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.]

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.

A Graph Showing Approximate Mass-Boundaries Between Planets, Brown Dwarfs, and Red Dwarf Stars

planet and brown dwarfs and red dwarf stars

 

I found the data for this graph from a variety of Internet sources, and it is based on a mixture of observational data, as well as theoretical work, produced by astronomers and astrophysicists. The mass-cutoff boundaries I used are approximate, and likely to be somewhat “fuzzy” as well, for other factors, such as chemical composition, age, and temperature (not mass alone), also play a role in the determination of category for individual objects in space.

Also, the mass range for red dwarf stars goes much higher than the top of this graph, as implied by the thick black arrows at the top of the chart. The most massive red dwarfs have approximately 50% of the mass of the Sun, or about 520 Jovian masses.

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

Itaumiped Has No Moon

Image

Itaumiped Has No Moon

You can make your own planet here, but you can’t have my name for it: http://planetquest.jpl.nasa.gov/system/interactable/1/index.html

This is simply one incarnation of Itaumiped, my own imaginary planet. Any time I need an imaginary planet, I use this anagram for “I made it up” as its name. Itaumiped’s star’s name, “Almausoped,” comes from “Also made up.”

I try to be prepared. After all, one never knows when one might need an imaginary planet — or star.

How To Destroy a Planet

Let me make it clear from the outset that this is a purely academic exercise. I don’t REALLY want to destroy any planet, let alone the one we live on, even though the beginning of this song — http://www.youtube.com/watch?v=rR5xTgMwpiM — is my cell phone ringtone. I will admit that much.

But, seriously, how would one destroy a planet? I don’t mean kill everything on it — I mean utterly obliterate the whole ball of rock.

Well, let’s assume it isn’t a rogue planet, but one like ours. It orbits a star, in a nearly-circular orbit, and rotates on its axis. Let’s say it’s the blue one in this picture:

…and we want to know a way to destroy it — just as an interesting puzzle to solve. That’s all — I promise.

Well, planets that are closer to their stars orbit faster, which is why the planet Mercury is the fastest-moving planet in our solar system. Speed up the orbital velocity, then, and a planet’s orbit will get smaller in diameter, for faster planets orbit more quickly. In the diagram (not even close to being at scale for anything real), this is why the red planet’s velocity vector is shown as being longer (faster) than that of the blue planet.

So, speed up a planet enough, and its orbit will decay until it falls into its star, which should destroy it most effectively.

So it’s really pretty simple. On the blue planet, pile up a bunch of nuclear weapons, rocket engines — whatever you can find that invokes Newton’s Third Law of Motion — and start blasting at the position where you see the orange triangle (sundown, local time), right on the equator (unless there is axial tilt involved; correct for that if there is, and keep the blast site in the plane of the ecliptic). Keep blasting as the planet rotates for one-fourth of a rotation (counterclockwise), until the blast site is at midnight local time, and then stop. Repeat at the next sundown, and so on.

All of this blasting will speed up the blue planet’s orbital velocity. Eventually, it will end up where the red planet is, if you do this gradually enough to maintain near-circularity of the orbit (get too eccentrically elliptical, and other results may occur). Keep up this madness, and the target planet will end up slowly spiraling into its star.

That’s it. Game over.

Please do not try this at home, though. All my stuff is here.

[Later edit:  I showed this to many of my science-geek friends, and they tore it to shreds. There are a lot of mistakes here. I considered taking it down, out of embarrassment, but have decided to leave it posted to remind myself that I can, and do, screw things up. I need such reminders!]