On Binary Planets, and Binary Polyhedra

Faceted Augmented Icosa

This image of binary polyhedra of unequal size was, obviously, inspired by the double dwarf planet at the center of the Pluto / Charon system. The outer satellites also orbit Pluto and Charon’s common center of mass, or barycenter, which lies above Pluto’s surface. In the similar case of the Earth / Moon system, the barycenter stays within the interior of the larger body, the Earth.

I know of one other quasi-binary system in this solar system which involves a barycenter outside the larger body, but it isn’t one many would expect: it’s the Sun / Jupiter system. Both orbit their barycenter (or that of the whole solar system, more properly, but they are pretty much in the same place), Jupiter doing so at an average orbital radius of 5.2 AU — and the Sun doing so, staying opposite Jupiter, with an orbital radius which is slightly larger than the visible Sun itself. The Sun, therefore, orbits a point outside itself which is the gravitational center of the entire solar system.

Why don’t we notice this “wobble” in the Sun’s motion? Well, orbiting binary objects orbit their barycenters with equal orbital periods, as seen in the image above, where the orbital period of both the large, tightly-orbiting rhombicosidodecahedron, and the small, large-orbit icosahedron, is precisely eight seconds. In the case of the Sun / Jupiter system, the sun completes one complete Jupiter-induced wobble, in a tight ellipse, with their barycenter at one focus, but with an orbital period of one jovian year, which is just under twelve Earth years. If the Jovian-induced solar wobble were faster, it would be much more noticeable.

[Image credit: the picture of the orbiting polyhedra above was made with software called Stella 4d, available at this website.]

Orcus and Vanth

There’s a binary dwarf-planet-candidate / large satellite pair, way out in the outer solar system, called Orcus and Vanth. Much like the “double dwarf planet” Pluto/Charon, and the other satellites in that system, Orcus and Vanth orbit the sun in a 3:2 resonance with Neptune, and this orbit crosses that of Neptune, as well. The Orcus/Vanth binary system is sometimes referred to as the “anti-Pluto,” because, unlike most “plutinos” (as such distant objects, in orbital resonance with Neptune, are called), Orcus and Vanth have a strange — and, so far, unexplained — relationship with the Pluto/Charon system. When Pluto and Charon are closest to the sun (perihelion), Orcus and Vanth are at their furthest from the sun (aphelion), and vice-versa. So far as I have been able to determine, this is not true for any other known plutinos. For more on the real Orcus and Vanth, please check this Wikipedia page.

Those are the scientific facts, as we know them . . . and now, it’s time for some silliness. On Facebook, recently, I mentioned that “Orcus” and “Vanth” really would make good names for comic book characters, but that I couldn’t decide what they should look like, nor what powers they should have. A discussion with some of my friends followed, and, together, we decided that Orcus should be a tough fighter-type, while “Vanth” sounded like a name for some sort of spell-caster. It didn’t take long before I decided I should visit one of the numerous create-your-own-comic-book-character websites, and go ahead and make quasi-anthropomorphized images of Orcus and Vanth — the characters, not the outer solar-system objects.

I used a website called Hero Machine for this diversionwhich you can find here. First, I created an image for a character named Orcus.

orcus

Unfortunately, I didn’t discover (until it was too late) that this website allows the user to change the background . . . and I didn’t want to re-make Orcus, so I went ahead and created an image of his companion, Vanth, instead.

vanth

I don’t have the time, nor the artistic talent, to write and illustrate actual comic book stories featuring this pair of characters . . . but perhaps someone will read this, and decide they want to take on such a project. That’s fine with me . . . but I want credit (in writing, each issue) for creating them, and, if the endeavor makes any money, I want at least 20% of the profits, and that’s if I have nothing more to do with creating Orcus and Vanth stories, beyond what is posted here. If I do have additional involvement, of course, we’ll need to carefully negotiate the terms of a contractual agreement. I consider 20% fair for simply creating images of this pair of characters, but actually co-creating stories would be something else altogether.

By the way, although Orcus certainly looks scarier, Vanth is actually the more formidable of the pair. She just pretends to play the “side-kick” role, in order to preserve the element of surprise, for situations when, during their adventures, Orcus finds himself in over his head, and Vanth then needs to really cut loose with the full extent of her abilities.

A Table of Known Masses for Numerous Objects in the Solar System, in Kilograms, Solar Masses, Jovian Masses, Terran Masses, and Lunar Masses

solar system object masses

The source of the information in the first two columns is this Wikipedia page. I calculated the numbers in the other columns, so any errors there are my own.

