# Why are gravitational perturbations stronger at larger semi-major axes?

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Why are mechanisms such as the Kozai-Lidov effect more prominent at at large semi-major axes?

If we had a binary system of a primary and secondary body, with the third perturber as the Sun. Why is it that the secondary feels more perturbations from the gravitational influence of the sun the further away it is from the primary?

"Why is it that the secondary feels more perturbations from the gravitational influence of the sun the further away it is from the primary?"

In brief, it is because the net perturbing acceleration on the secondary is just the (vector) difference between (a) the accelerative attraction towards the perturbing body experienced by the secondary, and (b) the accelerative attraction towards the perturbing body experienced by the primary.

Thus, the closer the secondary is to the primary, the more nearly equal in size and direction are those two attractions towards the perturber, and the closer to zero is their vector difference. Another result is that the the more similar are the changes in velocity in size and direction produced by the perturbing accelerations in the primary and the secondary, and the closer to zero is the resulting perturbation in their motions relative to each other.

This has been known for a long time as a consequence of Newton's 6th corollary to the laws of motion: "If bodies, anyhow moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves after the same manner as if they had been urged by no such forces."

Because there can be immense variety in the possible trajectories of the primary and secondary apart from the perturbations, any detailed illustrations can speedily develop massive and intricate trigonometrical expressions.

But in all cases, including that of the Kozai-Lidov effect, the scale of the effect, however intricate its form, depends on the size of the net perturbing accelerations.

A highly simplified configuration can at least show by example how a net perturbing force, affecting the relative motion of primary and secondary, is almost directly proportional to the first power of the distance between primary and secondary -- though it also depends of course on further factors due to changes in angular configuration.

The diagram below indicates some highly simplified configurations.

Suppose first that the primary (E) and secondary (M) are for one instant in line with the perturbing body (S), with M at M1 between E and S. Let s stand for distance ES, and d for distance EM (with d << s). Suppose also that the mass of E and M are negligibly small relative to the mass of S (though not negligible relative to each other).

With these approximations, the accelerative attractions of S on M and S on E are respectively $k/(s-d)^2$ and $k/s^2 ,$ and the net perturbing acceleration on M is the difference $( k/(s-d)^2 - k/s^2 ) .$

Putting $s(1-d/s)$ for (s-d), and using the binomial expansion of $1/(1-d/s)^2$ , one sees that the terms in $k/s^2$ cancel, leaving the net perturbing force as $k/s^2 * (2d/s)$ , plus terms in higher powers of d/s, i.e in $k d^2/s^4$ and so on.

Where d is very much smaller than s, the higher-power terms in d/s can be neglected, and then the net perturbing acceleration on M at M1 in the chosen example-configuration is closely approximated by $+2 k d / s^3$ , away from E and towards S.

If instead the configuration has M at M2 so that E is in line between M and S, then the net perturbing acceleration on M clearly becomes $~ -2 k d / s^3$ , i.e. away from E and away from S.

If instead M is at M3, with the line EM3 at right-angles to ES, and if also the angle ESM3 can be treated as small enough that its cosine can be approximated as 1 and its sine as d/s , then it is easily found that the net perturbing acceleration on M at M3 is approximately $k d/s^3$ towards E, again neglecting higher powers of d/s.

If M is at an intermediate position M4, and D represents the angle ESM4, it can be seen by using a little more trigonometry that the net perturbing acceleration on M at M4, under the assumptions already made, has a component parallel to line ES of approximately $+2 k d cos D / s^3$, and a component perpendicular to line ES (acting always towards line ES) of approximately $k d sin D / s^3$.

All the components are proportional to the E-M separation d, to the extent that the omitted series of terms in higher powers of d/s can be treated as negligible as has been done here.

## Spacecraft Trajectories

In even the earliest lunar and planetary missions, it was necessary to keep track of the spacecraft's trajectory and issue commands for onboard functions both engineering and scientific. Gradually a humans-and-machines art developed, represented today by large rooms full of people and displays backed by buildings full of computers and data systems. Initially centered in main theaters, as missions have become more complex, these facilities have become dispersed, providing work spaces for the many specialized flight management and scientific teams working during a mission. With the Internet and other modern communications available, scientists can now reside at their home institutions and participate in missions in real time.

The latest trend is toward increasing onboard autonomy, which holds the promise of reducing the large staffing needed round the clock to control missions. Some degree of autonomy is needed anyway in deep space, simply because of the round-trip signal times to distant spacecraft, tens of minutes for Mars and Venus, and many hours in the outer solar system.

Operations have become more and more dependent on software whose design and verification now constitute one of the main cost items in each new mission's budget. With the maturing of the operations art have come numerous stories of remarkable rescues when a distant robot (or, as in Apollo 13, a human crew) got into trouble, but there are also instances where a mistake on Earth sent a mission to oblivion.

