Star Struck

Orbital Mechanics Made Simple

Today’s lesson in orbital mechanics.

I was spending New Year’s Day looking over an old 2017 “Year In Space” calendar before recycling it when I found this statement concerning the Spitzer Space Telescope.

Spitzer travels in Earth’s orbit but at a slower pace; by 2063, Earth will have “lapped” the telescope and Spitzer will reenter the atmosphere.

OK, former Sumerlin astronomy students and fellow space nerds. What is wrong with this statement? Think about it before scrolling past the video that gives you a subtle hint.

If it travels in Earth’s orbit around the Sun, it will travel at the same speed as the Earth, not “at a slower pace.” If it moves more slowly than the Earth, it must be farther from the Sun than the Earth. As an example, Mars’s mean orbital speed is 24 kilometers per second as compared to Earth’s 30 km/s.

So Spitzer travels in an orbit that is similar to Earth’s, but slightly farther away from the Sun. For the serious space geeks, let’s go a little deeper.

Spitzer was launched in 2003, and according to the statement above will be lapped by the Earth 60 years later. So in that time, Spitzer will have completed 59 solar orbits while the Earth has completed 60. Its orbital period is 60/59 of a year, or 371 days. (We’re not trying to be super precise here.) There is a simple formula derived by Johannes Kepler 400 years ago that lets us determine Spitzer’s location.


Where P is the orbital period in years and A is the mean distance from the Sun in astronomical units (AU). One AU is the mean distance of the Earth from the Sun.


This puts Spitzer in a Sun-centered orbit that is roughly 1.69 million kilometers (1.05 million miles) farther out than Earth’s. For comparison, the Moon orbits about a quarter of a million miles away from Earth.

Who said orbital mechanics was complicated?

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The Great American Eclipse of 2017

Only two weeks away now! Rather than write a lengthy post, I’m just going to address the main questions I’ve been getting over the past few months.

The sun is going to be 90% obscured at my home. Is it worth traveling to the path of totality?

Yes, yes, a thousand times yes! Unless you have witnessed it, you cannot imagine what an awe-inspiring sight this is. Just as no picture of the Grand Canyon can do it proper justice, no video or photograph of a total solar eclipse can convey what it is like to gaze at a black hole in the sky where the sun shone an hour before. As one wit remarked, “Seeing a partial solar eclipse and saying you’ve seen an eclipse is like standing outside an opera house and saying you’ve seen an opera.”

Where will I be able to see a total eclipse?

The two best interactive maps I have found are here and here. The times on the Google map are given in UT (Universal Time), which is four hours ahead of EDT (Eastern Daylight Time), five hours ahead of Central Daylight Time, etc.

Will I see the same thing anywhere within that dark shadow?

The closer you are to the center line of the path of totality, the longer the period of totality. That time is roughly two and a half minutes. Which will seem like ten seconds.

When is it safe to look directly at the sun?

During totality, the sun’s surface is completely obscured by the moon, and it is perfectly safe to look directly at it. The few seconds immediately before and immediately after totality will allow you to witness the “diamond ring” effect, as the last visible portion of the sun’s surface peeks through lunar mountain ranges at the edge of the moon’s disc. Other than those few seconds before and after totality, and during totality itself, you need special eclipse glasses to safely look at the partial phases.

diamond ring

So where can I get these special eclipse glasses?

They are selling out fast. Amazon has many available from multiple vendors. But you need to be careful…

I heard some of these aren’t safe. How do I know if mine are?

Go here.

Clear skies and safe viewing, everyone!


Posted in Sky Phenomena Tagged with:

How Big Is That Star?

Two of the most fundamental properties of any astronomical object are size and distance. How big is it, and how far away is it? The vast scales of cosmological distances are determined by a variety of methods, just as one uses different tools to measure the length of a piece of wood and the height of a building.

Clearly the distance of an object is tied to its apparent size. The distances to even relatively nearby stars reduces these behemoths to mere points of light, not spherical objects like our nearby star the Sun. So how do we know how big these objects are, or indeed how big any object in the sky is?

