The transit of Mercury occurs this morning. It’s great if you get a chance to watch Mercury pass in front of the Sun, but if you don’t you shouldn’t fret too much. 

While the transit is great to watch, and many amateur astronomers strive to check it off their observation list, it isn’t as monumental as the transit of Venus. Observations of the Venus transit occur rarely, in pairs every 121 and 105 years. Because of the relative ease of observation, the Venus transits have been used to determine the scale of the solar system. Observations of the transit of Venus were the only way we could determine the size of our solar system until we developed more modern methods such as radio telemetry.

In contrast, the transit of Mercury occurs more than a dozen times a century. Because of its more common occurrence, a transit of Mercury was the first solar system transit to be observed, by Pierre Gassendi in 1631. While there is some speculation the Mayans may have recorded a transit of Venus, the first officially recorded transit wasn’t until 1639. But other than holding the status of being first, Mercurial transits haven’t played a big role in the scientific understanding of our solar system. Mercury itself played a role in our understanding of general relativity, just not its transits.

But that shouldn’t diminish the fact that we can observe the passing of our home star’s smallest and closest planet across the Sun from our vantage point. An astronomical event doesn’t have to be scientifically useful to be interesting. It can simply be unusual, beautiful and challenging to observe. It’s part of doing astronomy for astronomy’s sake.

So if you get a chance to catch Mercury, you should consider yourself lucky to see a wonderful astronomical event.

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When Venus is in the western sky after sunset, it is known as the evening star. It is a time when Venus is most prominently seen. For most of 2015 Venus will be quite visible in the evening as it approaches greatest elongation in May.

Because Venus and Mercury orbit closer to the Sun than Earth, we never see them far from the Sun in the sky. They are either in the evening sky just after sunset, or in the morning sky just before sunrise. The angular distance of a planet from the Sun is known as elongation, and the largest angle a planet reaches before moving closer to the Sun (in the sky) is known as greatest elongation. The greatest elongation of Venus is pretty consistent, only varying between 45 and 47 degrees. Because of its more eccentric orbit, Mercury’s greatest elongation can vary between 18 and 28 degrees.

With a much smaller elongation, Mercury can be particularly difficult to observe. Even when Mercury is at greatest elongation it can be close to the horizon. That’s because it also depends upon the time of year it occurs. The planets tend to orbit the Sun in a similar plane, known as the ecliptic. The seasons occur because of the axial tilt of the Earth relative to the ecliptic. Near the winter or summer solstice, the ecliptic has about a 20 degree tilt relative to the horizon. Near the fall or spring equinox, the ecliptic is nearly horizontal with the horizon. So the best time to view Mercury is when it is at greatest elongation near one of the solstices.

Over the next few days, Mercury has fairly favorable viewing conditions. It’s elongation is relatively good, and it’s still close to the winter solstice. More importantly, it is near Venus in the sky as well. Venus is much easier to find, being brighter, so if you find what appears to be the brightest star in the sky just after sunset, then you’ve found Venus. If you look a bit to the right of Venus over the next few days, you’ll find what appears to be a dimmer star, and that will be Mercury. You can see the exact positions day by day in the video.

If you happen to have a clear evening, definitely track down Venus and Mercury. Venus is always a wonderful sight, and if you find Mercury, you’ll have found a planet most people haven’t seen in the sky.

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More general relativity today. This time a bit on how to calculate the perihelion advance of Mercury in general relativity. When you derive the central force equation for relativistic gravity you find there is an extra term not seen in Newton’s gravity. The extra term is small, but enough to make Mercury’s orbit (any orbit really, but we typically use Mercury as an example) deviate slightly from an ellipse. Since the deviation is small, you can make some broad approximations, get an approximate solution for Mercury’s orbit, then determine the perihelion advance for one orbit.

That’s all fine and good if all you want is an approximate solution, but what if you really want to grind through things and find an exact answer. It turns out you can’t get a general analytic solution for the GR case, but you can solve it computationally. Do this in Mathematica and you can get an interpolating function. Okay, so get your interpolating function and figure the shift for one orbit. Problem solved, right?

Well, not so fast. Just doing it for one orbit is not very accurate, so its better to calculate the solution over several hundred orbits. But then you run into a subtlety of relativistic orbits. It turns out the perihelion advance is only part of what is going on. There are actually two types of non-Newtonian behaviors. One is a secular deviation (the perihelion advance), and the other is a periodic deviation.

You can see both of them in the first figure above. I’ve plotted the deviation between Newton’s orbit and Einstein’s, and what you can see is there is a kind of periodic motion with increasing amplitude. If we just take an average of this difference, everything washes out. What we want to find is how the amplitude increases over time. So we have to do a fit to the average amplitude increase. The result is seen in the second figure, which gives the steady advance of the perihelion.

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One aspect of general relativity that always amazes me is the level of precision needed to distinguish it from Newtonian gravity. Take, for example, the advance of Mercury’s perihelion. When you count in the gravitational tugs from the sun and all the planets, Newton predicts Mercury’s perihelion will advance about 531.65 arcseconds per century. When we measure the orbit of Mercury, we find its perihelion actually advances 574.10 arcseconds per century. This means Newton’s prediction is off by about 42.45 arcseconds per century. I say “about” because there is an uncertainty in our observations of about 0.65. General relativity predicts an “extra” perihelion advance of 42.98, which agrees exactly with experimental observation.

