Who Orbits Who, and Where? Check it Yourself

So far the arguments given in recent posts give us a clear idea of how the Earth-Moon system works: Earth’s a spinning sphere of diameter about 8000 miles (13000 km), and the size of the Moon and its distance are known too (diameter about 1/4 Earth’s, and distance about 30 times Earth’s diameter). We also know that the Sun is much further than the Moon and larger than the Earth, though we don’t know more details yet.

What else can we learn just with simple observations? Since the stars’ daily motion is an illusion from the Earth’s spin, and since the stars do not visibly move relative to one another, our attention is drawn next to the motion of the objects that move dramatically relative to the stars: the Sun and the planets.  Exactly once each year, the Sun appears to go around the Earth, such that the stars that are overhead at midnight, and thus opposite the Sun, change slightly each day.  The question of whether the Earth goes round the Sun or vice versa is one we’ll return to.   

Let’s focus today on the planets (other than Earth) — the wanderers, as the classical Greeks called them.  Do some of them go round the Earth?  Others around the Sun?  Which ones have small orbits, and which ones have big orbits? In answering these questions, we’ll start to build up a clearer picture of the “Solar System” (in which we include the Sun, the planets and their moons, as well as asteroids and comets, but not the stars of the night sky.)

The Basic Patterns

If we make the assumption (whose validity we will check later) that the planets are moving in near-circles around whatever they orbit, then it’s not hard to figure out who orbits who. For each possible type of orbit, a planet will exhibit a different pattern of sizes and phases across its “cycle when seen through binoculars or a small telescope. Even with the naked eye, a planet’s locations in the sky and changes in brightness during its cycle give us strong clues. Simply by looking at these patterns, we can figure out who orbits who.

What exactly is a “cycle”?  We could choose different definitions (and later we will) but we’ll see in a moment that the most convenient for now is to define a cycle as beginning when a planet becomes invisible due to the Sun’s glare, and ending at a time when it is again invisible and its activity begins to repeat.

We can divide possible planets into four classes: near Earth-orbiters, far Earth-orbiters, near Sun-orbiters and far Sun-orbiters. By “near” I mean a planet whose orbit has a radius smaller than the Earth-Sun distance, and “far” implies an orbit with radius larger than the Earth-Sun distance; see the figure at right.

Four classes of almost-circular planetary orbits, centered either on the Earth or Sun and with radius less than or greater than the Earth-Sun distance.

The four classes of planets show quite different patterns. The Moon, being a near Earth-orbiter itself, illustrates one of them, but the other three are quite distinct. Just by studying these patterns through a small telescope, you can figure out which class a planet belongs to.

First, here is a list of facts about these classes. The reasons for these facts are made clear in four figures which follow the list.

  • An Earth-orbiting planet has a constant apparent size (like the Moon), while a Sun-orbiting planet appears smaller at the beginning and end of its cycle than at mid-cycle.
  • A near planet has dramatic phases, but a near Earth-orbiter is crescent at the start and end of its cycle and full mid-cycle (like the Moon), while a near Sun-orbiter is full at the start and end of its cycle and crescent near mid-cycle.
  • A far planet has limited phases and is never a crescent or half-full; it is always more than half full.
  • All planets are full and visible all night at mid-cycle (like the Moon), except near Sun-orbiters, which are invisible at mid-cycle.
  • Near Sun-orbiters are never overhead, nor seen at midnight; they are seen only in the west for a short time after sunset or in the east for a short time before sunrise. They are visible for the longest time on the two nights when they appear half full, once in the evening and once in the morning.
  • Near Earth-orbiters may sometimes be seen in front of the Sun’s disk as a cycle begins (as when the New Moon eclipses the Sun); near Sun-orbiters may do the same but only at mid-cycle (called a planetary “transit”). Far planets never are never seen in front of the Sun from Earth’s perspective.

