On August 28, 1993, the Galileo space craft passed within 2,400 kilometers of the asteroid 243 Ida.  Galileo passed through the asteroid belt, then returned for a gravity assist with the Earth, on its way to Jupiter.  The Solid State Imaging camera on Galileo took a number of images which were transmitted back to earth the following year.  The most remarkable thing about these pictures was the discovery of a tiny moon, Dactyl.  It appears to be orbiting the asteroid at a distance of 90 kilometers (56 miles).  One image showed Dactyl five and a half hours after the first view.  It showed Dactyl in the light of Ida shine, as the camera was pointing back toward the sun.  Seeing the little moon in various orbital positions allows for an estimate of its orbit.  Astronomers have found numerous other double asteroids and moons around asteroids in the last ten years.  It would seem that they are not a rare phenomena. 

Ida is classified as one of the S (stony) type asteroids.  This classification expects that it is largely made of mixtures of nickel-iron and silicates.  The possible orbit of Dactyl in the photos seems to restrict the density of Ida at 2.2 - 2.9 times the weight of water.  This is lighter than one would expect from an asteroid made of stone and metal.  Earths crust has lava rocks with a density of 2.8.  Porous sandstone on earth has a density of 2.3.  Perhaps the rocks of Ida and Dactyl are very porous with lots of cavities or cracks.  There was no evidence of ice or water which might contribute to its low density.

Ida is believed to be a member of the Koronis family of asteroids. This family may have come from a larger planetary body that was smashed in a collision. Ida's orbit is about 270,000,000 kilometers from the sun (1.8 AU). Ida orbits through the asteroid belt between Mars and Jupiter with a period 4.84 years. Ida is potato shaped 56 km long x 24 km x 21 km. I estimated the volume of Ida by a combination of cylinder and hemisphere volumes. A rough estimate of its volume would be a sphere with a radius of 16 kilometers. The photos showed that Ida rotates with a period of 4 hours and 38 minutes.

Dactyl is apparently made of a similar material and is egg shaped, 1.2 x 1.4 x 1.6 kilometers. If it were spherical, it would be about 1.4 kilometers in diameter. Spectral analysis by Galileo showed that Ida & Dactyl are composed of different ratios of similar compounds. The spectrum of Dactyl indicates less iron-nickel, and more volcanic type rocks. Ida and Dactyl have been battered by collisions leaving large and small craters. Were these collisions before, after, or concurrent with their orbital marriage? These are questions that cannot be answered with any certainty. In the picture above, Dactyl is slightly closer to the camera than Ida.

I am going to do some rough calculations involving Ida and Dactyl. These calculations are only approximations. The actual masses and orbital parameters are not known. These calculations are also gross simplifications. The capture of Dactyl is not a two-body problem. There are several ways that we could imagine the capture of Dactyl.

• 1. We could assume the capture of Dactyl only involved the gravity of Ida, Dactyl, and the Sun.  The attraction of the Sun is greater than that of the asteroid at distances greater than about 50 kilometers from Ida.  At 90 kilometers distance, the sun has four times as much attraction as Ida. The mass of Ida is so small, that the distant sun influences Dactyl's orbit much more than the close by asteroid.  The three body problem (the Sun, Ida, and Dactyl) is a complicated problem.  The tiny size of Ida makes the capture of Dactyl an extremely unlikely event.

• 2. It may be that the capture of Dactyl is impossible without a close encounter with a fourth body.  Perhaps this fourth body could somehow slow Dactyl down enough (act like a retro rocket).  This seems like an extremely unlikely event.

• 3. Maybe Dactyl and Ida were originally part of a larger planetary body.  What if this body was broken up in a catastrophic collision.  Perhaps they were ejected at similar velocities and in the same direction.  Perhaps this would allow them to interact gravitationally.  Most explosive events scatter their parts radially away from each other.  The breakup of a rack of billiard balls tends to scatter the balls in the general direction of the cue ball's travel.  Each chunk of a violent break up would likely depart at high speed.  In order for the breakup to occur, the energies involved have to be significant.  The parts would probably have too much energy (not enough time) to interact for a capture.

