How to Distinguish Between the Waxing and Waning Moon, At a Glance

Posted on 12 August 2014 by RobertLovesPi

This is a waxing moon, meaning the sunlit portion we can see is growing. The outer curve also makes this view of the moon shaped more like the letter “D,” compared to the letter “C.” For the useful mnemonic here, remember that “D” stands for “developing.” D-shaped moons are in the waxing part of their cycle of phases, growing larger for about two weeks.

Later in the waxing portion of the moon’s cycle of phases, it becomes a gibbous moon — but retains its “D-like” shape. It is still slowly getting larger, approaching the full moon state.

Here is another gibbous moon, but it is shaped more like the letter “C” than the letter “D,” and, in this mnemonic, “C” stands for “concluding.” This moon’s sunlit portion is shrinking, moving away from fullness, towards the new moon state — in other words, it is a waning moon. All “C-shaped” moons, as viewed from Earth’s Northern hemisphere, are waning moons.

This crescent moon more closely resembles a “C” than a “D,” which is how I know, at a glance, that its phase cycle is concluding, and it is a waning crescent, soon to become invisible as a new moon.

This last picture shows the most difficult configuration to figure out: the points of the crescent near the moon’s North and South poles both point up. Having them both point down would pose the same problem. Here’s the solution, though: check to see which crescent-tip appears higher in the sky. In this case, it is the one on the left. That shifts the curve at the bottom of the moon (the one that is an actual moon-edge, rather than the terminator) slightly left-of-center, making the visible moon-edge more closely resemble a “C” than a “D.” This crescent moon, therefore, is a waning crescent.

Later addition: as a commenter pointed out, below, this method does not work from Earth’s Southern hemisphere — in fact, in that half of the world, the “D”/”C” rule must be completely reversed, in order to work. To accomplish this, “D” could stand for “diminishing,” and “C” could stand for “commencing,” instead.

[Image/copyright note: I did not take these photographs of the moon. They were found with a Google-search, and I chose images with no apparent signs of copyright. I am assuming, on that basis, that these images are not copyrighted — but, if I am wrong, I will replace them with other images, upon request.]

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

Posted on 12 August 2014 by RobertLovesPi

In 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.

[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:

[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:

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

Posted on 9 August 2014 by RobertLovesPi

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.

Proposed Mechanisms for New and Different Types of Novae

Posted on 9 August 2014 by RobertLovesPi

The picture above shows a proposed model for the production of a sudden increase in the brightness of a star — or rather, what is apparently a single star, optically, but would actually be a suddenly-produced binary stellar system.

The yellow object is a star, the system’s primary, and it has high mass (at least a few solar masses), when its mass is compared to those of the brown dwarfs in the two highly elliptical orbits shown in blue. These brown dwarfs aren’t quite stars, lacking enough mass to fuse hydrogen-1, which requires 75 to 80 Jupiter masses, but one of them (the larger one) is close to that limit. The smaller brown dwarf has perhaps half the mass of the larger brown dwarf. Their high orbital eccentricities give them very long orbital periods, on the order or 100,000 years. In a very small fraction of orbits, both brown dwarfs will be near perihelion (closest point to the primary) at the same time, and, during those rare periods, the two brown dwarfs become much closer to each other than they are to the primary.

When the two brown dwarfs become close enough to each other, matter from the smaller one could be drawn, by gravity, into the larger brown dwarf, increasing its mass, at the expense of its smaller sibling. At some point, in such a system, the larger brown dwarf’s mass could then reach the threshold to begin fusing hydrogen-1, and “turn on” as a true star — a red dwarf. From Earth, this red dwarf would not be distinguishable from the system’s most massive star, shown in yellow, until much later, when the two moved further apart. There would, however, be a sudden increase in luminosity from the system as a whole. Unlike other types of novae, this increase in luminosity would not fade away quickly, for red dwarfs have very long lifespans. This would enable them, upon discovery, to be distinguished from other single-brightening stellar events. Confirmation could then come from resolution of the new red dwarf component, as it recedes from the primary, making detection easier.

For a variation on this mechanism, the primary star could be somewhat more massive, and the two large brown dwarfs could be replaced by two large red dwarf stars. The larger red dwarf could draw matter from the smaller one, until the larger red dwarf became large enough to cross a higher mass threshold, and brighten substantially, with its color suddenly changing to orange or yellow.

A problem for this model: no such events are known to have happened. If they do happen, a likely explanation for their rarity is the likelihood that such orbits would be unstable, in a large fraction of similar cases, preventing the stellar-brightening event from having time to happen — in all but a few cases, none of which humans have (yet) both seen, and understood. If one of these things goes off nearby, though, we will learn about it quickly, for it will make itself known.

For another possible mechanism, there is another option: remove the primary altogether, and let the two objects of near-threshold mass orbit their common center of mass directly. They could then create a new star, or brighter star, by the mechanism described, one which might even produce a detectable accretion disk. A actual merger of the two brown dwarfs, or red dwarf stars, would be a variation of this idea, and would presumably be more likely if the two objects had masses very close to each other, so that neither would have an advantage in the gravitational tug-of-war.