The most obvious is by direct imaging, taking a picture of the object, measuring its apparent size, and determining its actual size from our knowledge of its distance. Apparent size (angular size) is measured as an angle. For larger angles, your hand held at arm’s length provides a rough guide.

The relationship of distance, linear (actual) size, and angular (apparent) size is shown below.

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Back to the Moon!

The last time humanity ventured out of low Earth orbit (LEO) was in December 1972. The Apollo 17 mission, last in the series, took Gene Cernan, Jack Schmitt, and Ron Evans to the moon, where Cernan and Schmitt spent a little more than three days on the lunar surface. We have not been any farther above the Earth since then than the 335 mile orbit of the Hubble Space Telescope on several servicing missions. Now SpaceX and Elon Musk propose to send two people on a trip around the moon in late 2018, 46 years after Apollo 17.

It’s worth looking at the two missions to see the similarities and the differences, at what has changed (and what has not) in almost five decades.

The SpaceX mission does not involve a landing on the lunar surface, but rather a looping around the moon and returning home. The announced plans don’t even call for going into orbit. This lowers the hardware and fuel requirements considerably—no separate lander like the lunar module (below), and no extra fuel required to carry that lander all the way to the moon.

There is also no fuel required to slow the spacecraft down enough to put it into a lunar orbit, or to speed it up enough to take it out of orbit and return it to Earth. A spacecraft with two passengers has fewer life support requirements than the 3-person Apollo spacecraft. So the 2018 (or later, probably) mission will be considerably simpler than the Apollo missions of the late ‘60s and early ‘70s.

Have we ever had this kind of mission before—circling the moon without either landing or orbiting? Ironically, yes. The ill-fated Apollo 13 mission aborted on its way to the moon after an explosion destroyed much of their life support. To conserve the remaining oxygen and electrical power, they looped around the moon and returned to earth as fast as celestial mechanics would allow. The diagram below is to scale. According to Musk, the SpaceX mission will go much farther out into deep space beyond the moon, 300,000 to 400,000 miles from Earth. The moon is on average about 240,000 miles from Earth.

How about the rocket that will send them on their way? The Saturn 5 was, and still remains, the most powerful operational rocket ever built. To see it thunder away from its launch pad was a sight to behold.

Although there are several somewhat technical ways to measure the power of a rocket, one useful yardstick is the mass payload it can deliver to LEO. For the Saturn V, this was 155 tons.

The SpaceX rocket for this new mission has not yet flown in the configuration needed to get a crewed spacecraft to lunar space. The Falcon 9 Heavy consists of a Falcon 9 core with two strap-on boosters. This will be able to deliver 60 tons to LEO.

This image showing the two rockets to the same scale shows just how massive the Saturn V really was.


Since there is no intention of landing on the moon, the spacecraft only needs to provide crew quarters and life support. SpaceX has already sent uncrewed versions of its Dragon spacecraft to resupply the International Space Station, but this Dragon 2 upgrade is designed to carry a crew of up to seven persons.

I’ve been unable to find a lot of detailed information about the life support systems, but the Dragon spacecraft includes a “trunk” with solar cells for power that could certainly accommodate the necessary supplies.

The two persons involved approached SpaceX on their own, and reportedly offered up enough money to cover the costs. If they pass some health tests, their names will be announced. Money may not buy you happiness, but it surely can buy you an out of this world adventure!

Posted in human spaceflight, Spacecraft, The Moon Tagged with:


A short post about something that is all over the news now. No, not the election. I’m talking about the latest “supermoon”. This is a term I don’t believe I had ever heard before about five years ago. I don’t know who to blame for it, but it refers to a full moon that is—what?—bigger than average? I don’t know who decides what qualifies. OK, enough complaining and eye rolling. Anything that gets people looking at and thinking about the moon is a good thing!

So why would one full moon be bigger than another? The only way would be if the moon were closer to the Earth sometimes than it is at others, right? Correct! You win!

