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When an artificial satellite orbits the Earth, the time it takes to make one complete orbit (its period) depends upon the the size of its orbit, following Kepler’s laws. The larger the satellite’s orbit, the longer its orbital period. This means if you make the orbit just the right size, the period can be 24 hours. If you make the orbit circular, and align it with Earth’s equator, then as the Earth rotates on its axis the satellite will orbit the Earth at the same rate. For someone standing on Earth, the satellite will appear to be parked in the sky. While the stars, Sun and Moon rise and set, the satellite will appear stationary. Such an orbit is known as a geostationary orbit.

Example of geostationary orbits. Credit: Wikipedia

The idea for geostationary satellites is widely attributed to science fiction writer Arthur C. Clarke, but several people had proposed the idea before him. Clarke popularized the idea and its possible use for communication satellites in a 1945 article in Wireless World. The first geostationary satellite was launched in 1964. While the first satellites placed in geostationary orbits were communication satellites, their use is limited by the fact that it takes about half a second for a radio signal to travel from Earth to the satellite and back. Over the years broadcast satellites and imaging satellites have been placed in geostationary orbits. Weather satellites are often placed in the orbit because their fixed location makes it easy to observe weather patterns as they form. You can see this in the video above from the Himawari 8 weather satellite.

In principle a geostationary satellite could remain in such an orbit for quite some time, but gravitational tugs from the Sun and Moon, as well as the fact that Earth is not a perfect sphere means that satellites must make orbit adjustments every now and then. Since this takes fuel, this limits the effective lifetime of a geostationary satellite. Usually when a geostationary satellite reaches the end of its useful life the remaining fuel is used to put it into an orbit further from the Earth, and is sometimes known as a graveyard orbit. This clears the satellite from the geostationary belt around Earth’s equator.

Paper: Arthur C. Clarke. Extra-Terrestrial Relays: Can Rocket Stations Give World-wide Radio Coverage? Wireless World. October 1945.

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Today marks the point in Earth’s orbit where it is closest to the Sun, known as perihelion. It might seem odd for those in the northern hemisphere for this to occur in the middle of winter, but that’s because the seasons are due largely to the tilt of Earth’s axis relative to its orbital plane. The variation in Earth’s distance from the Sun is minor in comparison, because Earth’s orbit is almost (but not quite) a perfect circle.

A perfect circle (left) compared to Earth’s orbit (right).

The difference between perihelion (closest to the Sun) and aphelion (most distant) for Earth is just under 5 million kilometers. That’s quite a distance, but it is only about 3% of Earth’s average distance. Treating Earth orbit as a perfect circle is actually a good approximation. The same is true for the other planets, which is part of the reason why the whole “orbits are circles” idea of early astronomy lasted as long as it did. Of course that raises the question as to just why planetary orbits are so nearly circular.

There are likely several factors that played a role in forming circular orbits. One is that as early protoplanets form, more elliptical orbits would tend to cross the paths of other bodies, and collision with them would tend to produce more circular orbits. Then there is the fact that large planets such as Jupiter tend to form early on, and their gravitational influence will tend to clump smaller bodies into groups. It could also be due to orbital resonances between smaller bodies, which would tend to stabilize orbits. Then there are effects such as the Kozai mechanism, when tend to align orbits into a common plane. Then there is the fact that circular orbits are more stable against gravitational perturbations than elliptical orbits.

As we’re learning from computer simulations, the dynamics of solar systems are quite complex. But simulations do show that nearly circular orbits are quite common, and studies of exoplanets also show nearly circular orbits in many cases. So it seems that throughout the universe circular orbits are a good approximation for planets.

But it isn’t exact, which is why we can celebrate today as our closest day to the Sun. It also happens to be Isaac Newton’s birthday, which is perhaps fitting given his work on planetary orbits.

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Of all the inner planets, only Mercury hasn’t had a probe land on it. It likely won’t for quite some time. The reason isn’t because of lack of desire, or worthy science to be done, but because of a simple thing known as delta-v.

Delta-v, or change in velocity, is the amount of speed a spacecraft need to gain or lose in order to reach a destination. If you launch a spacecraft from Earth, it is basically moving at the speed of Earth in its orbit around the Sun. To get to another planet you have to either increase your speed to reach Mars and the outer planets, or lose speed to reach Venus or Mercury.

There’s two ways to get delta-v. One is by simply using fuel. Fire rockets in the right direction and you can speed up or slow down as necessary. The only downside is that the more you need delta-v, the more you need fuel, and that adds mass and cost to your spacecraft. Another way is to make a close fly-by of a planet. Basically, if you approach a planet in the direction of its orbit (coming up from behind, if you will), then the gravity between the planet and your spacecraft will cause the spacecraft to speed up at the cost of slowing down the planet by a tiny, tiny amount. Making a flyby in the opposite direction can cause your spacecraft to slow down. This costs you nothing in terms of fuel, but takes time because you need to orbit the Sun in just the right way.

Messenger’s complex orbit to reach Mercury. Credit: Wikipedia

So what does this have to do with Mercury? Since Mercury is close to the Sun, you have to lose a lot of delta-v to reach it. In fact, even with a very efficient orbit, the delta-v to reach Mercury is about the same as that needed to reach Jupiter. Of course we’ve reached Jupiter several times, put a probes in orbit, and even dropped a probe into Jupiter’s atmosphere. We only put an orbiter around Mercury in 2011 with the Messenger spacecraft.  Before that, there was only the Mariner 10 flyby of Mercury, so it’s been a while since we’ve visited the planet. It also took several gravitational assists with Earth and Venus just to get Messenger into Mercury orbit.