• 4. What if the break up of the planetesimal was caused by a close encounter with a large body within Roche's limit?  For a stone and metal planetesimal, a Roche's limit breakup would have to be at almost cloud top level with a large gravitational object like Jupiter.  There can be no doubt that the broken parts could travel off slowly together.  The Jupiter sized planet would most likely do the capturing, however.  An example of this type of breakup was the long string of comets known as Shoemaker-Levi 9.  The capture and eventual collision with Jupiter provided the most spectacular fireworks seen in many centuries.

The escape velocity for a body can be calculated by a formula.

V_esc = (2 G m / r)^1/2

I estimated the equivalent radius of Ida as 16,000 meters, the density as 2.9 (2,900 kilograms per cubic meter), and the mass of Ida as 5.8 x 10¹⁶ kilograms.  If these values are close, the escape velocity from the surface should be ~22 meters / second.

The escape velocity for earth is 11,180 meters/second.  A slow baseball pitch travels 26 meters/second (60 mph).  A fast pitch travels 40 meters/second (90 mph).  If your space suit gave you the required freedom of motion, you could throw a baseball at an initial velocity greater than the escape velocity.  The baseball would end up in a solar orbit independent of Ida.

Imagine a baseball thrown from Ida with an initial velocity of 23 meters/ second.  It will slow down gradually due to the pull of Ida but will never stop and fall back to the surface.

If the baseball departs with an initial velocity less than 22 meters/second, it will eventually slow down and return.

What if the initial velocity of our throw was exactly the escape velocity?  The escape velocity varies inversely with the square root of the distance from Ida.  As the baseball moves away from Ida, a unique escape velocity exists for that particular point in space relative to Ida.  Throughout its flight, the baseball will always be at the escape velocity for that point.  It will slow down due to the gravity of Ida, but it will never slow down enough to return.

Weight and Gravity on Ida

The weight of an object on earth depends on the value of gravity at the surface and the centrifugal force due to the rotation of the earth.  If we ignore the centrifugal force, the equation for g is:

g = G m / d²

G is the gravitational constant, m is the mass of the planet, and d is the distance from the center of the earth.  For earth, g = 9.8 meters/second². You might be more familiar with this expressed as 32 feet/second².

If we assumed a radius of 16 kilometers for a spherical Ida, the g on Ida is

g = 5.8 * 10¹⁶ kg * G / (16000 meters)²

or about .015 meters/second ² at the surface.  This is about half an inch per second ².

A little jump on Earth.

Imagine that you bend your knees and with all your effort jump straight up.  On earth you might be able to jump with an initial velocity of 3 meters/second. The distance formula for your jump would look like:

h = - 4.9 t² + 3t

where h is the height from the ground, t is the time in seconds from the beginning of the jump, -4.9 / sec² is due to the acceleration of gravity, and 3 meters/second is the initial upward velocity.  The first derivative gives the velocity at any time throughout the jump.

v = -9.8 t + 3

Your velocity is initially upwards (+), but is eventually stopped by the acceleration of gravity.  If you solve the first derivative for v=0, you find that when t= 0.306 seconds your upward velocity stops, and then you begin to fall back down to the earth.  Plugging this value, t= .306, back into the distance formula, you find that you can jump up .46 meters (1.5 feet) from the earth's surface.

A little jump on Ida

Have you seen the pictures of the astronauts on the moon doing their little shuffle jump?  Now let us try the same thing on Ida.  Feet together, knees bent, we spring up like a jack in a box.  Our height formula on Ida would look something like this:

h = -0.0075 t² + 3t

The first derivative, the velocity would look like this:

v = -0.015 t + 3

Solving this for v = 0 gives t = 200 seconds.  Our jump would last 3 1/3 minutes and we would go up 300 meters or about 1000 feet off the ground.  The final velocity upon return to the surface would be only 3 meters/second, even though we fell back from a height of 1000 feet.  If we landed on our feet, we should not be hurt because the "g" on Ida is so low.