## 1. Introduction

Modified Newtonian Dynamics (MOND) has been proposed in [15] as an alternative to the dark matter paradigm (see [16] ). At the non-relativistic level, the best formulation of MOND is the modified Poisson equation (see [2] ),

where ρ is the density of ordinary (baryonic) matter, U is the gravitational potential, g = ∇ U is the gravitational field and g = ∥ g ∥ its ordinary Euclidean norm. The modification of the Poisson equation is encoded in the MOND function μ ( y ) of the single argument y ≡ g / a 0 , where a 0 = 1.2 × 10 − 10 m / s 2 denotes the MOND constant acceleration scale. The MOND function μ ( y ) tends to 1 for y ≫ 1 in a Newtonian strong-field regime, and tends to y for y ≪ 1 in a weak gravitational field regime. According to [14] , [3] and [4] the most important effect of MOND in the Solar System is the External Field Effect (EFE) which produces two corrections (parametrized by two quantities Q 2 and Q 4 ) to the Newtonian potential which increase with the distance to the Sun. In other words, objects with a large semi-major axis are more sensitive to the effects of perturbations induced by MOND formalized by a modified Poisson equation.

Hence, we study comets with large semi-major axes to determine the magnitude of the effects of MOND theory. Indeed, the comets are good candidates because they not only go far from the Sun on a very eccentric orbit but also come back close to the earth to be observed accurately. When the comets approach the Sun, their gravitational orbit is affected by the sublimation of ices from their nucleus surface. The outgassing triggers non-gravitational forces that significantly modify the orbit of the comet close to the Sun (under 3 AU). These non-gravitational forces have been modeled for the first time in [12] and then improved in [13] . Other more physical approaches for the non-gravitational forces have been developed in [19] , [18] , [6] and [11] . These last models take into account outgassing from only a few areas on the nucleus which describes more accurately the observations made by space probes.

The model developed in [13] to compute the non-gravitational forces is both sufficient to study cometary orbits and more easily implemented than the more sophisticated model. This model is used to generate cometary ephemeris and gives a good estimate of the non-gravitationnal effect for cometary orbits. These non-gravitationnal forces are obtained by fitting the astrometrical data but it is important to take into account all of the small effects, such as relativistic terms, to estimate correctly the outgassing (see [10] ). That is why the main goal of this paper is to quantify what would be the MOND perturbation on comets if this theory is validated and what is the maximum order of magnitude of this effect.

In [9] and [8] , the authors used the formalism developped in [3] and [4] to constrain the quantity Q 2 with the collected data of the Cassini spacecraft mission. Even though the authors claimed that the range of values of Q 2 are drastically restricted with that set of data, we choose to keep all the different values of Q 2 in order to obtain the extreme variations of the comet orbits as in [3] and [4] .

The plan of the paper is as follows :

In section 2 , we present a brief reminder about the Gauss equation of the perturbed two body problem and the implementation of the non-gravitational and MOND perturbations. Section 3 shows the consequence in terms of secular variation of the orbital elements due to the non-gravitational and the MOND perturbations of three comets. We conclude in section 4 and give some prospects.

## The Discovery of Uranus

After a few nights, he found it was moving slowly relative to the stars, and thus was not a star.

Within a few months, astronomers had confirmed it was not a comet, but was in fact a new planet orbiting beyond Saturn!

To appear resolved in his small telescope, it had to be bigger than the Earth.

The new planet was eventually named Uranus, and made Herschel's fame, starting his career as one of the most distinguished astronomers of the 18th Century.

The new planet had a semi-major axis of a=19.2 AU. The prediction from the Titius Bode Law was a=19.6 AU!

All of a sudden, astronomers began to wonder if there wasn't something to the Titius-Bode Law after all.

In particular, was there really a "missing" 5th planet at a=2.8AU between Mars and Jupiter?

## Translational Motion

### 3.2 Effect of Gravity on the Vertical Jump

The weight of an object depends on the mass and size of the planet on which it is located. The gravitational constant of the moon, for example, is one-sixth that of the Earth therefore, the weight of a given object on the moon is one-sixth its weight on the Earth.

From Eq. 3.11 , the height of the jump on the Earth is

The force F m that accelerates the body upward depends on the strength of the leg muscles, and for a given person this force is the same on the moon as on the Earth. Similarly, the lowering of the center of gravity c is unchanged with location. With F ′ the reaction force on the moon = F m + W ′ , the height of the jump on the moon ( H ′ ) is

Here W ′ is the weight of the person on the moon (i.e., W ′ = W / 6 ). The ratio of the jumping heights at the two locations is

That is, if a person can jump to a height of 60 cm on Earth, that same person can jump up 3.6 m on the moon.

## Acksblog

The main asteroid belt (MAB) occupies a donut-shaped (toroidal) region far from the Earth, between the orbits of Mars and Jupiter. The asteroids which comprise it are not as closely packed as movies on the TV depict them. Although there are perhaps one million asteroids larger than 1 km in the main belt, the space they occupy is so vast that the average distance between them is > 500,000 kilometers. They all orbit the Sun in the same sense (counter-clockwise) as the planets.