There are many other objects of known mass in the solar system, but I tried not to skip any, as I worked from larger-mass objects down toward those of smaller mass. Skipping some was necessary, though, for there are many objects (the likely dwarf planet Sedna is but one example) for which the mass is simply unknown. The next one I encountered after the asteroid Pallas did not have a name, but merely an alphanumerical designation, so I decided to stop there.

On the Direction of Motion of Spinning Polyhedra, the Rotating Earth, and Both the Rotation and Orbital Revolution of Other Objects in the Solar System

twistedIn which direction is the polyhedron above rotating? If you say “to the left,” you’re describing the direction faces are going when they pass right in front of you, on the side of the polyhedron which faces you. However, “to the left” won’t really do . . . for, if you consider the faces hidden on the side facing away from you, they’re going to the right. What’s more, both of these statements reverse themselves if you either turn your computer over, or stand upside-down and look at the screen. Also, if you do both these things, the situation re-reverses itself, which means it reverts to its original appearance.

Rotating objects are more often, however, described at rotating clockwise or counterclockwise. Even that, though, requires a frame of reference to be made clear. If one describes this polyhedron as rotating clockwise, what is actually meant is “rotating clockwise as viewed from above.” If you view this spinning polyhedron from below, however, it is spinning counterclockwise.

Since I live on a large, spinning ball of rock — of all solid objects in the solar system, Earth has the greatest mass and volume, both — I tend to classify rotating objects as having Earthlike or counter-Earthlike rotation, as well. Most objects in the Solar system rotate, and revolve, in the same direction as Earth, and this is consistent with current theoretical models of the formation of the Solar system from a large, rotating, gravitationally-contracting disk of dust and gas. The original proto-Solar system rotated in a certain direction, and the conservation of angular momentum has caused it to keep that same direction of spin for billions of years. Today, it shows up in the direction that planets orbit the sun, the direction that most moons orbit planets, and the direction that almost everything in the Solar system rotates on its own axis. Because one direction dominates, astronomers call it the “prograde” direction, with the small number of objects with rotation (or revolution, in the case of orbital motion) in the opposite direction designated as moving in the “retrograde” direction.

So which is which? Which non-astronomical directional terms, as used above when describing the spinning polyhedron there, should be used to describe the prograde rotation of Earth, its prograde orbital revolution around the sun, and the numerous other examples of prograde circular or elliptical motion of solar system objects? And, for the few “oddballs,” such as Neptune’s moon Triton, which non-astronomical terms should be used to describe retrogade motion? To find out, let’s take a look at Earth’s revolution around the Sun, and the Moon’s around the Earth, for those are prograde is well. This diagram is not to scale, and the view is from above the Solar, Terran, and Lunar North poles.

animation

[Image found reblogged on Tumblr, creator unknown.]

Prograde (Earthlike) motion, then, means “counterclockwise, as viewed from above the North pole.” To describe retrograde (counter-Earthlike) motion, simply substitute “clockwise” for “counterclockwise,” or “South pole” for “North pole,” but not both. Here’s the spinning Earth, as viewed from the side:

just_earth_800

[Image source: http://brianin3d.wordpress.com/2011/03/17/animated-gif-of-rotating-earth-via-povray/ ]

If you’ll go back and check the polyhedron at the top of this page, you’ll see that its spin is opposite that of this view of the Earth, and it was described as moving clockwise, viewed from above. That polyhedron, and the image of Earth above, would have the same direction of rotation, though, if either of them, but not both, were simply viewed upside-down, relative to the orientation shown.

Stella 4d, the software I use to make rotating polyhedral .gifs (such as the one that opened this post), then, has them spin, by default, in the same direction as the Earth — if the earth’s Southern hemisphere is on top! As I live in the Northern hemisphere, I wondered if that was deliberate, for the person who wrote Stella 4d, available at www.software3d.com/Stella.php, lives in Australia. Not being shy, I simply asked him if this were the case, and he answered that it was a 50/50 shot, and simply a coincidence that it came out the way it did, for he had not checked. He also told me how to make polyhedral .gifs which rotate as the Earth does, at least with the Northern hemisphere viewed at the top:  set the setting of Stella 4d to make .gifs with a negative number of rotations per .gif-loop. Sure enough, it works. Here’s an example of such a “prograde” polyhedron:

negative spin

Surface Gravitational Field Strengths for Numerous Solar System Objects

It isn’t difficult to find rankings for the most massive objects in the solar system, rankings of objects in terms of increasing distance from the sun, or rankings of objects by radius. However, ranking objects by surface gravitational field strength is another matter, and is more complicated, for it is affected by both the mass and radius of the object in question, but in different ways. If two objects have different masses, but the same radius, the gravitational field strength will be greater for the more massive object. However, increasing the radius of an object decreases its surface gravitational field strength, in an inverse-square relationship.