An object on earth's surface weighs about 650 times as much as the same object on Ida's surface.  Imagine your 170 pounds on earth showing up as only 4 ounces on Ida.  It is important to recognize the difference between mass and weight.  You have the same mass on Ida (perhaps more since you would have to be wearing a space suit).  Your inertia (the measurement of mass) is still the same.  Because you weigh less, does not mean you can jump up with more than an initial velocity of 3 meters / second.  It still requires the same amount of force to accelerate your body mass on Ida as it does on Earth.

The Dactyl orbit

Dactyl is probably not in a circular orbit, but we will assume it is for simplicity.  We can estimate the volume of Dactyl as 1.4 x 10⁹ meters ³, and the mass as 4 x 10¹² kilograms.

V = [G( m _Dactyl +m _Ida )/90000]^1/2

The mean orbital velocity, at a distance of 90 kilometers from the center of Ida, should be approximately 6.5 meters/second.  The orbital velocity will vary considerably in an elliptical orbit.  It will increase at periapsis and decrease at apoapsis.  If Dactyl's orbit were circular, its period would be about 24 hours.

On the Dynamics of Capture

The outer moons of Saturn and Jupiter are thought to be captured.  Some of these moons rotate backwards, or have orbits that are out of the orbital plane of the inner moons.  Surely if Jupiter can capture a moon, Ida should be able to capture its little moon.  This is NOT the case however.  The more massive a planet, the easier it is for it to capture a moon.  The smaller Ida has a great disadvantage in capturing a moon of its own.  Here is why.

Let us go back to the escape velocity.  If we threw a baseball at more than 22 meters/second from Ida it would escape into a solar orbit.  If we threw the baseball at the escape velocity, ~22 meters/second, it's velocity would always equal the escape velocity at each altitude.  The velocity of the baseball decreases with increasing altitude as Ida's gravity continues to act on the baseball.  We can calculate the escape velocity for our baseball as:

V_esc = v_{esc surface} / [ (r + h)/ r ]^1/2

r=radius of Ida h=height above surface of Ida.  The escape velocity at the surface is assumed to be 22 meters/second. At the distance of Dactyl's orbit (90 km), the escape velocity is 9.3 meters/second.  In order for Ida to capture Dactyl 90 km away, their relative speed must not exceed 9.3 meters/second.

On the Capture of Dactyl

Imagine that Ida was to capture Dactyl today. We have estimated the mean orbital speed of Dactyl at 6.5 meters/second and the escape velocity at the 90-kilometer orbit as 9.3 meters/second. The escape velocity is the mean orbital velocity times 1.414 or (2)^1/2. Notice the narrow range of velocities. We have a window less than 3 meters/second wide between the orbital velocity and the escape velocity.  This is an extremely small window to pass through in order to achieve an orbit.

Let us try two scenarios for the capture of Dactyl. Let us assume that Dactyl was originally an Apollo asteroid. It was on a highly elliptical orbit about the sun. Let us further assume that its distance from the sun had reached a maximum, (aphelion). At that point it would be moving very slowly in relation to the sun. Imagine that Ida happens to pass close to that point at that particular time. Ida would be moving at right angles to the path of Dactyl at about 22,000 meters/second in relation to the sun. Their relative velocity is still 22,000 meters/second. Even though Dactyl was almost stationary in relation to the sun, its relative speed in relation to Ida is 3000 times too fast for a capture.

What if Dactyl were on an almost identical orbit with Ida around the sun. They are both traveling 22,000 meters/second in relation to the sun and have very low speeds in relation to each other. Let us assume that at great distance their relative speed was about -1 meter/second. As the pair slowly approached each other, the relative velocity would increase. Ida is about 10,000,000 times more massive than Dactyl, so Dactyl would begin to change its speed and also its orbit around the sun. At 500 km the acceleration of gravity due to Ida is .00001 meters / second². After 81,000 seconds (22.5 hours) the acceleration due to Ida would have increased the velocity of approach by much more than the escape velocity. Even though the accelerations are very small, the slow approach would allow the accelerations to accumulate. It would seem that we would inevitably miss the orbital eye of the needle. Dactyl would inevitably escape the minuscule gravitational embrace of Ida. The tug between Ida and the Sun would also cause a change in Dactyl's solar orbit.