The primary clue to their recent creation is the relatively large average inclination of their orbits. The orbits of all the planets lie within about 3 degrees of the orbit of the Earth, which astronomers use to define zero inclination, thus the solar system as a whole is quite flat. However, the main belt asteroids’ orbital inclinations range up to some 28 degrees, with 90% between 0 and 20 degrees. Astronomers believe that these are rocky bodies out of which the terrestrial planets accreted 4.6 billion years ago, but that because of the gravitational influence of Jupiter, they were prevented from accreting to form a planet-like body.

The question which modern astronomy avoids is: If they are 4.6 billion years old, why have they not settled into the flat plane of the solar system, like all of the planets?

Cyclic Catastrophism

In the CC scenario the main belt asteroids were ejected from Jupiter at high inclinations by an enormous jet of hot gases within the last 6,000 years, thus have not had time to settle into the plane of the solar system. The planet Venus was created, as were all the more ancient terrestrial planets, by a high energy impact on Jupiter 6,000 years ago. Because Jupiter is a frozen, solid, methane gas hydrate body, the impact triggered a fusion explosion so enormous that a plasma cloud, thousands of times the size of Jupiter itself, rebounded and the heavy elements within that roiling hot cloud formed the hot planet Venus we see today.

The nuclear fusion explosion out of which Venus was created, also resulted in a continuous fusion furnace in the impact crater on Jupiter, continued to burn so fiercely that it sent a jet of flaming gases a million and a half miles out into space, diminishing only slowly over the last six thousand of years. The jet swept around as the giant planet rotated, cooling, combining, and freezing as the gas expanded out into space. The most well known satellites of Jupiter, the Galilean moons, formed hot at the time of Venus’ birth in their current synchronized orbits, but were repeatedly coated with material from the jet for millennia. Indeed, the slowly diminishing jet is manifested in the puzzling differences and composition of these four bodies. The jet material striking the outer moons had frozen into large chunks of ice before impacting and causing craters. The jet was so hot at the radius of Europa in the first few millennia, that only the heavy elements could condense on it forming a rocky-iron core, then with further cooling the great mass of water, which comprised the bulk of the jet was later able to condense to form its ocean surface, which, because of its recent formation, is still fluid beneath the icy surface.on Europa. Io received the most and the hottest effect of the jet and has always been too hot for water to condense on it. These moons are not heated by gravitational tugging, but because of their recent fiery birth.

Fig 2 Main belt asteroid 253 Mathilde photographed by Near

However, because the jet was located at 22.5 degrees south latitude, the position currently well marked by the Great Red Spot, much of the jetted material missed the Galilean moons and accreted (froze) to form all of the main belt asteroids, similar to the bodies which produced the impact craters on Ganymede and Callisto. As a result of their formation from the body of Jupiter, the main belt asteroids have the same proportion of the elements as the giant planet. 253 Mathilde, shown in Fig. 2 has a density of only 1.3, very similar to Jupiter’s 1.33. It is porus because it formed from vapor in space. The image shows that Mathilde is a single solid low density body – not a ‘loose rubble pile’ . They are primarily water but have a proportional abundance of all the heavier elements, including iron. As a result of their condensing/freezing while still within the magnetic field of Jupiter they each incorporate a permanent magnetic field. These same properties – low density and magnetic fields characterize all main belt asteroids observed up-close. The main belt asteroids Ida and Gaspra have displayed magnetic field effects. Since astronomers believe that meteorites, with densities of 3 g/cm^3 are bodies originally deflected from the Main Asteroid Belt, they have difficulty explaining the unexpectedly low measured densities measured to date, leading to the notionthat they are perhaps not rigid bodies, but are ‘rubble piles’ of rocks held together by gravity.

The Cyclic Catastrophism scenario very nicely explains the high orbital inclinations of the main belt asteroids. First, by the fact that they were only created recently and therefore have not had time to be slowly drawn into the plane of the solar system. More quantitatively, it also explains the range of their inclinations. Three factors come into play: (a) the orbital inclination of Jupiter itself 1.3 degrees (b) the inclination of its axis of rotation (obliquity), 3.13 degrees and (c) the latitude of the Great Red Spot, which marks the crater from which the the gas jetted for some six millennia, -22.5 degrees. The sum of all these factors, approximately 27 degrees, gives the possible range of inclinations of the bodies which formed from the jet shooting out of Jupiter. The figure above expresses in 𔃳D’ the orbital inclinations on the left scale as a function of the semi-major axis at the bottom, while the numbers of asteroids is given in the form of a color scale. Note that the inclinations shown are limited to the precise range consistent the cyclic catastrophism scenario. The gaps in the semi-major axes are due to gravitational resonances with Jupiter which excluded certain orbits.

