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Throughout the solar system, countless objects of widely varying sizes, bizarre geometries, and diverse compositions orbit the Sun independently or attend gas giant planets in swarms.  Those which overwhelmingly consist of rock and metal are called asteroids, and threaten Earth-bound human civilization at the same time they hold out the promise of a free, unimaginably wealthy, and unbounded future history.  They bear a stark message for mankind: Come to us in hope, before we come to you in fire.

The progress of our adventure so far (current in bold):

1.  The Sun
2.  Mercury
3.  Venus
4.  Earth (Vol. 1)
5.  Earth (Vol. 2)
6.  Earth (Vol. 3)
7.  Earth (Vol. 4)
8.  Earth (Vol. 5)
9.  Earth (Vol. 6)
10.  Luna
11.  Mars (Vol. 1)
12.  Mars (Vol. 2)
13.  Mars (Vol. 3)
14.  Phobos & Deimos
15.  Asteroids (Vol. 1)
16.  Asteroids (Vol. 2)
17.  Asteroids (Vol. 3)
18.  Ceres
19.  Jupiter
20.  Io
21.  Europa
22.  Ganymede
23.  Callisto
24.  Saturn
25.  Mimas
26.  Enceladus
27.  Tethys, Dione, and Rhea
28.  Titan
29.  Iapetus
30.  Rings & Minor Moons of Saturn
31.  Uranus
32.  Moons of Uranus
33.  Neptune
34.  Triton
35.  The Kuiper Belt & Scattered Disk
36.  Comets
37.  The Interstellar Neighborhood
Topics covered in Vol. 2 of the Asteroids sub-series:
I.  Context
II.  Population Groups, Families, and Spectral Types
III.  History
IV.  Properties
i) Orbital and Rotational Features
ii) Size and Mass Characteristics

iii) Surface Features
V.  Past Relevance to Humanity
VI.  Modern Relevance to Humanity
VII.  Future Relevance to Humanity
VIII.  Future of The Asteroids
IX.  Catalog of Exploration
IV.  Properties

1.  Orbital and Rotational Features:

Here is an animated graphic depicting the three-dimensional orbital motion of Earth's only known Trojan, 2010 TK7:

Most people who don't have a detailed physical understanding of a Trojan orbit and the Lagrange points associated with it will find it a difficult concept to grasp, so I will paraphrase an explanation given in a comment posted by eyesoars to the previous volume that was far better than my own feeble attempt to describe the phenomenon in comments:

1.  When an asteroid shares a solar orbit with a planet, it moves around the Sun at the same speed as the planet, so it experiences no net gain or loss around the orbit relative to the planet over a full cycle.

2.  However, the planet exerts a gravitational pull on the asteroid, so if the object is ahead of the planet - a "leading" Trojan - gravity from the planet will slow it down, and if the object is behind the planet (a "trailing" Trojan), gravity will speed it up.  In orbital mechanics, speeding up or slowing down along an orbit changes the orbit: Moving faster leads to a bigger orbit (i.e., you move outward relative to the Sun), and moving slower leads to a smaller orbit (inward relative to the Sun).  


3.  But as an orbit widens, the energy pumped into widening it puts it on a higher level in the gravity well, transforming kinetic energy into gravitational potential energy.  This results in slower speed along the orbit.  So, as counterintuitive as it is, speeding up along an orbit takes you outward and slows you down to a speed commensurate with that orbit.  If you are already at the speed corresponding to an orbit and slow down along it, the orbit will get smaller and the lost gravitational potential energy will transform into kinetic energy, causing you to speed up along the inner orbit.  This is why it takes so much energy to move between planets: Getting to a planet's location isn't enough unless the intent is to smash into it at hypervelocity - you also have to match its orbital speed.

4.  Likewise, in orbital mechanics, when a trailing Trojan is pulled forward into a wider orbit at higher initial speed, the initial gain it gets toward the planet is then reversed because the object slows down and is passed by the planet.  So relative to the planet, the object begins to lose ground along its new, outward orbit.


5.  The amount of energy given by the planet's gravity to the asteroid at this point is not enough to make its entire orbit wider in a circular sense, only to bend it into a more elliptical shape with an aphelion (far point from the Sun) outside the orbit of the planet.  Thus, as the asteroid drifts away relative to the planet and the latter's gravity diminishes, the object crosses the orbit of the planet and moves closer to the Sun.