Public Schools in the United States Should Rename the “Free Lunch”

Posted on 8 August 2014 by RobertLovesPi

If you live in the USA, you are probably familiar with the phrase “free lunch,” or “free and reduced lunch,” as used in a public-school context. For those outside the USA, though, an explanation of what that phrase means, in practice, may be helpful, before I explain why a different name for such lunches should be used.

The term “free and reduced lunch” originated with a federal program which pays for school lunches, as well as breakfasts, with money collected from taxpayers — for students whose families might otherwise be unable afford these meals. The program’s eligibility requirements take into account both family income and size. There’s a problem with it, though: the inaccuracy of the wording used, especially the troublesome word “free.” The acronym above, “TANSTAAFL,” is familiar to millions, from the works of Robert A. Heinlein (science fiction author), Milton Friedman (Nobel-Prize-winning economist), and others. It stands for the informally-worded phrase, “There ain’t no such thing as a free lunch,” which gets to the heart of the problem with the terminology we use when discussing school lunches. (Incidentally, I have seen an economics textbook use the phrase “TINSTAAFL,” in its place, to change “ain’t no” to “is no.” I do not use this version, though, for I am unwilling to correct the grammar of a Nobel laureate.)

The principle that “free lunches” simply do not exist is an important concept in both physics and economics, as well as other fields. In physics, we usually call it the Law of Conservation of Mass and Energy, or the First Law of Thermodynamics. This physical law has numerous applications, and has been key to many important discoveries. Learning to understand it, deeply, is an essential step in the education of anyone learning physics. Those who teach the subject, as I have in many past years, have an even more difficult task: helping students reach the point where they can independently apply the TANSTAAFL principle to numerous different situations, in order to solve problems, and conduct investigations in the laboratory. It is a fundamental statement of how the universe works: one cannot get something for nothing.

TANSTAAFL applies equally well in economics, where it is related to such things as the fact that everything has a cost, and those costs, while they can be shifted, cannot be made to simply disappear. It is also related to the principle that intervention by governments in the economy always carries costs. For example, Congress could, hypothetically, raise the federal minimum wage to $10 per hour — but the cost of doing so would be increased unemployment, especially for those who now have low-paying jobs. Another possible cost of a minimum-wake hike this large would be a sudden spike in the rate of inflation, which would be harmful to almost everyone.

To understand what people have discovered about the fundamental nature of physical reality, physics must be studied. To understand what is known about social reality in the modern world, economics must be studied. Both subjects are important, and understanding the TANSTAAFL principle is vital in both fields. Unfortunately, gaining that understanding has been made more difficult, for those educated in the United States, simply because of repeated and early exposure to the term “free lunch,” from childhood through high school graduation. How can we effectively teach high school and college students that there are no free lunches, when they have already been told, incessantly, for many years, that such things do exist? The answer is that, in many cases, we actually can’t — until we have first helped our students unlearn this previously-learned falsehood, for it stands in the way of the understanding they need. It isn’t a sound educational practice to do anything which makes it necessary for our students to unlearn untrue statements.

I am not advocating abolition, nor even reduction, of this federal program, which provides essential assistance for many families who need the help. Because I am an American taxpayer, in fact, I directly participate in funding this program, and do not object to doing so. I do take issue, however, with this program teaching students, especially young, impressionable children in elementary school, something which is untrue.

We need to correct this, and the solution is simple: call these school lunches what they actually are. They aren’t free, for we, the taxpayers, pay for them. Nothing is free. We should immediately replace the phrase “free and reduced lunch” with the phrase “taxpayer-subsidized lunch.” The second phrase is accurate. It tells the truth, but the first phrase does the opposite. No valid reason exists to try to hide this truth.

A 240-Atom Fullerene, and Related Polyhedra

Posted on 5 August 2014 by RobertLovesPi

The most well-known fullerene has the shape of a truncated icosahedron, best-known outside the world of geometry as the “futbol” / “football” / “soccer ball” shape — twenty hexagons and twelve pentagons, all regular. The formula for this molecule is C₆₀. However, there are also many other fullerenes, both larger and smaller. One of my favorites is C₂₄₀, simply because I sometimes make class projects out of building fullerene models with Zome (available at www.zometool.com), and the 240-atom fullerene is the largest one which can be built using Zome. Here’s what it looks like, as molecular models are traditionally colored.

This polyhedron still has twelve pentagons, like its smaller “cousin,” the truncated icosahedron, but far more hexagons. What’s more, these hexagons do not have exactly the same shape. If this is re-colored in the traditional style of a polyhedron, rather than a molecule, it looks like this. In this image, also, the different shapes of hexagons each have their own color.

Like other polyhedra, a compound can be made from this polyhedron and its dual. In this case, the dual’s faces are shown, below, as red triangles. The original fullerene-shape is in purple for the pentagonal faces, and orange for the hexagons.

In the base/dual compound above, it can be difficult to tell exactly what this dual is, but that can be clarified by removing the original fullerene. What’s left is called a geodesic sphere — or, quite informally, a ball made of many triangles. The larger a fullerene is, the more hexagonal rings/faces it will have, and the more triangles will be found on the geodesic sphere which is its dual. For the 240-atom fullerene shown repeatedly, above, here is the dual, by itself, with different colors indicating slightly different triangle-shapes. (An exception is the yellow and green triangles, which are congruent, but have different colors for aesthetic reasons.)