The semi-major axes of the main belt asteroids varied considerably due to the rotation (spin) of the planet. Jupiter now rotates with a period of about ten hours, but rotated considerably faster, perhaps in seven hours before the Venus impact. The slowing of Jupiter’s rotation which has been taking place recently was due to the angular momentum ejected by the enormous jet over the last six millennia. The monotonic ‘tail’ end of this rotational slowing, which continued until about 1930, is currently imagined to represent the ‘drift’ of the Great Red Spot, and is not recognized as the rotational slowing of Jupiter.

Bodies which froze from the jet when directed parallel to Jupiter’s orbital motion vector were given a higher orbital velocity and thus a larger semi-major axis, in fact these bodies may comprise the Kuiper belt. The streams ejected in the opposite direction would have attained lower semi-major axes, resulting in highly eccentric orbits of bodies which decay and eventually impact the Sun (Kreutzer ‘comets’), causing sunspots. The permanent magnetic fields of these bodies have a profound effect on the Suns magnetic field, and result in the known presence of water in sunspots. Those bodies ejected in the intermediate directions comprise the majority of the observed main belt asteroids.

Upper left drawing is Jupiter with jet extending upward several planetary diameters

Apparently the jet shooting from Jupiter was still sufficiently large to be observed in the ninth century at which time a drawing of it was included in an Arabic text. This drawing was a classification of (temporary) comets in terms of planetary (permanent) characteristics at that date. The title of the document is not known, but was probably a short epistle on comets, not a longer text which is termed a Kitab al-Mughni. A really interesting aspect of this depiction of Jupiter is the implication that an Arab culture had astronomical telescopes in the ninth century.

The icy main belt asteroids have nothing whatsoever to do with the rocky-iron near earth asteroids, which are the result of tens of thousands convulsions within priori-Mars as it orbited the Earth only 32,000 km distant, up until 700 BC.

## Comet ISON News Links

Comet ISON will pass Mars at around 1600 GMT (noon in the eastern USA) today at a distance of about 7 million miles. Unfortunately, it is right beside the moon this morning in Leo, so it would be quite difficult to see even with a telescope. (The moon will be out-of-the-way after Oct 5).

I made a video and a Mathematica demonstration showing the path of ISON through the solar system, but there is a much nicer interactive viewer at solarsystemscope.com.

## Collapsing orbit due to gravitational wave

Your equation looks interesting, but I am not able to interpret its format. Can you possibly please reformat it?

When I first looked at your post, the format of the equations were unclear on my screen for some unknown reason. I apologize for my confusion. I can now see equations about which I have a few questions.

Is G the constant of Gravity? Are there any assumptions about it's units?

I am guessing V represents a coordinate corresponding to a volume, and the integral is over a volume space Σ. Is this correct? What confuses me it the relationship between Σ and the space involving the orbits of interest.

I get that the integral gives a value for matrix element Iij based in the factors xi and xj. What I do not fully get is the relationship between Σ and the two variables xi and xj.

I do not understand the notion of the two vertical bars with t-||r||.

I am unsure of my understanding the subscript k.
I am guessing k = i = j. Is this correct?
If so, then Qkk = (2/3) Ikk. Is this correct?

I am also guessing that T00 is the element of a tensor corresponding to t,t. Is this correct? Also, can you refer me to a source in which the definition of this T tensor is defined in detail?

Where do the masses of the two objects fit into these equations?

I would much appreciate any help you can give me.

i) ##G## is the gravitational constant [you can set ##G=1## if you like].
ii) ##dV## is the spatial volume element, e.g. for example simply ##dV = dx^1 dx^2 dx^3## in Cartesians.
iii) ##Sigma## is some subset of ##mathbb^3## containing the bodies of interest, over which you do the volume integral.
iv) ##||mathbf|| := sqrt<(x^1)^2 + (x^2)^2 + (x^3)^2>## if ##mathbf = (x^1, x^2, x^3)^T## and ##t - ||mathbf||## can be called the retarded time.
v) ##k## is summed over, whilst ##i## & ##j## free [c.f. Einstein summation convention].

vi) ##T^<00>## is just the energy density. For example if you have two particles orbiting each other, with positions ##mathbf_1(t)## and ##mathbf_2(t)## and masses ##m_1## and ##m_2##, then you could write ##T^<00>(mathbf,t) = m_1 delta^<(3)>(mathbf - mathbf_1(t)) + m_2 delta^<(3)>(mathbf - mathbf_2(t))##.

[N.B. ##delta^<(3)>(mathbf - mathbf) equiv delta(x^1 - u^1) delta(x^2 - u^2) delta(x^3 - u^3)##].