And that is the case because the moon’s orbit around the Earth is not a circle, but an ellipse. The point in the lunar orbit when it is closest to the Earth is called perigee; the most distant point in the orbit is apogee. The diagram below is exaggerated to illustrate the point.

And here is an image showing the extremes of the moon’s apparent size at these two points in its orbit.

What’s so special about the upcoming full moon? The time at which it will be full is only two and a half hours after the time it reaches perigee. It will not be full when it is this close again until 2034. The exact moment for this November’s full moon phase is 8:52 am on November 14th, but either Sunday night the 13th or Monday night the 14th should be equally special.

Enjoy! And if you hear any howling, watch from inside.



Posted in Sky Phenomena, The Moon Tagged with:

Mars or Bust!

Ready to go to Mars? Elon Musk is. So is Robert Zubrin. And so is NASA. But I wouldn’t book my flight quite yet. There are some significant challenges to be overcome before any human footprints are left on the Red Planet’s surface.

What are the problems involved in sending humans to Mars, landing them there, and returning them safely to Earth? After all, we sent 12 men to the surface of the moon and back more than 40 years ago. Why not Mars? A few comparisons will be helpful.

1) The moon is 240,000 miles (385,000 kilometers) away, and the longest Apollo mission (the final one, Apollo 17) lasted a little over 12 ½ days.


Mars is, at its closest, 34 million miles (55 million km) away from Earth. But the realities of interplanetary travel require a long and curving path from here to there and back again. How far it is and how long it takes depends on the relative positions of the two planets in their orbits around the sun. As an example, the Curiosity Rover currently exploring Gale Crater traveled 352 million miles (566 million km) in 253 days.

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Posted in human spaceflight, Mars, Solar System, Spacecraft

Jupiter and the Fourth of July

There are currently 24 active spacecraft exploring the solar system beyond low Earth orbit, ranging from the relatively nearby–Lunar Reconnaissance Orbiter is mapping our moon–to the far-flung. Voyager 1 has actually left the solar system and is currently 135 astronomical units from the Earth. (One astronomical unit is the average distance between the Earth and the sun; 150 million kilometers or 93 million miles.) But one is rapidly approaching a rendezvous with our largest planet, Jupiter. The Juno probe will fire its rocket engine on July 4th to place itself into orbit.

So what’s different about this mission? We’ve been to Jupiter before, more than 20 years ago, when the Galileo spacecraft explored not only the planet but its many moons as well. Remember this iconic picture of Io, the pepperoni pizza moon of Jupiter? This image is courtesy of the Galileo orbiter.


And this image of the planet itself, with Io at its side:


Somewhat less well remembered is a probe that the main spacecraft released into the Jovian atmosphere, floating down on a parachute until increasing heat and pressure caused instrument failure. Scientists had a pretty good idea of what they would find as the probe descended: successive cloud layers of ammonia, ammonium hydrosulfide, and water.

Jovian atmosphere

Surprise! The clouds just weren’t there—especially the water layer. What’s up? Are our models that far off? Or did the probe just happen to hit a dry and cloudless area? It’s as if we sent a probe to Earth expecting a water planet, and it landed in the Sahara Desert. How representative are our results?

This is the question that Juno is designed to answer. Jupiter emits microwave radiation from its hot interior. Water absorbs microwave energy (how your tea is heated in a microwave oven), so a microwave receiver on the spacecraft can map any water clouds below. The closer to the clouds we fly, the more clearly we can see. Doing so also allows us to peer deep inside Jupiter’s interior in ways that we can never see inside our own Earth.

But flying close to Jupiter definitely has its challenges. Jupiter’s magnetic field is by far the strongest of all the planets, and that results in intense radiation belts around its equator. These are super-sized versions of the Van Allen radiation belts around our own planet. (The International Space Station flies below these. Only the Apollo astronauts on their way to and from the moon passed through them, and then very quickly.) Juno’s orbit is polar, flying over the poles, dipping beneath the radiation belts, and moving most rapidly at its closest approach.


