that for an object orbiting another object in a Keplerian orbit

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

Here v is the relative velocity of the two bodies, r is the distance between their centers of mass, and a is the semi-major axis of the orbit. The constant μ is the standard gravitational parameter. It equals the product of the gravitational constant G and the combined mass M of the two bodies.

In practice it is often accurate enough to let M be the mass of the larger object; this works for GPS satellites circling the earth but would not be adequate to describe Charon orbiting Pluto.

For a circular orbit, r = a and so v² = μ/r. Then r is constant and v is constant.

More generally a is constant but r is continually changing, and the vis-viva equation says how r and v relate at any given instant. This shows, for example, that velocity is smallest when distance is greatest.

For an object to escape the orbit of another, the ellipse of its orbit has to become infinitely large, i.e. a → ∞. This says that if v is escape velocity, v² = 2μ/r. For a rocket on the surface of a planet, r is the radius of the planet. But for an object already in a high orbit, escape velocity is less because r is larger.

Related posts

GPS satellite orbits

Posted on by John

GPS satellites all orbit at the same altitude. According to the FAA,

GPS satellites fly in circular orbits at an altitude of 10,900 nautical miles (20,200 km) and with a period of 12 hours.

Why were these orbits chosen?

You can determine your position using satellites that are not in circular orbits, but with circular orbits all the satellites are on the surface of a sphere, and this insures that certain difficulties don’t occur [1]. More on that in the next post.

To maintain a circular orbit, the velocity is determined by the altitude, and this in turn determines the period. The period T is given by

T = 2\pi \sqrt{\frac{r^3}{\mu}}

where μ is the “standard gravitational parameter” for Earth, which equals the mass of the earth times the gravitational constant G.

The weakest link in calculating of T is r. The FAA site says the altitude is 20,200 km, but has that been rounded? Also, we need the distance to the center of the earth, not the altitude above the surface, so we need to add the radius of the earth. But the radius of the earth varies. Using the average radius of the earth I get T = 43,105 seconds.

Note that 12 hours is 43,200 seconds, so the period I calculated is 95 seconds short of 12 hours. Some of the difference is due to calculation inaccuracy, but most of it is real: the orbital period of GPS satellites is less than 12 hours. According to this source, the orbital period is almost precisely 11 hours 58 minutes.

The significance of 11 hours and 58 minutes is that it is half a sidereal day, not half a solar day. I wrote about the difference between a sidereal day and a solar day here. That means each GPS satellite returns to almost the same position twice a day, as seen from the perspective of an observer on the earth. GPS satellites are in a 2:1 resonance with the earth’s rotation.

(But doesn’t the earth rotate on its axis every 24 hours? No, every 23 hours 56 minutes. Those missing four minutes come from the fact that the earth has to rotate a bit more than one rotation on its axis to return to the same position relative to the sun. More on that here.)

Update: See the next post on the mathematics of GPS.

[1] Mireille Boutin, Gregor Kemperc. Global positioning: The uniqueness question and a new solution method. Advances in Applied Mathematics 160 (2024)

Increasing speed due to friction

Posted on by John

Orbital mechanics is fascinating. I’ve learned a bit about it for fun, not for profit. I seriously doubt Elon Musk will ever call asking me to design an orbit for him. [1]

One of the things that makes orbital mechanics interesting is that it can be counter-intuitive. For example, atmospheric friction can make a satellite move faster. How can this be? Doesn’t friction always slow things down?

Friction does reduce a satellite’s tangential velocity, causing it to move into a lower orbit, which increases its velocity. It’s weird to think about, but the details are worked out in [2].

Note the date on the article: May 1958. The paper was written in response to Sputnik 1 which launched in October 1957. Parkyn’s described the phenomenon of acceleration due to friction in general, and how it applied to Sputnik in particular.

Related posts

[1] I had a lead on a project with NASA once, but it wasn’t orbital mechanics, and the lead didn’t materialize.

[2] D. G. Parkyn. The Effect of Friction on Elliptic Orbits. The Mathematical Gazette. Vol. 42, No. 340 (May, 1958), pp. 96-98

Music of the spheres

Posted on by John

The idea of “music of the spheres” dates back to the Pythagoreans. They saw an analogy between orbital frequency ratios and musical frequency ratios.

HD 110067 is a star 105 light years away that has six known planets in orbital resonance. The orbital frequencies of the planets are related to each other by small integer ratios.