The Patterns, Illustrated

Now, some figures that illustrate these statements. A few comments first:

  • As usual, they are not drawn to scale, and instead are drawn to emphasize the basic geometry.
  • In each case, the Sun and Earth are indicated as if they were fixed, which isn’t true of course. But the geometry of what we see of a planet is always relative to where the Sun is in the sky (since the Sun’s location determines when sunset and sunrise and midnight occur). It’s therefore both convenient and sufficient, for current purposes, to draw where a planet is located relative to the line between the Earth and Sun.
  • The apparent size of an Earth-orbiting planet stays the same (as long as its orbit is near-circular, which we are assuming from the beginning.) That’s why the Moon’s apparent diameter doesn’t change much, even as it goes through its phases from crescent to full. But the apparent size of a Sun-orbiting planet changes over its cycle, simply because its distance from Earth changes over time.
  • Each depiction of a planet on its cycle shows the half that’s lit by the Sun as lighter in color than the dark half, and also shows a line that divides the half visible from Earth from the half that is hidden from Earth. The overlap of the lit half and the visible half determines the phase that we see in our telescopes. When the lit and visible halves coincide, the planet appears as a full disk in the sky; when they barely overlap, the planet is a crescent; and so on.
  • Although I’ve drawn each cycle as though the planet first moves into the evening sky, it’s possible in principle that a planet move in the opposite direction, and is first seen in the morning sky. That won’t affect our ability to tell what class it’s in, since apparent sizes and phases follow a pattern that doesn’t care whether the cycle proceeds in one direction or the other.
A near Earth-orbiter, shown at various points along its cycle (not to scale!), which begins at right. The planet’s lit half is depicted in a lighter color; the dotted line passing through the planet separates the side visible from Earth from the hidden side. The appearance of the planet at certain points in the cycle is shown in the square boxes. This planet’s cycle is like the Moon’s; at a fixed distance from Earth, it remains of fixed apparent size, and its phases go from crescent to full back to crescent. The crescent phases are visible briefly after sunset or before sunrise while the full phase is visible all night long and is overhead at midnight. Occasionally, as its cycle begins, this planet may block part of the Sun’s disk.

A far Earth-orbiter, at a fixed distance from Earth, remains of fixed apparent size, but its phases are very different from the Moon; they start and end near full, never reach half-full, and are full mid-cycle when the planet is visible all night long and overhead at midnight.
A near-Sun orbiter has phases almost exactly opposite the Moon — full at the start and end of the cycle, and crescent near mid-cycle. However, it is never overhead, nor visible at midnight, and appears half lit at a moment called “greatest elongation”, when it is visible for the longest period after sunset or before sunrise. Its apparent size is smallest as the cycle starts and ends and is largest near mid-cycle; right at mid-cycle it is both unlit and hidden by the Sun’s glare. Occasionally, as mid-cycle, this planet may block part of the Sun’s disk.
A far Sun-orbiter has phases not so different from a far Earth-orbiter, but its distance from Earth varies considerably, so its apparent size is smallest as the cycle starts and ends and is largest mid-cycle, when it is full, visible all night, and overhead at midnight.

The Evidence

Already, just by looking at Venus in the sky, you may guess it’s a near Sun-orbiter. It’s small and dim early and late in its cycle, and it’s bright and high in the sky in the evening somewhat after a quarter cycle and in the morning somewhat before its cycle reaches third quarter. Near mid-cycle it remains bright but is visible for less and less time after sunset or before sunrise, and at mid-cycle it disappears. It’s never overhead and never visible all night long. Of our four classes of planets, only a near Sun-orbiter has these properties.

If you look at images of Venus through a small telescope, so that you can see its apparent size and phase clearly as it goes through its cycle, you’ll see clearly that this guess is correct. A half-cycle of photos of Venus can be seen here, or here.

I haven’t found any similar sets of photos of Mercury online (let me know if you are aware of any, please.) But Mercury has a crescent phase too, and, like Venus, it’s never overhead. In fact its maximum height in the night sky is much less than that of Venus (and it’s never visible for more than about an hour after sunset or before sunrise) which indicates that not only is it a near Sun-orbiter like Venus, it orbits closer to the Sun than Venus does.

Mars, on the other hand, shows a completely different pattern. At mid-cycle, the Red Planet is visible all night, and brilliant, so clearly it is not a near Sun-orbiter. Its true nature is most easily determined by telescopes, which show that it is always more-than-half lit and that its size changes dramatically during its cycle. I haven’t found yet a good sequence of telescope photos that show Mars through a complete full cycle (about two Earth years) or half cycle. The best I’ve found show about six months on one or both sides of mid-cycle (here, or here.) These show you that Mars is full and large at mid-cycle, and a bit less than full (but never half-full) and smaller away from mid-cycle. That’s enough already to show it’s a far Sun-orbiter, though it would be nice to see telescope photos that indicate Mars becomes nearly full again, and quite a bit smaller still, at the beginning and end of its cycle. (By the way, Mars’ cycle does proceed in the opposite direction from what’s shown in the figure.)