MTW has an approximation of the formula given above in a post by ergosphere, which is known as the quadrupole formula. See page 978, section $36.2, "Power radiated in terms of internal p ower flow". In non-geometric units, MTW's formula, which is derived in the context of one body orbiting another is: ##P_## is the power (energy/unit time) radiated away by gravitational waves. It's a weak field approximation, so among other assumptions we are assumed there is no significant gravitational time dilation. The formula will work approximately for things like the Hulse-Taylor binary, it won't apply (nor will the quadrupole formula apply) in the strong-field regime of a pair of inspiraling black holes. ##P_0## is a constant, equal to c^5/G, G being the gravitational constant. Numerically, in SI units it's ##approx 3.62 , 10^<52>## watts. The large size of ##P_0## means that the ratio of the ##P_## to this constant in the weak field regime is small, much less than one. ##P_## is the internal power flow of the system. It's described as the product of: The argument uses Kepler's law (this is a weak field approximation!), which in this context is: That's why it involves the ratio of the square of the radius of the orbit and the cube of the orbital period. ##P_## is equivalent to the non-spherical part of the energy of the system, ##approx m v^2## per unit time, i.e. the non-spherical part of the power of the system. The units work out, the square of a power divided by a constant power is a power. This is an approximation of an approximation. There are various constant factors omitted. If you still want to understand the quadrupole formulation better, the ##I_## is rather similar to the moment of inertia tensor, if you are familiar with the moment of inertia tensor. Then some diagonal elements are subtracted to make the trace of ##I_## vanish, making it the so-called "reduced" quadrupole tensor. Finally, to use the quardrupole formula, we need to take the third time derivative of this tensor. And to get the total emitted power, we'd have to project the resulting tensor expression to various angles from the source, then integrate. The complexities make it a bit hard to understand the physics, the formulation in terms of internal power flow makes the physics a bit more understandable. However, if you are interested in cases other than GW's emitted by a circular orbit, it's unclear to me how well MTW's formula would apply. ## Answers and Replies Yes and no. You are correct that the distance between two masses depends on ##a_1+a_2##. However, as you note, one of those accelerations is utterly negligible. Furthermore, if you work in an inertial frame then the accelerations of each object genuinely are independent of their mass. It's only if you adopt the (accelerating) rest frame of one object that the other's acceleration depends on its mass. So, in short, there are (usually unwritten) caveats to the "gravitational acceleration is independent of mass" claim. It's not wrong, but it has some assumptions about how you are doing the measuring of acceleration. Thanks for the awnser. But why not? Isn’t the force on m1 the same as on m2 because of Newton’s third law? Yes and no. You are correct that the distance between two masses depends on ##a_1+a_2##. However, as you note, one of those accelerations is utterly negligible. Furthermore, if you work in an inertial frame then the accelerations of each object genuinely are independent of their mass. It's only if you adopt the (accelerating) rest frame of one object that the other's acceleration depends on its mass. So, in short, there are (usually unwritten) caveats to the "gravitational acceleration is independent of mass" claim. It's not wrong, but it has some assumptions about how you are doing the measuring of acceleration. Take this case, something more familiar maybe. Two people are pulling a rope - one to the left and one to the right so that the rope is in tension. Each exerts a force F on their end of the rope to have no acceleration. The 'total' force does not become 2F. The force on the left person from the rope is F. The force on the rope by the left person is F, but of the opposite sign, or direction. The force on the right person from the rope and on the rope by the right person follow the same statics. Nowhere does a force of 2F come into play. First of all forces are vectors. The gravitational interaction between two forces is given by the force on particle 1,$vec_<12>=-frac<|vec_1-vec_2|^2> frac_1-vec_2><|vec_1-vec_2|> = - G m_1 m_2 frac_1-vec_2><|vec_1-vec_2|^3>.$The force on particle 2 is$vec_<21>=-vec_<12>,$in accordance with Newton's 3rd Law. The equations of motion are$m_1 ddot>_1=vec_<12>, quad m_2 ddot>_2=vec_<21>.$Now, because ##vec_<12>+vec_<21>=0##, it's a good idea to introduce the center-of-mass coordinates$vec=frac<1> (m_1 vec_1+m_2 vec_2).$Using the equations of motion leads to$ddot>=0.$The center of mass is thus moving with constant velocity. To describe the motion further it's obviously a good idea to introduce the relative position vector ##vec=vec_2-vec_1##. Using ##vec## and ##vec## you can easily express ##vec_1## and ##vec_2## via these new coordinates, and the equation of motion for ##vec## turns out to be$mu ddot>=-G m_1 m_2 frac><|vec|^3>=vec(vec), qquad (*)$where the "reduced mass" is defined by$mu=frac.\$
Thus the further solution of the equations of motion is reduced to the motion of one "quasi-particle" with the reduced mass ##vec## with the force given on the right-hand side of (*).

## Why are gravitational perturbations stronger at larger semi-major axes? - Astronomy

To be able to answer this question, we need define what a planet is, or is not.

### What is a Planet ?

We now know that the Earth too is a planet, and that all these planets orbit the Sun.

The Moon, and the planetary satellites, orbit planets, so one might start by defining planets as objects that orbit the Sun directly.

A star is an object that generates energy stably by fusion of Hydrogen. Objects more massive than about 0.076 solar masses can do this.