The planets, starting from the star, are labeled b, c, d, e, f, and g. In 9 “years”, from the perspective of g, the planets complete 54, 36, 24, 16, 12, and 9 orbits respectively. So the ratio of orbital frequencies between each pair of consecutive planets are either 3:2 or 4:3. In musical terms, these ratios are fifths and fourths.

In the chord below, the musical frequency ratios are the same as the orbital frequency ratios in the HD 110067 system.

Here’s what the chord sounds like on a piano:

hd11067.wav

Related posts

Set of orbits with the same average distance to sun

Posted on by John

Suppose a planet is in an elliptical orbit around the sun with semimajor axis a and  semiminor axis b. Then the average distance of the planet to the sun over time equals

a(1 + e²/2)

where the eccentricity e satisfies

e² = 1 − b²/a².

You can find a proof of this statement in [1].

This post will look at the set of all orbits with a fixed average distance r to the sun. Without loss of generality we can choose our units so that r = 1.

Clearly one possibility is to set a = b = 1 so the orbit is a circle. The distance is constantly 1, so the average is 1.

We can also maintain a distance of 1 by reducing a but increasing the eccentricity e. The possible orbits of average distance 1 satisfy

a(1 + e²/2) = 1

with 0 < b ≤ a ≤ 1. A little algebra shows that

b = √(3a² – 2a),

and that 2/3 < a ≤ 1. As a approaches 2/3, b approaches 0.

Let’s put the center of our coordinate system at the sun and assume the other focus of the elliptical orbits is somewhere along the positive x-axis. When e is 0 we have a unit circle orbit. As e approaches 1, the orbits approach a horizontal line with the sun on one end.

Related posts

[1] Sherman K. Stein. “Mean Distance” in Kepler’s Third Law. Mathematics Magazine, Vol. 50, No. 3 (May, 1977), pp. 160–162

Dutton’s Navigation and Piloting

Posted on by John

This morning Eric Berger posted a clip from The Hunt for Red October as a meme, and that made me think about the movie.

I watched Red October this evening, for the first time since around the time it came out in 1990, and was surprised by a detail in one of the scenes. I recognized one of the books: Dutton’s Navigation and Piloting.

Screen shot with Dutton's Navigation and Piloting

I have a copy of that book, the 14th edition. The spine looks exactly the same. The first printing was in 1985, and I have have the second printing from 1989. So it is probably the same edition and maybe even the same printing as in the movie. I bought the book last year because it was recommended for something I was working on. Apparently it’s quite a classic since someone thought that adding a copy in the background would help make a realistic set for a submarine.

My copy has a gold sticker inside, indicating that the book came from Fred L. Woods Nautical Supplies, though I bought my copy used from Alibris.

Here’s a clip from the movie featuring Dutton’s.

Dutton’s has a long history. From the preface:

Since the first edition of Navigation and Nautical Astronomy (as it was then titled) was written by Commander Benjamin Dutton, U. S. Navy, and published in 1926, this book has been updated and revised. The title was changed after his death to more accurately reflect its focus …

The 14th edition contains a mixture of classical and electronic navigation, navigating by stars and by satellites. It does not mention GPS; that is included in the latest edition, the 15th edition published in 2003.

Related posts

Oval orbits?

Posted on by John

Johannes Kepler thought that planetary orbits were ellipses. Giovanni Cassini thought they were ovals. Kepler was right, but Cassini wasn’t far off.

In everyday speech, people use the words ellipse and oval interchangeably. But in mathematics these terms are distinct. There is one definition of an ellipse, and several definitions of an oval. To be precise, you have to say what kind of oval you have in mind, and in the context of this post by oval I will always mean a Cassini oval.

Ellipses and ovals each have two foci, f1 and f2. Let d1(p) and d2(p) be the distances from a point p to each of the foci. For an ellipse, the sum d1(p) + d2(p) is constant. For an oval, the product d1(p) d2(p) is constant.

In [1] the authors argue that just as planetary orbits are nearly circles, they’re also nearly ovals. This post will look at how far the earth’s orbit is from a circle and from an oval.

We need a way to specify which oval we want to compare to the ellipse of earth’s orbit. We’ll do this by equating the major and minor semi-axes of the two curves. These are usually denoted a and b, but the same variables have a different meaning in the context of ovals, so I’ll denote them by M for major and m for minor.

The equation of an ellipse is

(x/M)² + (y/m)² = 1

and the equation of an oval is

((x + a)² + y²) ((x – a)² + y²) = b².

Setting x = 0 in the equation of an oval tells us

m² = b – a²

and setting y = 0 tells us

M² = b + a².