Gravitational field strength is measured in N/kg, which are equivalent to m/s², the units for acceleration. The terms “gravitational field strength” and “acceleration due to gravity,” both of which are symbolized “g,” are actually synonymous. I prefer “gravitational field strength” because referring to acceleration, when discussing the weight of a stationary object on the surface of a planet, can cause confusion.

Use of the numbers given below is easy:  given the mass of a thing (an imaginary astronaut, for example), in kilograms, simply multiply this figure by the given gravitational field strength, and you’ll have the weight of the thing, in newtons, on the surface of that planet (or other solar system object).  If, for some odd reason, you want the weight in the popular non-metric unit known as the “pound,” simply divide the weight (in newtons) by 4.45, and then change the units to pounds.

How is surface gravitational field strength determined? To explain that, a diagram is helpful.

gravity

The large green circle represents a planet, or some other solar system object, and the blue thing on its surface, which I’ll call object x, can be pretty much anything on the solar system object’s surface. There are two formulas for Fg, the force of gravity pulling the planet and the thing on its surface toward each other. One is simply Fg= mxg, a form of Newton’s Second Law of Motion, where “g” is the gravitational field strength, and mx is the mass of the object at the surface. The other formula is more complicated:  Fg= (Gmxmp)/r².  This is Newton’s Law of Universal Gravitation, where “G” (not to be confused with “g”) is the universal gravitational constant, 6.67259 x 10-11 Nm²/kg², and mp and r are the mass and radius of the planet (or other solar system object). Because they each equal Fg, the expressions mxg and (Gmxmp)/r² can be set equal to each other, yielding the equation mxg = (Gmxmp)/r², which becomes g = (Gmp)/r² after mis cancelled. The mass of the object on the surface is not needed — “g” is simply a function of mp and r.

There is a problem, however, with the idea of “surface” gravitational field strength — and that is the fact that the five largest objects in the solar system, the sun and the gas giants, all lack visible solid surfaces. One cannot stand on Jupiter — if you tried, you’d simply fall inside the planet. Therefore, for Jupiter, picture a solid platform floating at the top of the visible clouds there, and place the test object on this solid platform. Under those conditions, multiplying the test object’s mass by the Jovian value of “g” will, indeed, yield the weight of the object there, as it could be measured by placing it on a bathroom scale, at rest on the floating platform. For the other gas giants, as well as the sun, the idea is the same.

The objects included in the list below are the sun, all eight major planets, all dwarf planets (and dwarf planet candidates) with known values of “g,” all major satellites, some minor satellites, and a few of the largest asteroids. Many more objects exist, of course, but most have values for “g” which are not yet known.

Here are the top five:

Sun/Sol,      274.0 N/kg

Jupiter,          24.79 N/kg

Neptune,       11.15 N/kg

Saturn,          10.44 N/kg

Earth/Terra,    9.80 N/kg

The top five, alone, make me glad I undertook this project, for I did not realize, before doing this, that our planet has the highest surface gravitational field strength of any object in the solar system with a visible solid surface.

The next five include the rest of the major planets, plus one Jovian moon.

Venus,    8.87 N/kg

Uranus,   8.69 N/kg

Mars,      3.711 N/kg

Mercury, 3.7 N/kg

Io,           1.796 N/kg

The third set of five are all planetary moons, starting with earth’s own moon. The others are Jovian moons, except for Titan, which orbits Saturn.

Moon/Luna, 1.622 N/kg

Ganymede,  1.428 N/kg

Titan,           1.352 N/kg

Europa,        1.314 N/kg

Callisto,        1.235 N/kg

The fourth set of five begins with the largest dwarf planet, Eris, and includes two other dwarf planets as well.

Eris,        0.827 N/kg           (dwarf planet)

Triton,     0.779 N/kg          (Neptune’s largest moon)

Pluto,      0.658 N/kg           (dwarf planet)

Haumea, 0.63 N/kg             (dwarf planet)

Titania,   0.38 N/kg             (largest moon of Uranus)

The fifth set of five includes the remaining dwarf planets with known values of “g.”

Oberon,          0.348 N/kg   (moon of Uranus)

1 Ceres,         0.28 N/kg       (dually classfied:  dwarf planet and largest asteroid)

Charon,          0.278 N/kg    (largest moon of Pluto)

Ariel,               0.27 N/kg      (moon of Uranus)

90482 Orcus, 0.27 N/kg      (probable dwarf planet)

The sixth set of five are dominated by Saturnian moons.