This image is of a nearby M dwarf, Gliese 229. The faint object to the right is a companion, Gliese 229B, which appears to be a brown dwarfs. Note that Gleise 229 is not as big as it appears here: it is a point of light. It appears large on the detector because it is bright, and the image is stretched to bring up the faint details, including the Brown Dwarf. (Click here for the HST press release.)

### The Discovery of the Outer Solar System

In 1781, William Herschel discovered Uranus. He noted it as a possible planetary nebula because it was greenish and presented a resolved disk (stars are still points of light as seen through telescopes, with very few exceptions). He reobserved it later, and found that it had moved, and so could not be a planetary nebula. Follow-up observations yielded an orbit, and showed that it was more distant that Saturn, and nearly as large as Saturn. It was clearly a new planet.

In 1772, J.E. Bode published a note earlier stated by J.D. Titius, remarking on a simple mathematical relation for the semi-major axes of the planetary orbits. Mathematically, the formula is a = (2 n X 3 + 4) / 10 where a is the semimajor axis of the orbit and n is an index, beginning at -1 for Mercury, 0 for Venus, 1 for Earth, 2 for Mars, 4 for Jupiter, and 5 for Saturn. If you set 2 -1 = 0, the predicted semi-major axes agree with the true semi-major axes to within a few percent. The Titius-Bode law is empirical: there is no physical reason why it should hold, but it has proven of some use as a predictor.

After its discovery in 1781, it was found that Uranus, at 19.2 astronomical units, satisfied the Titius-Bode law for n=6.

On January 1 1801, G. Piazzi discovered an object, Ceres, that orbited between Mars and Jupiter, in the spot where the Titius-Bode law predicted an object for n=3. Ceres is fairly faint, and we now know that it is small, with a radius of 993 km. Following Piazzi's discovery, an number of other objects, including Juno, Pallas, and Vesta, were discovered to have similar orbits. There are nearly 10,000 objects now known which have orbits between Mars and Venus: collectively, these are the asteroids.

In the 1840s, the position of Uranus deviated from its predicted orbit by one minute of arc. Urbain Leverrier, working in Paris, and J.C. Adams, working in London, independently predicted the existence of another planet whose gravitational pull would affect, or perturb, Uranus's orbit. This lead to the discovery of Neptune, which had been seen but not recognized as a planet, by Galileo, some 243 years earlier. Neptune is at a distance of 30 AU, in disagreement with the Titius-Bode law prediction of 40 AU (for n=7).

Even after accounting for perturbations from Neptune, Uranus appeared to show some residual perturbations. This lead to the search for Planet X. In 1930, after many grueling years of blinking plates in the search for a moving object, Clyde Tombaugh, working at the Lowell Observatory, identified the object we now know as Pluto. With a mean distance of 40 AU, Pluto satisfies the Titius-Bode law for n=7. Pluto turned out to be a lot fainter, and smaller, than expected, and could not have been responsible for the perturbations in Uranus' orbit. Ironically, later analysis showed that the perturbations that led to the search for Pluto were not real, and were likely due to observational error.

Further information about Pluto is available at the URL referred to in Christine Lavin's song.

What of the Titius-Bode law? It is not a true law of nature, like Newton's laws. There seems to be no physical reason for it. That there is some mathematical regularity to the spacing of objects in the inner solar system seems to be a natural consequence of the way planets form, and gravitationally interact with each other. If planets get too close together, perturbations either cause them to collide, or one to get ejected from the solar system. Mathematical modelling of this process shows that you always end up with distances between the planets that can be approximated as a geometric series.

### Characteristics of the Planets

• They orbit in or near the ecliptic. Excluding Pluto, the greatest inclination to the ecliptic is 7 o (Mercury).
• They have nearly circular orbits. With the exceptions of Mercury and Pluto, the largest orbital eccentricity is less than 10%.

A second salient characteristic of planets is their density. With the exceptions of Mercury and Venus, all the planets have satellites. We can use Newton's laws to determine the masses of the planets. Masses of Mercury and Venus are determined most accurately from tracking their gravitational pull on spacecraft we've sent there. Planetary volumes come from measuring their apparent size and knowing their distance from the earth (using Kepler's laws and a bit of geometry).

The density of a planet gives away its bulk composition. We know that gas is low density even a highly compressed gas like the Sun only has a density of 1.4 gm/cc. Liquids and ices have densities near 1 (the density of water is 1 gm/cc note, however, that liquids will evaporate at low pressures). Rock has densities of about 3 metals have higher densities. The terrestrial planets are rocky with metallic cores the Jovian planets are gaseous with rocky/metallic cores and icy mantles.

### Terrestrial Planets

Terrestrial planets are small and rocky. They have metallic cores. They were not large enough to sweep up significant atmospheres as they formed, so any atmospheres they have today are from gas trapped in the planet as it formed, and released, mainly through volcanic and other tectonic activity.