So

b = (M² + m²)/2

and

a² = (M² – m²)/2.

For the earth’s orbit, M = 1.00000011 and m = 0.99986048 measured in AU, astronomical units. So or oval has parameters

a = 0.011816102

and

b = 0.99986060.

If you plot Kepler’s ellipse and Cassini’s oval for earth’s orbit at the same time, you can’t see the difference.

Planet orbits are nearly circular. If we compare a circle of radius 1 AU with Kepler’s ellipse we get a maximum error of about 1 part in 10,000.

But if we compare Cassini’s oval with Kepler’s ellipse we get a maximum error of about 1 part in 100,000,000.

 

In short, a circle is a good approximation to earth’s orbit, but a Cassini oval is four orders of magnitude better.

It would be difficult to empirically distinguish an ellipse from an oval as the shape of earth’s orbit, but theory is clearly on Kepler’s side since his ellipses fall out of Newton’s laws. Cassini’s error was more qualitative than quantitative.

More orbital mechanics posts

[1] Kepler’s ellipse, Cassini’s oval and the trajectory of planets. B Morgado1 and V Soares. 2014 Eur. J. Phys. 35 025009 DOI 10.1088/0143-0807/35/2/025009

Solar Day vs Sidereal Day

Posted on by John

How long does it take the earth to complete one rotation on its axis? The answer depends on your frame of reference. A solar day is the time it takes for the sun to appear at the same position in the sky. A sidereal day is the time it takes for a distant star to appear in the same position. These are not the same.

This post will illustrate the difference between a solar day and a sidereal day. To make things a little simpler, assume the earth has a perfectly circular orbit around the sun, and that the earth’s axis of rotation is perpendicular to the orbital plane, i.e. there is no axial tilt.

Sidereal day

Imagine an astronomer Alice observing our solar system from the vantage point of a distant star. How distant? We’ll show below that it doesn’t make much difference, but we’ll assume for now that she is “infinitely” far away, which means “far enough away that we don’t have to worry about exactly how far.”

The time it takes for Alice to observe one rotation of the earth on its axis is 1 sidereal day = 24 sidereal hours. We will suppose that from Alice’s perspective it takes 360 (sidereal) days for the earth to orbit the sun.

Here comes the sun

Let’s set the origin of our coordinate system at the sun. Assume that at time t = 0 the earth is located at (1, 0) and that an observer Bob is at the bottom of a deep well on the equator looking up at the sun. The time between Bob’s observations of the sun is one solar day.

Twenty four (sidereal) hours later, the earth is located at (cos 1°, sin 1°) and Bob’s well is parallel to the x-axis, but not looking directly at the sun. He will be looking at the sun a few minutes later when the earth’s rotation brings the sun into view.

Just how long will Bob have to wait to see the sun again? About 4 minutes, because the sun is 1° away from his line of sight, and he needs the earth to turn 1°, which takes 24/360 hours, or 4 minutes. This is not exactly correct though, because the earth has moved in its orbit during that 4 minutes.

So a solar day is about 4 minutes longer than the time it takes for the earth to rotate on its axis (from Alice’s perspective). Even though we’ve made several simplifying assumptions, our estimate only differs from the exact value by about 4 seconds.

Equations

Let the distance from the center of the earth to the sun be 1 and the radius of the earth be ε. Let t be time in (sidereal) days. The position of the center of the earth as a function of time will be

(cos t°, sin t°)

and Bob’s position is

(cos t° − ε cos 360t° , sin t° − ε sin 360t°).

Bob sees the sun at time t = 0. When will he see the sun next? When the slope of the line from the center of the earth to his position equals the slope of the line from the sun to the center of the earth, i.e.

tan t° = tan 360t°

The value of ε doesn’t matter.

There are two solutions to this equation, one when Bob is facing the sun and another when he is on the opposite side of the earth from the sun. We know t ≈ 1 and the solution near 1 is on the correct side of the sun, i.e.

t = 1.002786 days = 1 day + 4.011142 minutes.

Distant stars

Now let’s imagine at that time t = 0, while Bob is looking up at the sun, on the opposite side of the earth Charlie at the bottom of another well looking up a star a distance R away, located at (R + 1, 0).

Charlie’s position as a function of t is

(cos t° + ε cos 360t° , sin t° + ε sin 360t°).