I made these four rotating images using Stella 4d: Polyhedron Navigator. To try this program for yourself, simply visit www.software3d.com/Stella.php. At that site, there is a free trial download available.

On “Digging to China”

Posted on 4 August 2014 by RobertLovesPi

When I was a little kid, my sister and I dug a big hole, in our front yard, and simply called it “the digging-hole.” It looked a lot like the hole shown above, except for the fact that, during daylight hours, our digging-hole usually included two small, dirt-covered, determined children, armed with plastic shovels. We tried, for years, to dig that hole as deep as possible. My personal goal, of course, was the Earth’s molten core, not India, and certainly not China.

Why do Americans so often talk about digging a hole straight down to China, anyway? Even if the Earth were solid all the way through its interior, digging straight down, from almost anywhere in the contiguous 48 states of the USA, would not put you in China, nor even India (which is, at least, closer to being correct than is China), but at the bottom of the Southern Indian Ocean. Salty water would suddenly rush into your newly-dug tunnel, killing you instantly, as soon as you got close to enough to the other side for the extreme water-pressure there to finish your digging project for you. The only exceptions to this watery doom would be coming out of the tunnel on one of the islands in that ocean, which would require great precision to hit deliberately.

Also, the fact that China and the USA are both Northern-hemisphere nations easily rules China out as the hypothetical “solid-earth” destination for Americans who dig straight down, and all the way through. If you could go through the center of the earth from North of the equator, you’d have to end up South of the equator. Isn’t that obvious? Don’t people look at globes?

“Evolution is just a theory.” Please STOP saying this!

Posted on 4 August 2014 by RobertLovesPi

Why?

Well, just to get started, these three things are also “just” theories:

1. Germs are the cause of many diseases.
2. Everything you have ever touched is made of atoms.
3. The spinning earth doesn’t fling us into outer space because of gravity.

Would any reasonable person actually think the phrase “just a theory” makes sense for any of these three things? Use of this phrase, for evolution, the Big Bang, or anything else, indicates one thing: the person talking does not understand the meaning of the word “theory.” Theories are the best science has to offer, and science is the foundation of modern civilization. These theories are based on the repeated testing of hypotheses, using experiment, to explain what we observe — so they are evidence-based explanations, not mere guesses, as the annoying phrase “just a theory” implies.

Evolution is every bit as well-established a theory as the three examples cited above. All theories are subject to further testing, which is an important self-correcting mechanism in science. No theory is beyond revision or replacement, if new experimental evidence calls for it. However, that fact doesn’t make any particular theory invalid — it simply helps explain why science works. It also works just as well whether people believe in it, or approve of it, or agree with it — or not.

If you want to disprove the theory of evolution, just find a fossilized rabbit in a one-billion-year-old rock, as J.M.S. Haldane famously observed. It will only take one such finding to accomplish your goal, and you can publish your results, and become famous – if you can find such a fossil. For your own safety, though, please do not hold your breath while looking.

Craters and Slopes Near the South Pole of the Moon Adorn the Faces of a Rhombic Enneacontahedron

Posted on 3 August 2014 by RobertLovesPi

The images on the faces of this polyhedron are based on information sent from NASA’s Lunar Reconnaisance Orbiter, as seen at http://lunar.gsfc.nasa.gov/lola/feature-20110705.html and tweeted by @LRO_NASA, which has been happily tweeting about its fifth anniversary in a polar lunar orbit recently. I have no idea whether this is actually an A.I. onboard the LRO, or simply someone at NASA getting paid to have fun on Twitter.

To get these images from near the Lunar South Pole onto the faces of a rhombic enneacontahedron, and then create this rotating image, I used Stella 4d: Polyhedron Navigator. There is no better tool available for polyhedral research. To check this program out for yourself, simply visit www.software3d.com/Stella.php.

Surface Gravitational Field Strengths for Numerous Solar System Objects

Posted on 1 August 2014 by RobertLovesPi

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.

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 F_g, the force of gravity pulling the planet and the thing on its surface toward each other. One is simply F_g= m_xg, a form of Newton’s Second Law of Motion, where “g” is the gravitational field strength, and m_xis the mass of the object at the surface. The other formula is more complicated: F_g= (Gm_xm_p)/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^-11Nm²/kg², and m_pand r are the mass and radius of the planet (or other solar system object). Because they each equal F_g, the expressions m_xg and (Gm_xm_p)/r² can be set equal to each other, yielding the equation m_xg = (Gm_xm_p)/r², which becomes g = (Gm_p)/r² after m_xis cancelled. The mass of the object on the surface is not needed — “g” is simply a function of m_pand 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, 9.806 65 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 (2^nd 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 (5^th most massive asteroid)

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

Tethys, 0.147 N/kg (Saturnian moon)

52 Europa, 0.14 N/kg (7^th 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 (4^th 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 (6^th largest moon of Uranus)

Amalthea, 0.020 N/kg (5^th 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