Still, Messenger is just an orbiter. There’s no lander component to the mission. That again has to do with delta-v. The reason we’ve been able to put so many landers and rovers on Mars is that it has a mild atmosphere. We can use that atmosphere to “air-brake” a spacecraft, slowing it enough to make a landing without crashing. Mercury has no atmosphere, so any lander would have to be slowed by fuel alone, which is a hefty challenge.

There are currently no solid plans to put a lander on Mercury. Delta-v is a harsh mistress.

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A plutino is an asteroid-sized body that orbits the Sun in a 2:3 resonance with Neptune. They are named after Pluto, which also orbits the Sun twice for every three orbits of Neptune. It is thought that Pluto and the other plutinos were clustered into this resonance during the migration of the outer planets during the early solar system.

We know from observing several exoplanetary systems that large planets are often close to their star. This agrees with computer models that indicate larger planets will tend to drift inward toward their star due to drag during the accretion stage of the solar system. But in our solar system the large planets are all outer planets, which is unusual. It’s generally thought that orbital resonances caused the large planets to move outward.

The most popular model, known as the Nice (pronounced neese) model, posits that Jupiter was roughly at its current distance when it entered a 1:2 resonance with Saturn. The resulting resonance drove Neptune (initially closer than Uranus) to the outer edge of the solar system, and pushed pushed Uranus and Saturn outward to their current positions. Because of this shift in the large planets, some smaller objects such as Pluto were pushed into a resonance with Neptune, becoming the plutinos.

One interesting aspect of plutinos is that binary plutinos are rare. Pluto, which is Charon companion (seen above) is one of the exceptions, but then Pluto is the largest plutino. One other binary plutino was discovered in 2012. This is in contrast to the asteroid belt and the outer Kuiper belt, where binary objects are more common. So what is it about plutinos that makes binaries rare?

It has been thought that perhaps some effect of their original clustering might be the reason, but now a new paper in Astronomy and Astrophysics shows that it might be the 2:3 resonance itself that is the cause.

The authors ran computer simulations of 4-body models. That is, they put the Sun and Neptune in their current position, then ran models with different binary bodies in plutino orbits. What they found is the 2:3 resonance tends to gradually increase the eccentricity of the binaries orbits to the point where they become unstable. The resonance also makes it difficult to form binary systems for the same reason.

If this model is correct, then not only would binary plutinos be rare (just as we observe), but the binary plutinos that do exist would be older binaries brought into 2:3 resonance after Neptune had reached its current orbit.

This is just one model, and a rather simple one at that. It will take more computational analysis to see if the model is robust. But if it holds up it would explain why most plutinos are forever alone.

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Orbital dynamics, that is, the motion of planets and stars about each other, is deeply dependent on computational modeling. The basic motion of one planet or star about another (the so-called two body problem) is fairly simple, and can be summarized by Kepler’s laws of planetary motion, the motion of multiple planets and stars is extraordinarily complex. In fact while the two-body problem is almost trivial to solve, the three-body problem has no exact general solution. As soon as you have three or more masses in your system, the motion can be highly chaotic.

There are, however, some broad patterns we see in many-body systems, such as resonances and the formation of gaps in the asteroid belt. There are also interesting effects that have some surprising consequences. One of these effects is known as the Kozai mechanism.

Within orbital dynamics there are certain properties such as energy and angular momentum that are conserved (or approximately conserved). This means mathematically that certain properties of a system are constants. The Kozai mechanism deals with a quantity related to angular momentum known as Lz. This quantity is a constant, and it relates how elliptical an object’s orbit is (its eccentricity) to the orientation or tilt of its orbit (its inclination).

Credit: Gravity Simulator

You can get an idea of how these are related by imagining a flexible hula hoop. If you hold the hoop in front of you, aligned vertical to the ground, then the hoop will look like a circle to you. If you tilt the hoop at an angle to the ground, then it will look like a squashed oval to you. Now, if you wanted to move the hula hoop vertical again, but keep it looking like an oval, then as you tilt it vertical you have to squash the hoop itself. To keep the apparent shape constant, if you change the angle, you also have to change how much the hoop is squashed. The Lz property is similar, in that the eccentricity and inclination can change in relation to each other, but the “apparent shape” (Lz) stays constant.

The Kozai mechanism does just that. Basically it is a resonance between a smaller object and a larger object that causes the smaller object’s orbit to change its inclination to become more aligned with the larger object, and doing so at the cost if increasing the eccentricity of its orbit. The effect is a little hard to visualize, but you can see an example in the image above. The central dot is a star, and the orbits of two planets are shown. Outside of view is a second star orbiting at a different angle from the planets. You can see how the orbit of the outer planet is gradually shifted due to the second star.

One of the ways the Kozai mechanism comes into play is between the orbits of Neptune and Pluto. Gradually Pluto’s orbit is being tilted toward the orbital plane of Neptune. As a result, the eccentricity of Pluto’s orbit increases. This can explain why Pluto’s orbit is so elliptical, while most of the other planets are not. Another consequence of this mechanism is that the point of Pluto’s closest approach to the sun (its periapsis) is shifted relative to Neptune’s, which is part of the reason why Pluto will never collide with Neptune even though it crosses Neptune’s orbit.

A similar resonance should occur with other minor planets similar to Pluto (the Plutoids). This means that their orbits should be clustered rather than randomly distributed. We already see that with other minor planets such as Makemake.

All due to an interesting orbital resonance.

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