6. As per the mechanics described above, because the asteroid now has a smaller orbit than the planet, its gravitational potential energy is transformed into kinetic energy and the object acquires greater speed than the planet.  It thus gains on the planet and moves closer to it on the inside track.  An identical process occurs for a leading Trojan as a trailing one, only the planet's gravity works to slow it down rather than speed it up, but the result is the same.


As I've tried to show in the above illustrations, Trojan orbits are "kidney" shaped, and the two "triangular" Lagrange points, L4 and L5, are their centers of motion - points in empty space that can be orbited stably.  The stability of these orbits is the reason asteroids are commonly found in them, and also why a number of space colonization proposals involve placing manned stations in them - particularly with respect to the Earth-Moon Lagrange points.  A few unmanned solar observatory probes are currently in Sun-Earth Trojan orbits.  

The other Lagrange points of a system (L1, L2, and L3) do not involve the same phenomena as L4 and L5 and provide only unstable orbits, which is why natural objects are generally not observed there, although they are useful for plotting low-energy spacecraft trajectories.  I have shown Lagrange point diagrams in previous entries in this series, but I will reiterate here for the sake of clarity and completeness - this illustration shows the Earth-Moon system and depicts the geometric relationship among the EML (Earth-Moon Lagrange) points:


The detailed description given above involves only the elements of motion that occur in a flat orbital plane, but as seen in the animation at the beginning, the reality is much more complicated - actual Trojan configurations are in 3D with significant motions occurring both above and below the orbital plane of the planet.  However, the description above more or less covers the basic principle of triangular Lagrange points and the Trojan orbits associated with them.  The other inner solar system asteroids and the Main Belt move in far more intuitive ways, but it's still worth seeing their relative dimensions and populousness illustrated in motion:

Another quirky topic is the Hilda group asteroids whose dynamic motions occur along a three-lobed configuration between the Main Belt and the orbit of Jupiter - I recommend going full screen to see the motions of individuals objects:

The three "lobes" of the Hilda group asteroids are a very striking feature, which is caused by a particular resonance with Jupiter.  Two of the lobes correspond to the radial positions of the Jupiter Trojan swarms, but are inward of them, and the third is on the opposite side of the Sun from Jupiter but also inward of its orbit.  I should note that the triangular configuration only occurs as a collective result of all objects in this particular resonance, and that individual Hildas do not in fact move along such a path - their orbits are standard ellipses whose perihelion and aphelion migrate relative to Jupiter along with the overall configuration.  Another animation of Hildas with larger-sized elements to make the motions clearer, which in the second half also shows how the Jupiter Trojans move:

I should note that because Jupiter is held immobile in the animation above, some of the Hilda motions depicted (e.g., "curlicues" at each lobe) are illusionary and do not occur when the motion of Jupiter is taken into consideration.  Individual objects entering the lobes are approaching aphelion, and their motion there is no different from the far point of any other elliptical orbit.  Nonetheless, the relationship among Hildas conforms to the overall triangular arrangement with the relative positions of individual objects constantly changing and yet regularly periodic, so there are important far-future implications for the potential utility of these objects that I discuss in a later section of this entry.

We will now look at the orbital and rotational behavior of our example asteroids introduced in Volume 1: 25143 Itokawa, 951 Gaspra, 433 Eros, 243 Ida and its moon Dactyl, 253 Mathilde, 21 Lutetia, and 4 Vesta.  Charts of orbital periods and rotational periods:  

Asteroid Orbital Period

Asteroid Rotational Period

There is nothing shocking the orbital periods given the objects' respective locations in the solar system (e.g., Itokawa is near Earth, and has a period between 1 and 2 years), but in terms of rotation, Mathilde is quite an outlier: While every other example object rotates in a matter of hours, this asteroid takes more than 17 days to complete a revolution.  Typically the interpretation of a radically slow rotational period is that a massive impact occurred at an angle and negated most of an object's angular momentum, which is not hard to believe given Mathilde's gaping maw of a crater that dominates an entire face of the body.  Even less shocking is Itokawa's relatively long period (12.1 hours), given its obvious composition as a pile of rubble: Significantly faster rotations would only have further weakened the hold of its already miniscule gravity and limited its ability to accumulate.