For Jupiter and Saturn, it’s even harder to find photos that show a cycle. The changes in Jupiter and Saturn are less than for Mars; the two planets are always nearly full, and their apparent sizes vary less than Mars’ size does. Nevertheless, this variation that’s present gives evidence that they too are far Sun-orbiters, further out than Mars. Additional evidence in favor of this interpretation is that both of these planets are at their most brilliant mid-cycle, when they are visible all night and overhead at midnight. (Again, their cycles proceed in the opposite direction from what’s shown in the figure.)

A Sun-orbiting planet at a huge distance from the Sun (such as Neptune) has a very similar orbit to an Earth-orbiting planet at the same distance. Apparent sizes and phases are similar, so to distinguish them requires more precise techniques. Not to scale.

To see and study Uranus and Neptune really requires a telescope; that’s part of why these planets were discovered relatively recently (1781 and 1846). They too are always nearly full and vary slightly in size. Because these variations are relatively small, it becomes more difficult to tell whether whether they are Sun-orbiters or Earth-orbiters, especially for Neptune. Indeed, as shown in the figure, the difference between the two classes becomes very small when the radius of a planet’s orbit is much larger than the Earth-Sun distance. This opens a loophole which we’ll close later.

The Classical Greeks Erred on Venus

Classical Greek astronomers were drawn to a different and incorrect hypothesis about Venus and Mercury (though there were dissenters). Instead of having them orbit the Sun or the Earth, Ptolemaic astronomy had them in a fifth class, each orbiting a empty point between the Sun and the Earth.

The phases and sizes of a planet orbiting a point between Earth and Sun are completely different from those of a near Earth-orbiter, as you can see in the figure above. If telescopes had been invented two thousand years ago, Greek astronomers would have known their model of the solar system was wrong, because the phases and apparent sizes of Venus and Mercury would have been wildly off. But the telescope wasn’t invented until 1608. It quickly spread across Europe in 1609, and by 1610 Galileo Galilei was building his own. By late that year he had made a telescope good enough for observations of Venus, and by early 1611 he’d seen enough: its sizes and phases prove it’s a near Sun-orbiter, demonstrating the Ptolemaic model of the planets is wrong. Soon after, observations of Mars showed it too is a Sun-orbiter rather than an Earth-orbiter, again contradicting Ptolemy.

The phases of Venus if Ptolemy’s model of the planets had been correct; it would never be half lit or near-full. With a small telescope, Galileo quickly saw that Venus’s phases are those of a near Sun-orbiter, contradicting Ptolemy.

Summing Up

Using the phases and apparent sizes of objects as seen through a small telescope, we, like Galileo, now have direct confirmation of what we learn in school: that the non-Earth planets orbit the Sun; that Mercury is the closest planet, followed by Venus, both with orbits smaller than the Earth-Sun distance; that beyond them lie Mars, Jupiter and Saturn, with orbits larger than the Earth-Sun distance. Larger telescopes and more care would reveal that Uranus and Neptune lie beyond Saturn. (Pluto requires professional telescopes; we won’t be looking there.)

What about the Sun and the Earth, though? What can we say about them?

Unlike the Moon, the Sun’s disk is always full. That’s because, unlike all the planets and moons that surround it, it’s the light source that’s by far the most important in our vicinity. It glows on its own, whereas all planets and moons glow mainly through reflected light. So from all perspectives, at all times, it acts as a near-sphere radiating light, and thus appears on the sky as though it were a disk. The same effect would be found if you put a round light bulb near a round piece of fruit. No matter where you put them and no matter where you stand, the bulb will always be a complete sphere, while the fruit will show phases of light and shadow (since the half of the fruit that faces you may only be partly lit.)

Meanwhile, like the Moon, the Sun doesn’t change size during its cycle, indicating the Earth-Sun distance never changes by much. And yet, over a year, the Sun moves around, relative to the stars. Something’s moving in a circle. But who orbits who? Does the Earth go round the Sun?  Does the Sun go round the Earth? Or do they both go round a common point?

Compared to what we just did for the other planets (and for the Moon), none of these questions is easy to answer. Just ask the classical Greeks, who argued about it and drew the wrong conclusion.  You’d think something so basic would be simple. But it’s not.

2 responses to “Who Orbits Who, and Where? Check it Yourself

  1. I think you mean *far Sun-orbiter* for Mars, Jupiter and Saturn. (Currently it says _far Earth-orbiter_ …)