Rhea,          0.265 N/kg         (Saturnian moon)

4 Vesta,      0.25 N/kg           (2nd largest asteroid)

Dione,         0.233 N/kg        (Saturnian moon)

Iapetus,     0.224 N/kg         (Saturnian moon)

Umbriel,    0.2 N/kg             (moon of Uranus)

The seventh set of five are mostly asteroids.

704 Interamnia,  0.186 N/kg          (5th most massive asteroid)

2 Pallas,                0.18 N/kg            (3rd most massive asteroid)

Tethys,                 0.147 N/kg          (Saturnian moon)

52 Europa,           0.14 N/kg            (7th most massive asteroid)

3 Juno,                 0.12 N/kg            (large asteroid, w/~1% of mass of the asteroid belt)

Starting with the eighth group of five, I have much less certainty that something may have been omitted, although I did try to be thorough. My guess is that most future revisions of this list will be necessitated by the discovery of additional dwarf planets. Dwarf planets are hard to find, and there may be hundreds of them awaiting discovery.

Enceladus,    0.114 N/kg          (Saturnian moon)

Vanth,           0.11 N/kg             (moon of probable dwarf planet 90482 Orcus)

10 Hygiea,    0.091 N/kg          (4th most massive asteroid)

15 Eunomia, 0.08 N/kg            (large asteroid, with ~1% of mass of asteroid belt)

Miranda,       0.079 N/kg          (moon of Uranus)

Here is the ninth group of five:

Nereid,      0.072 N/kg  (Neptunian moon; irregular in shape)

Proteus,    0.07 N/kg    (Neptunian moon; irregular in shape)

Mimas,      0.064 N/kg  (Saturnian moon / smallest gravitationally-rounded object in                                                                                the solar system)

Puck,         0.028 N/kg  (6th largest moon of Uranus)

Amalthea, 0.020 N/kg  (5th largest Jovian moon)

Finally, here are “g” values for the two tiny moons of Mars, included because they are nearby, and are the only moons Mars has to offer. A more exhaustive search would reveal many asteroids and minor satellites with “g” values greater than either Martian moon, but smaller than Amalthea, the last solar system object shown in the last set of five.

Phobos, 0.0057 N/kg

Deimos, 0.003 N/kg

Pie Chart for Main-Belt Asteroid Masses

Image

Pie Chart for Main-Belt Asteroid Masses

I looked up enough asteroid masses to use them to make this pie chart. I now have three reactions.

First: oh, that’s why only Ceres is round!

Second: who was stupid enough to name an asteroid Europa? That name is taken!

Third: wow — those small ones sure do make up a lot of the total!

Astronomy Update

Image

Astronomy Update

The brightest star in the picture above is Mintaka, from Orion’s Belt. We just adopted a cat, and named her Mintaka. I think that calls for an astronomy update — just the basics, stuff that everyone should know much of.

  1. Starting with our solar system:
    1. Planets:
      1. Mercury, smaller than earth’s moon, no atmosphere to speak of, no moons, tidally locked with the sun
      2. Venus, no moons, hottest planet in solar system due to thick carbon dioxide atomsphere
      3. Earth, one planet-sized moon, only known location of life
      4. Mars, thin carbon dioxide atmosphere, two small, irregularly-shaped moons
      5. (Many asteroids in main asteroid belt, between orbits of Mars and Jupiter)
      6. Jupiter, largest of four gas giants and everything else in the solar system except the sun, 67 known moons (four are planet-sized, and three of those four have known or suspected sub-surface water oceans — Europa, Callisto, and Ganymede)
      7. Saturn, gas giant with most extensive ring system in the solar system, 150 known moons and moonlets, including one planet-sized moon, Titan, with a thick atmosphere and possible subsurface ocean, and another moon with a known subsurface water ocean, Enceladus)
      8. Uranus, planet with axis of rotation closest to the ecliptic, 27 known moons
      9. Neptune, one large moon, Triton, among 14 known moons
    2. Dwarf Planets:
      1. Ceres, only dwarf planet in the asteroid belt
      2. Pluto/Charon double dwarf planet system
      3. Haumea
      4. Makemake
      5. Eris, largest dwarf planet
      6. Sedna
  2. Other known solar systems:
    1. 1,795 exoplanets
    2. 461 exoplanets in multiplanet systems
    3. Total of 1,114 exoplanet systems, all within our galaxy
  3. Nearest known star, other than the sun: Proxima Centauri, 4.2 light-years distant
  4. Number of stars in the Milky Way, our galaxy:  ~300 billion
  5. Number of galaxies in the known universe:  ~100-200 billion