The orbital trajectory is shifted by the fact that Jupiter is gaseous and not perfectly round. It rotates so rapidly and is fluid enough so that it bulges at the equator.

Jupiter equatorial bulge

This shifts the orbit over time so that Juno will eventually pass through the equatorial belt. Its scientific instruments are shielded inside a titanium box, but even so the intense radiation will likely kill them. It’s a tough environment in which to operate!

But while it does, Juno promises to give us new insight into the interior of Jupiter, the better to understand the origin and evolution of our home group of planets.

Posted in Planets, Solar System, Spacecraft Tagged with: ,

Mars Opposition 2016

That increasingly bright and obviously red object rising in the southeast late at night (around 10:30 pm EDT from Lynchburg) is Mars. It will rise ever earlier as it moves into position exactly opposite the sun in our sky on May 22, when it will rise around 8:30 EDT. This every-26-month event is a Mars opposition, and this is a reasonably good one.

What makes one opposition “better” than another? Therein lies a tale of orbital peculiarities that allowed the true nature of our solar system to come to light.


The orbits of planets around the sun are not perfect circles, they are in fact ellipses, circles that have been pulled and stretched. The ellipse below varies from circularity far more than any planetary orbit in our solar system, but it illustrates the point.


Earth’s orbit is elliptical, but not very much so. We are roughly 3 million miles (5 million kilometers) closer to the sun in January than we are in July, with an average distance of 93 million miles (150 million kilometers). But Mars! Mars has the second most (after Mercury) elliptical orbit of the eight planets (sorry, Pluto lovers) and that means that not all Mars oppositions are created equal.

The wonderful diagram below shows the positions of both Earth and Mars for all oppositions between 2012 and 2027. The distances between the two planets are given in astronomical units (AU) where one AU is that average distance between Earth and the sun. Mars, further from the sun than the Earth and therefore moving more slowly around it, takes 687 days for one orbit. The oppositions will occur at different places around that orbit, and only when the faster-moving Earth has caught up to the more stately motion of its sister planet.


The opposition of 2027 is an example of a “bad” opposition. Mars is near its aphelion (farthest distance from the sun, marked by the orange A), and so the distance between the two planets is 0.6780 AU, or 63 million miles (101 million kilometers). By contrast, the 2018 opposition is a very “good” one. Mars and Earth line up almost exactly halfway between perihelion (closest distance to the sun) for Mars and aphelion for Earth. The planetary distance is 0.3862 AU: 36 million miles (58 million kilometers). Quite a difference!

The difference between these two oppositions is seen in the greater brightness of Mars in our skies with the nearer opposition, and the greater apparent size of its disc. In the days before we had robots roaming its surface, Mars was eagerly scanned with the most powerful telescopes of the day at each opposition, particularly at very favorable ones.

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The 29th of February

A relatively rare event—an extra day in February, the every-four-years February 29th—greets us again this Monday. Why does this happen? As you might expect from my posing the question on this blog, the answer is wrapped up in astronomy.

In fact, our whole calendrical system is based on astronomy. The year is based on the time required for Earth to complete one circuit of the sun. The month is (loosely in a solar calendar, exactly in a lunar calendar) the time between repeating lunar phases, known as the moon’s synodic period. A day is the time of one rotation of Earth on its spin axis. Even a week of seven days is based on the seven naked-eye objects known to the ancients as “planets”: the sun and the moon along with Mercury, Venus, Mars, Jupiter, and Saturn.

But these units of time don’t fit neatly into each other. There are about 12.4 synodic periods of the moon in a year, not exactly 12. We compensate for this with months that are mostly longer than the 29.5 days of one synodic period. And there are 365.2425 days in a year, not 365. Hence a periodic February 29th.

You’ll notice that the “extra” time in a year is pretty close to one fourth of a day. So every four years, we add an extra day to February.