Gravitational settling of the planet (heavy atoms sinking, with lighter atoms rising) leads to the formation of a dense core and generates heat from the release of gravitational potential energy. In addition, the decay of naturally-occuring radioactive isotopes, mainly Thorium and Uranium, is an important heat source in planets. This heat must be radiated away through the planetary surface. This heat in the planetary core (the Earth's core is molten nickel-iron) drives convection in the mantle, which generates volcanic anjd seismic activity, and causes the continental plates to move. This is called plate tectonics.

The amount of heat generated is proportional to the mass, or volume (the densities are all similar), and so is proportional to the cube of the radius. The luminosity, or the rate at which the planet can radiate away this heat, is proportional to the surface area, or the square of the radius. Therefore larger planets retain more heat, and remain tectonically active longer.

Mercury is heavily cratered. The smallest of the terrestrial planets, and the closest to the Sun, Mercury has no significant atmosphere. This is attributable to the fact that neither its low gravity nor high temperature are conducive to retaining an atmosphere. Mercury is not tectonically-active.

The other terrestrial planets, Venus, Earth, and Mars, all have atmospheres, and all have evidence of tectonic activity. Earth is tectonically-active now Mars may have been as recently as half a billion years ago. All three show evidence of craters, and erosion from wind and water.

Venus has a thick atmosphere of carbon dioxide with sulfuric acid clouds. It has no liquid water. Venus may once have had water: the Venusian atmosphere today is the result of a runaway greenhouse effect.

The terrestrial atmosphere is unique in the solar system because of its high oxygen content. Oxygen is highly reactive, and is present only because it is continuously generated by plants. Earth is also unique in having liquid water on its surface.

• its low gravity, and
• the lack of tectonic activity, which replenishes atmospheric gasses

### Jovian Planets

Jupiter has a composition close to that of the Sun. The other Jovian planets are smaller, and have retained less hydrogen. All have densities close to 1 gm/cc, reflecting their gas and ice composition. The satellites of the Jovian planets have densities in the 1-3 gm/cc range, reflecting their compositions of ice and rock.

### Formation of the Planets

The clear correlation between the density of a planet and its proximity to the Sun can be explained in terms of the condensation sequence.

At temperatures above about 1500K, there are no solids: everything is gaseous. In the inner disk, about where Mercury is today, the temperature was close to 1000K. At these temperatures only metals and highly refractory minerals, like metal oxides and silicates, could condense out of the gas. Further out in the solar nebula, near where Earth and Mars are today, less refractory minerals, including feldspars, triolite and carbonaceous compounds, could condense. These have lower densities than the highly refractory minerals. At about 5 AU, where Jupiter is today, ices could form. The most common elements (aside from hydrogen and helium) are oxygen, carbon, and nitrogen their compounds, water, methane, and ammonia, are the main constituent of the common ices. Ices and carbonaceous compounds, which evaporate at low temperatures, are called volatiles.

Because there was more material to condense at low temperatures, the Jovian planets grew large rocky cores (Jupiter's core is about 10 Earth masses), which could collect lots of ice. The gravitational pull of these masses, and the low temperatures, meant that the Jovian planets could sweep up large gaseous atmospheres of hydrogen and helium. The overall composition of Jupiter is close to the solar composition, while the terrestrial planets have insufficient gravity to retain large atmospheres. At the temperature and gravity of the Earth, hydrogen escapes into space because of its thermal velocity.

The planets grew through collisions of small particles. At low relative velocities, colliding particles often stick. This is especially true if they are coated with ices. As particles stick, they grow, and their gravitational pull increases. This makes it more likely that future collisions will occur. A gravitational runaway occurs, with particles colliding and sticking, and objects called planetesimals growing. Computer simulations show that within a million years or so after the grains begin sticking, one ends up with billions of kilometer-sized planetesimals. These too continue to collide, sometimes sticking together and sometimes fragmenting.

Over the next hundred million years, these planetesimals accrete together to form planetary embryos (objects the size of the Moon or Mars). Eventually, gravitational forces result in all the embryos either colliding or being ejected from the solar system, leaving the small number of planets we see today.

Look here for a summary of this process written by a planetary astronomer.

### Origin of the Moon

The Earth is unique among the terrestrial planets in that it has a large satellite. Mercury and Venus do not have satellites, and the two small moons of Mars, Phobos and Deimos, are most likely captured asteroids.