When will Charlie see his star next? When the line from the center of the earth through his position has the same slope as the line from Charlie to the star. This happens when

tan 360t° = sin t° / R,

the right side above being the tangent of the angle at R of a right triangle with base on the x-axis and hypotenuse running from the star to Charlie. As R goes to infinity, the right side goes to 0, and the solutions for t are integer numbers of days.

Charlie will see the star at time slightly less than 1, so let t = 1 − x. So we need to solve

tan 360(1 − x)° = − tan x° = − sin (1 − x)°/R.

Using the approximation sin θ ≈ θ ≈ tan θ for small angles θ (in radians) we have

−2πx ≈ −2π(1 − x)/(360R)

and so

x = 1/(360R + 1).

Now R is very large. For the nearest star, Proxima Centauri, R is about 270,000 AU. So x is on the order of 10−8 or smaller. This is why it doesn’t matter which star Alice is located near: a sidereal day is essentially the same whatever distant star you use as your reference point.

Sphere of influence

Posted on by John

Apollo 11 orbit diagram via NASA

Suppose a spaceship is headed from the earth to the moon. At some point we say that the ship has left the earth’s sphere of influence is now in the moon’s sphere of influence (SOI). What does that mean exactly?

Wrong explanation #1

One way you’ll hear it described is that the moon’s sphere of influence is the point at which the earth is no longer pulling on the spaceship, but that’s nonsense. Everything has some pull on everything else, so how do you objectively say the earth’s pull is small enough that we’re now going to call it zero? And as we’ll see below, the earth’s pull is still significant even when the spaceship leaves earth’s SOI.

Wrong explanation #2

Another explanation you’ll hear is the moon’s sphere of influence is the point at which the moon is pulling on the spaceship harder than the earth is. That’s a better explanation, but still not right.

The distance from the earth to the moon is about 240,000 miles, and the radius of the moon’s SOI is about 40,000 miles. So when a spaceship first enters the moon’s SOI, it is five times closer to the moon than to the earth.

Newton’s law of gravity says gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance. The mass of the earth is about 80 times that of the moon. So at the moon’s SOI boundary, the pull of the earth is 80/25 times as great as that of the moon, about three times greater.

Correct explanation

So what does sphere of influence mean? The details are a little complicated, but essentially the moon’s sphere of influence is the point at which it’s more accurate to say the ship is orbiting the moon than to say it is orbiting the earth.

How can we say it’s better to think of the ship orbiting the moon than the earth when the earth is pulling on the ship three times as hard as the moon is? What matters is not so much the force of earth’s gravity as the effect of that force on the equations of motion.

The motion of an object between the earth and the moon could be viewed as an orbit around earth, with the moon exerting a perturbing influence, or as an orbit around the moon, with the earth exerting a perturbing influence.

At the boundary of the moon’s SOI the effect of the earth perturbing the ship’s orbit around the moon is equal to the effect of the moon perturbing its orbit around the earth. It’s a point at which it is convenient to switch perspectives. It’s not a physical boundary [1]. Also, the “sphere” of influence is not exactly a sphere but an approximately spherical region.

The moon has an effect on the ship’s motion when it’s on our side of the moon’s SOI, and the earth still has an effect on its motion after it has crossed into the moon’s SOI.

Calculating the SOI radius

As a rough approximation, the SOI boundary is where the ratio of the distances to the two bodies, e.g. moon and earth, equals the ratio of their masses to the exponent 2/5:

r/R = (m/M)2/5.

This approximation is better when the mass M is much larger than the mass m. For the earth and the moon, the equation is good enough for back-of-the-envelope equations but not accurate enough for planning a mission to the moon. Using the round numbers in this post, the left side of the equation is 1/5 = 0.2 and the right side is (1/80)0.4 = 0.17.

Context

Everything above has been in the context of the earth-moon system. Sphere of influence is defined relative to two bodies. When we spoke of a spaceship leaving the earth’s sphere of influence, we implicitly meant that it was leaving the earth’s sphere of influence relative to the moon.

Relative to the sun, the earth’s sphere of influence reaches roughly 600,000 miles. You could calculate this distance using the equation above. A spaceship like Artemis leaves the earth’s sphere of influence relative to the moon at some point, but never leaves the earth’s sphere of influence relative to the sun.

Related posts

[1] The sphere of influence sounds analogous to a continental divide, where rain falling on one side of the line ends up in one ocean and rain falling on the other side ends up in another ocean. But it’s not that way. I suppose you could devise an experiment to determine which side of the SOI you’re on, but it would not be a simple experiment. An object placed between the earth and the moon at the SOI boundary would fall to the earth unless it had sufficient momentum toward the moon.