But wait. It isn’t exactly one fourth—it’s a little less. After 400 years of adding a day every four years, we would have added a total of about three extra days; we have to compensate somehow. We do so by not adding February 29th in three out of four century years. We only have a leap day in century years that are exactly divisible by 400.

Here’s how it works. 1896 was a leap year, as was 1904. But 1900 was not. It is a “century year”, but it is not divisible by 400. The year 2000, however, was a leap year.


The effect of this on the time of the northern summer solstice—the exact moment when the sun reaches its northernmost point in the sky—can be seen in the graph. Notice how the trends for the 18th, 19th, and 20th centuries all moved a little lower, corresponding to later dates. The summer solstice reached its latest point in 1903, roughly 3 pm on June 22nd, Greenwich Mean Time (GMT). And note how this trend would have continued if the year 2000 had not been a leap year. 2003’s summer solstice would have been even later than 1903’s; instead, it was around 8 pm on June 21st.

As we moved into the second half of the 20th century, the precision of our clocks improved to the point that the motions of the Earth, its revolution around the sun and its rotation on its axis, were shown to be too variable. Time itself was redefined in terms of the frequency of a particular atomic energy transition. For those who care to know, the official definition of a second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. One minute is 60 times this, and one hour is of course 60 minutes.

Periodically, a leap second is added to keep atomic clock time in sync with what is known as mean solar time. 26 such leap seconds have been added since this began in 1972, the last coming on June 30, 2015.

Do we ever subtract a leap second? No, the extra time is necessary because the rotation of the Earth is very slowly but inexorably decreasing. Each day is ever so slightly longer than the day before.

So use the extra time, whether an extra day or an extra second, to good advantage! I plan to have a nice lunch with a new friend on this particular February 29th.


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That Einstein Was a Smart Guy

A long time ago in a galaxy far, far away…

Two black holes collided and merged, releasing unimaginable quantities of energy in the form of gravitational waves. Last year these waves were detected on Earth by an exquisitely sensitive pair of instruments near Hanford, Washington and Livingston, Louisiana, and last week that detection was announced to the world. Unless you have given up all media for Lent, you have surely heard of it by now.

So what’s the big deal? Why are scientists so excited? Why is there talk of a Nobel Prize? What are gravitational waves, anyway?

Einstein and Gravity

We’ll start the story at a time much more recent than the ancient event whose signals only reached Earth after more than a billion years of travel: almost exactly a century ago, in fact. The name of Albert Einstein was known to the world of physicists, but had not yet become synonymous with scientific genius to the public at large. After publishing his Special Theory of Relativity in 1905—this is the one that predicts clocks running at different rates as they approach the speed of light, along with other non-intuitive results—Einstein worked for years to generalize his theory. The 1905 work was “special” because it didn’t really incorporate the effects of gravity into its equations. This is not much of a problem except in very strong gravitational fields.

The General Theory of Relativity published in 1915 took full account of the effects of gravity. The Special Theory had already shown that space and time could no longer be considered as independent of each other, that they merged into a single entity best described as four-dimensional space-time.

I’ve never met anyone who could visualize four dimensions. To help us understand what is going on, it’s helpful to drop back to our familiar three dimensions, and visualize a stretchable rubber sheet. For the sake of convenience, we’ll put a grid on it. You’ll see why in a minute.

Here you go. Kind of boring, isn’t it? This is “flat” space-time, with no gravitational effects to be seen. Light traveling from one side to the other will follow a straight path along one of the grid lines. And what causes gravitational effects? Well, matter will do it—the more the better. And what matter does to space-time is to change its shape. In the presence of matter, space-time is curved.

The greater the mass, the greater the curvature of space-time. Einstein had described gravity in an entirely new way. Instead of its being some mysterious force that magically reached out from the sun to influence the planets, it was simply our description of how space-time was curved by the massive sun, and how the planets naturally responded to that curved space-time. In the classic words of John Wheeler: “Mass tells space-time how to curve, and space-time tells mass how to move.”

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