• Charles Darwin suggested that the Moon formed when the young, rapidly-spinning Earth, still molten, bifurcated into two objects. It was suggested that the Moon came from the Pacific Ocean basin. We know that this could not have happened the Pacific Ocean basin is the result of recent plate tectonics the moon does not have the composition of the Earth's mantle, and the Earth never rotated fast enough to split into two pieces.
• The Moon might have formed in a disk orbiting the proto-Earth, much like the Solar nebula gave rise to planets. This is thought to be the origin of the Jovian satellite systems. However, were this true the moon should have the same composition as the Earth.
• The moon might have formed elsewhere, and been gravitationally captured by the Earth. It is hard to understand how a capture would result in a lunar orbit that is nearly circular, nearly in the ecliptic, and prograde.
• The currently accepted hypothesis is that the Moon is the consequence of the last major collision in the early solar system between the proto-Earth and a Mars-sized planetary embryo. The two cores merged. Some of the debris was ejected into orbit around the Earth, where it cooled and condensed into the moon.The low density of the Moon is a consequence of the fact that most of the ejecta was from the rocky mantles, and included little of the metallic core material.

### The Minor Planets

1 Ceres is the largest of the Asteroids (the 1 is the numbering scheme all asteroids get a number once they are identified many are later named). They belong to a small number of classes, based either on composition (determined from spectra), or from their orbital characteristics. The Apollo asteroids have eccentric orbits which cross Earth's orbit. (this means that they could collide with the Earth). The total mass of all the asteroids is less than the mass of Earth's Moon. Most meteorites originated in the asteroid belt. The Martian moons Phobos and Deimos appear to be captured asteroids, as do many of the small outer moons of Jupiter and Saturn.

951 Gaspra 241 Ida 253 Mathilde 443 Eros 5535 Annefrank

We know from ground-based observations that asteroids have irregular shapes, and that some are double. The Galileo and NEAR spacecraft flew by 3 asteroids, 951 Gaspra, 241 Ida, and 253 Mathilde. Gaspra measures about 19 X 12 X 11 km, and is an S-type asterois, which means it is made of metallic nickel-iron and with magnesium-silicates. Ida is a larger (58 X 23 km) S-type asteriod. It has a small satellite, Dactly, about 1 km across, which orbits about 90 km from the center of Ida. Mathilde is a C-type asteriod, the most common type, rich in carbonaceous materials. It is quite dark, reflecting only about 3% of the incident sunlight. Mathilde is the largest of these asteroids, with dimensions of about 59 X 47 km.
Annefrank is small, only 8 km long, and dark, reflecting 10-20% of sunlight hitting it. Stardust passed within 3300 km of Annefrank on November 2 2002.

The Near Earth Asteroid Rendezvous (NEAR) spacecraft flew by and then later orbited 433 EROS for about a year beginning on February 14 2000. EROS is a 40 x 14 x 14 km rock that rotates in about 5 hours. It orbits the Sun every 1.8 years, on an orbit with a perihelion of 1.13 AU and an aphelion of 1.78 AU. It was discovered in 1898.

NEAR later landed on the surface.

• The full view.

• Some up close views. Each image is about 550 meters across, taken from a height of about 13 kilometers.

• Close up of some boulders, from 250 meters.

• The final picture, from 120 meters up.

2060 Chiron (note the asteroid nomenclature) is the largest of the Centaurs, asteroid-sized objects orbiting between Saturn and Uranus. Chiron has a radius of 170 km. There are about 10 of these currently known. Chiron shows a comet-like coma when near the Sun (its orbit is elliptical, ranging from 8 to 18 AU), and is also classified by the IAU as a comet.

The Trans-Neptunian Objects (TNOs), of which about 100 are known, are found, as their name implies, out beyond Neptune. They have orbits similar to that of Pluto. These may represent the largest of the Kuiper belt objects. It has been suggested that Pluto/Charon is the largest of the TNOs. It is also likely that Triton, the largest satellite of Neptune, is a captured TNO. The TNOs seems to be of mostly icy composition.

### And What About Sedna and Eris?

Sedna, discovered in 2004, was the most distant large object known in the Solar System at that time. Its size is uncertain, but it is probably about half the size of the Moon. Its orbit is highly eccentric, taking it from about 76 AU from the Sun (it is now about 86 AU out) to about 1000 AU in a 12,000 year orbit. It is not a member of the more distant Oort cloud. It may have been flung out of the Kuiper belt early in the history of the Solar System. Last year brought the announcement that Eris (2003 UB313), 68 AU from the Sun and with a radius of 1200 km, was even larger.

### Is Pluto a Planet?

The issue reached a head in February 1999 when Brian Marden, the director of the Central Bureau for Astronomical Telegrams, suggested that Pluto be honored with minor planet designation 10,000. This would ensure Pluto's recognition as the largets of the Trans-Neptunian Objects (TNOs), but would have the side effect of demoting Pluto from major to minor planet status. His proposal was incorporated in a Minor Planet Electronic Circular M.P.E.C. 1999-C03.

Reaction was swift. On the previous day, the (IAU) had issued a press release on the matter (apparently Marsden's editorial was written a few days earlier, and published later than the press release which contradicts it). Marsden's response was given the next day, in M.P.E.C. 1999-C10.

The official position of the IAU is given in this press release. Scroll down to Resolution 5A.

A PDF presentation I made on this topic is available here.

So, what's the answer? This is a case where there is no right answer, as long as you can support your case.