Actual motions of rotation can be quite complicated in an asteroid - especially low-mass, irregularly-shaped ones - because motion can occur in more than one axis.   However, all of our example asteroids have relatively simple, single-axis motions, and whatever extra rotational motion occurs would be in the form of precession and/or nutation of that axis.  Rotational videos, flyby videos, and images of our example asteroids:


Ida Rotation


Each object's axis is different relative to the shape of its mass.  As the video shows, Itokawa rotates in a flat spin around an axis somewhat off-center from the middle of its length.  Gaspra, as a somewhat flatter body, pinwheels around an axis through its broader face.  Eros tumbles end-over-end with its broader profile sweeping through its rotation.  Ida, like Gaspra and Itokawa, rotates in a flat spin around an axis through its broader profile.  I haven't found any useful information about the rotational configuration of Mathilde, and the flyby video is understandably not very helpful given the object's rather long rotational period.  Lutetia doesn't quite fit into any kind of shape category, so its axis bears no special description.  Vesta, as the second most massive body in the Main Belt, is a somewhat warped, oblong shape that rotates around an axis through its flatter profile such that its equatorial radius is much larger than its polar radius.  

2.  Size and Mass Characteristics:

Charts for mass, largest dimension (i.e., size), and equatorial surface gravity - although the last of those is much less meaningful for the smaller, irregular bodies with bizarrely-shaped gravity fields:

Asteroid Mass

Asteroid Size

Surface Gravity

The mass chart above is in orders of magnitude, meaning that each increment is ten times larger than the previous.  So Vesta, being of order 20, has a mass of magnitude ten billion times larger than Itokawa at order 10.  Earth, at order 24, has a mass of magnitude 10,000 times larger than Vesta.  Clearly this object is a cut above the rest in terms of mass, size, and gravity.  To give a sense of the spread of gravities, a 150-lb person would weigh 4.5 milligrams on Itokawa and take seven and a half minutes to fall 1 meter, but would weigh 3.9 lbs on Vesta and take only 8.8 seconds to fall that distance.  The other objects would follow a distribution similar to the relationships shown in the surface gravity chart.  

Below are rough visual size comparisons.  For individual comparisons, I only compare objects of roughly comparable size- mouse over to see the names of objects if unfamiliar and unlabeled:

Asteroid Comparison

Asteroid Comparison 2

Itokawa vs. ISS

Itokawa vs. Sears Tower

Itokawa and Methone (Saturn moon)

Itokawa and Comet Braille

Itokawa and Dactyl

Itokawa and Annefrank

Itokawa and Steins

Gaspra and Manhattan

Gaspra and Methone (Saturn moon)

Gaspra and Steins

Gaspra and Annefrank

Gaspra, Deimos, and Phobos

Gaspra and Halley's Comet

Gaspra and Calypso (Saturn moon)

Gaspra and Telesto (Saturn moon)

Gaspra and Helene (Saturn moon)

Gaspra and Thebe (Jupiter moon)

Gaspra and Pandora (Saturn moon)

Gaspra and Prometheus (Saturn moon)

Gaspra and Epimetheus (Saturn moon)

Eros and Cape Cod

Eros, Deimos, and Phobos

Eros and Calypso (Saturn moon)

Eros and Telesto (Saturn moon)

Eros and Helene (Saturn moon)

Eros and Pandora (Saturn moon)

Eros and Thebe (Jupiter Moon)

Eros and Prometheus (Saturn moon)

Eros and Epimetheus (Saturn moon)

Eros and Janus (saturn moon)

Ida and Hawaii

Ida, Deimos, and Phobos

Telesto (Saturn moon), Dactyl, and Ida

Helene (Saturn moon) and Ida

Pandora (Saturn moon) and Ida

Ida and Thebe (Jupiter moon)

Ida and Prometheus (Saturn moon)

Ida and Epimetheus (Saturn moon)

Lutetia and Sicily and Mathilde

Enceladus (saturn moon) and Vesta

Miranda (Uranus moon) and Vesta

Originally posted to Troubadour on Sat Jun 23, 2012 at 09:03 PM PDT.

Also republished by SciTech.

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