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.]

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.

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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

Important Safety Guidelines from Your Gravity Company, GravCorp, Inc.

floating-people-068

Please read these safety guidelines carefully. Also, we recommend displaying them prominently, securely fastened to the sturdiest wall in your home, in the event that your gravitational service is ever shut off for non-payment of your GravCorp gravity bill.

Because your friends at GravCorp care about you and your family’s safety, GravCorp will never shut your gravity off abrupty, but does so gradually, over the 24-hour period following the end of the shut-off date (prominently printed in red, bold type) on your gravity shut-off notice. It is best to evacuate early during this period. [Tip:  when you notice that you weigh noticeably less than you did the day before, that is your signal to leave.] We are not responsible for anything that happens if you fail to heed this advice, but we do have some safety guidelines to help those who, through no fault of ours, fail to leave their homes in a timely manner.

Once gravity shut-off is complete, if you are still inside your home, follow these safety rules carefully:

1. Be certain to keep moving at all times. Stationary humans have been known to die from lack of oxygen in the absence of gravity, due to the buildup of a spherical cloud of exhaled carbon dioxide, centered in the region of their mouths and noses. If you still have electrical service while your gravity is shut off, however, you can also avoid this danger by turning on all the electric fans in your home, such as the ceiling fan in the picture above. 

2. Should you choose to go outside, exercise extreme caution to avoid serious accidents (most of which are likely to be fatal). If you still have telephone or Internet service, we recommend paying your past due GravCorp account balance (plus the $135 reconnect fee) by phone or Internet, from inside your home.

3. Keep all liquids inside containers, for inhalation of even part of a floating ball of water, or other liquid, can cause death by drowning.  [Tip:  don’t forget to seal all toilets — both bowl and tank — using approved, waterproof sealing methods and materials.]

4. Act quickly to pay your past due bill, plus the $135 reconnect fee, or have a pressure suit on and pressurized, for the air above you is already beginning to escape into space.

5. Remain calm, do not panic, and consider setting up automatic bank drafts to pay your gravity bill, effortlessly, each month. It’s convenient, safe, and saves you money on postage. (An annual $3.14 convenience fee will be charged to your GravCorp account, on or near July 1st each year, for this optional service.)

 

[Image credit:  The picture above was found at http://www.thedistractionnetwork.com/going-to-bed/.]

Your Toes Are Younger Than Your Head

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Your Toes Are Younger Than Your Head

Unless, like a bat, you sleep upside-down, your toes are younger than your head.

Why?

Because, having spent more time slightly closer to the center of the earth, they have endured a slightly stronger gravitational field strength. This, in turn, due to relativistic time dilation, slows time down for your toes, relative to your head. With a slower passage of time during all periods when you were upright, less time has passed for them — and so they are younger.

Image credit: http://www.bestpodiatristnyc.com/british-hammertoes-are-wonky-toes/#sthash.xKDfxbgJ.dpbs