The difference between Newton’s model and Einstein’s amounts to 28 millionths of a degree each orbital revolution. Put another way, Mercury makes one orbit every 87.969 days, but it reaches its perihelion about a half second later than Newton says it should. The difference between Newton and Einstein is less than a human heartbeat in time.

The most amazing thing about all this? This deviation from Newton was first accurately measured by Urbain Le Verrier in 1859.

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Mercury is the closest planet to the Sun.  You might think that means it is also the hottest planet, but that award goes to Venus with its thick, heat-trapping atmosphere. Mercury can be listed as one of the hottest planets in the solar system, but also one of the coldest.  This is because Mercury doesn’t have an atmosphere to speak of. So the side of Mercury facing the Sun can reach temperatures of 700 K (800 F/430 C), while the dark side of Mercury cools to 100 K (-280 F/-170 C). Mercury is an icy-hot world.

It was long thought that Mercury might be tidally locked due to its proximity to the Sun, with a single side facing the Sun.   We now know that isn’t the case.  Instead, the planet rotates in a 3:2 resonance with its orbit.  This means that two Mercurian years last three Mercurian days.  Because Mercury’s orbit is rather elliptical, the path of the Sun through Mercury’s sky is rather complex.  Mercury rotates at a relatively constant rate, but its orbital motion speeds up and slows down due to its orbital eccentricity.  This means that the motion of the Sun is generally East to West, as it is on Earth, but there are times when the Sun can be seen to stop, move West to East for a while, then stop and return to its general East-West motion.

Image of a scarp crossing craters. Credit: NASA

We also know that despite the Sun’s heat, the planet is gradually cooling.  Mercury is not a large world, and so its interior can’t maintain its internal heat.  As the planet slowly cools, it also gradually shrinks.  We see evidence of this in long scarps across its surface.  These look like long cliffs, but they cut across craters and other features, indicating that they were formed at a later time.  As the planet cools and shrinks, its surface wrinkles, forming these scarps.

Being so close to the Sun, it’s rather difficult to get to.  More difficult than Mars or even the outer planets.  The first probe to orbit Mercury was Messenger in 2011, and it was only then that we could get a complete map of its surface.  We’re still learning a great deal about Mercury.

Up next: Venus

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The average distance between stars in our solar neighborhood is about one parsec, or just under 4 light years.  While this is a massive distance on our scale, from a star’s perspective it isn’t overly large.  Since stars are always moving relative to each other, there are bound to have been “close encounters” with other stars over the roughly 5 billion year history of our solar system.

By close I mean distances on the order of 1/20 of a parsec or less.  The radius of our solar system is about 1/5000 of a parsec, so even such close encounters have no chance of colliding with a planet.  Still, the close approach of a star would gravitationally pull on the planets ever so slightly, which means the planets would shift in their orbits by a tiny amount.  Kind of like the flutter of a butterfly’s wings makes a tiny bit of breeze.

This raises an interesting question about the stability of our solar system.  Would it be possible for a small tug from a visiting star to throw planets out of their orbits over millions of years.  Is it possible to have a planetary “butterfly effect”?

Correlation function for Mercury

One way to test this would be to run simulations of the solar system over millions of years, but unless you want to pay for time on a supercomputer cluster that isn’t very practical.  Another way is to run a simulation over a few hundred years, and then look at how statistically similar the motion remains over time.  In mathematics, this measure of similarity is known as a correlation function.  In the figure above I’ve plotted the correlation function for Mercury over about a century.  You can see there is a periodic oscillation (which is due to the precession of Mercury’s orbit over time) but you can also see that the overall amplitude of the oscillation is decreasing over time.  This basically means that Mercury’s orbit is slightly losing its consistency over time.

If you look at the rate of that decay, you can calculate what is known as the correlation time.  This is a measure of how long (based on your model) you could expect the orbit to be stable.  In the case of Mercury this turns out to be about 30 million years.  This does not mean that Mercury will tumble out of its orbit after that time, but rather than its orbit will remain largely unchanged for at least that long.

Correlation times of the planets

If you calculate the correlation time for all the planets, you get times on the order of tens of millions to billions of years, which means our solar system seems to be pretty stable.  If you do the same calculation for Pluto, however, you get a time of only about 10,000 years.  The orbit of Pluto is not nearly as stable as those of the classical planets.

Although we only have about a dozen planets to work with (if you count Pluto, Ceres, etc.), you can look at some of the factors that make a planet more or less stable.  For example in the second figure I plotted the correlation time of the planets with the eccentricity of their orbits (how elliptical they are).  You can see that the more elliptical the orbit, the less stable the orbit appears to be.

Not unexpectedly, it turns out that the most stable planets are ones that are reasonably massive, with fairly circular orbits close to their star.  Small mass planets far from their star in elliptical orbits are much less stable.  This means that when we look for planets around other stars we would expect to find lots of large planets orbiting in circular orbits near their star, which is exactly what we are finding.

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