Notice that at about 300 kilometers distance, the approach velocity begins to increase rapidly (that is the rate of change of distance from Ida).

After 76000 seconds (21 hours), we begin rapidly increasing speed and  exceed the escape velocity.  The velocity is negative because it is approaching Ida.

The sheparding moon of Saturn, because they are similar in mass, are thought to do an orbital dance.  As a slightly faster moon approaches its partner, it speeds up and changes the radius of its orbit around Saturn.  The lead moon also slows down and changes its orbit.  The result is that astronomers expect the moons to change places as they whirl around Saturn.  Dactyl is far too small to alter the orbit of Ida appreciably, and so they will not change places in their solar orbit.

You might ask, how do we get a satellite into orbit around a distant planet?  We fire retro rockets at just the right time to slow it down and so permit a capture.  Dactyl would need a huge bank of rockets to slow it down since its mass is perhaps 4 x 10¹² kilograms.  Of course we must assume that Dactyl has been captured by some natural process.  Somehow they have gotten married, because they certainly appear to be orbiting the sun as a pair, like the earth and its moon.

In 1999 another moon was discovered around the asteroid Eugenia. The photos were taken with a ground based telescope on Mauna Kea Hawaii.  The diameter of Eugenia is about 215 kilometers and its moon is about 13 kilometers.  As of this writing, astronomers have detected eight other moons around asteroids.  The largest is about 150 kilometers.  The smallest is a 100 meters rock orbiting the tiny near earth asteroid 2000 UG₁₁ every sixteen hours.

William of Occam's Razor: the Assumptions

Can we solve this mystery, not with changing the variables in our formulas, but by examining our assumptions about the nature of the universe?  Questioning assumptions is an extremely dangerous business.  When you question fundamental assumptions you are questioning everything at once.  Western people have settled the matter of fundamental assumptions many centuries ago, and those questions are no longer asked.  We are stout hearted, so we will ask the most ridiculous of questions.

What is the little assumption that the Greek philosophers invented 2300 years ago that affects everything all at once?  If this assumption were false, would the universe still appear as it does to us today?  If this assumption were false, could many of the mysteries of astronomy be "explained?"  Is there evidence from archaeoastronomy which violates the fundamental assumption?  Could we "explain," in a non-mathematical way, why Ida captured Dactyl?

Remember that an assumption is an idea accepted without proof.  A fundamental assumptions is accepted "a priori", apart from any evidence or experiment.  It is accepted before a system of knowledge is even invented.  Fundamental assumptions are foundational to almost everything you think you know.  Fundamental assumptions cannot be proved true.  They can be falsified, however.  Think about it!

Why should you think about fundamental things, especially at the level of assumptions?  Isn't it ridiculous to examine something that no one ever questions?  Historically, even higher level assumptions are rarely questioned.  According to Thomas Kuhn, only when a replacement paradigm can explain things in a better way, does a thinking revolution occur.  There are many reasons why you should test the fundamental assumption.  Dactyl is only one small reason.  Every time you encounter a great cosmic or solar system mystery --- such as the missing mass of the universe -- ask yourself this:  Is there a simple idea at the level of a fundamental assumption that solves this problem in a simple non complicated (read non mathematical) way?  Think about it!

If you do start thinking about the fundamental assumption, I suggest keeping it to yourself.  There is nothing more foolish sounding than trying to explain the fundamental assumption to people who have been trained all of their lives to accept it by faith.

I will leave you with a quote from the Bible.  It was written by the wise man Solomon almost 3000 years ago.  "When I gave my heart to know wisdom and to see the task which has been done on the earth (even though one should never sleep day or night), and I saw every work of God, I concluded that man cannot discover the work which has been done under the sun.  Even though man should seek laboriously, he will not discover; although the wise man should say, 'I know,' he cannot discover."

Notice that even the solar system, "under the sun," can never be understood.  If the fundamental assumption were false, people would think they know, but they would always be wrong.  If the fundamental assumption were false, science itself would fail even while amazing us with the power of its mathematical calculations.  Think about it!

Copyright August 1997 by Victor J. McAllister  Do not republish without the approval of the author.
Last modified February 5, 2004.

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