Kozai and the Crab

The star 55 Cancri hosts a system of five known planets of diverse nature, ranging from the small 55 Cancri e close to the star (3% of Jupiter’s mass, around 10 times Earth’s, at 0.016 AU) to the massive and distant 55 Cancri d (4 times Jupiter’s mass at 5.7 AU). When many planets are detected in a system, it is important to verify that the planets are in a stable configuration that will not be disrupted on relatively short time-scales. While this is true for 55 Cancri there is a hitherto overlooked complication in this system: the existence of a binary companion to the primary star. While distant, the companion could still potentially disrupt the planetary system. A recent paper by Kaib, Raymond & Duncan attempts to determine whether the planetary system is stable to perturbations from this companion.

The particular dynamical effect that could destabilise the planets is known as the Lidov-Kozai effect, after its discoverers. In the simplest case, this occurs in systems comprising a star, a companion such as a planet or binary star, and a massless test particle such as a comet or asteroid, when the ratio of semi-major axes is very large, and the mutual inclination of their orbits exceeds a certain critical value. If the conditions are met, the test particle experiences very large oscillations in eccentricity and inclination. There is a constant of motion, \cos I \sqrt{1-e^2}, where I is the inclination and e is the eccentricity, so the eccentricity and inclination oscillations are in phase. The time-scale for them to occur is approximately P_\mathrm{Kozai}=P_\mathrm{Kep}\frac{m_\star}{m_\mathrm{b}}\frac{a_\mathrm{b}^3}{a_\mathrm{pl}^3}, where P_\mathrm{Kep} is the orbital period of the planet, m_\star and m_\mathrm{b} are the masses of the primary star and binary companion, and a_\mathrm{pl} and a_\mathrm{b} are the semi-major axes of the planet and binary companion.

Eccentricity evolution under the Lidov-Kozai mechanism

Eccentricity evolution of a planet experiencing Lidov-Kozai cycles. Note how the eccentricity is driven to very high values (0.8, in this case). From Malmberg, Davies & Chambers (2007).

In the 55 Cancri system, there is a massive binary companion star, and the planets, being much less massive, would effectively behave as test particles. Since there is no reason to assume that the companion’s orbit is in the same plane as the planets’, their mutual inclination could be large, and the planets would then undergo Lidov-Kozai cycles. Such cycles would be devastating to the 55 Cancri planetary system, since the planetary eccentricities would increase to very large values, their orbits would intersect, and the planets could collide or scatter each other onto very different orbits. In contrast, in the system as currently observed, the planets’ eccentricities are all fairly low, less than 0.1 , and the system has remained stable for around 10 billion years. This suggests that the Lidov-Kozai effect is not at work in this system.

The authors investigate why this is so, integrating the equations of motion for the planets and the companion star interacting gravitationally. Since the companion star’s orbit is unknown (only the distance from the star projected onto the plane of the sky can be measured), they integrate many different systems with different orbits of the binary companion. They also reduce the five planet system to a four or two planet system in order to follow the evolution for longer, since the time that the equations can be integrated for is limited by finite computing power. The qualitative behaviour is, however, the same in all cases.

They find that the system is stable in the majority of runs (84%), and the planets do not undergo the large eccentricity oscillations associated with the Lidov-Kozai effect. However, there are still large inclination oscillations, as shown below:

Inclination evolution of planets in the 55 Cancri system

Inclination evolution of planets in the 55 Cancri system. the points show the inclinations of four planets in the 4-planet integrations. The line shows the inclination of the fifth planet in the 5-planet integrations. All planets evolve in phase, and the addition of the fifth planet has no effect on the nature of the solution. From Kaib, Raymond & Duncan (2011).

The planets’ inclinations vary enormously, but all change in phase: effectively, the system behaves as a rigid body, locked together. The inclined binary companion excites inclination oscillations in the planets. If each planet were experiencing Lidov-Kozai cycles alone, the periods of the oscillations would all be different, since the period depends on the planet’s semi-major axis (see the equation above). Here, however, the planets are sufficiently close that they transfer their change in inclination to each other on much shorter time-scales, and thus all behave the same way under the companion’s perturbations. The eccentricity oscillations are suppressed because the Lidov-Kozai effect depends on a sensitive resonance between the precession rates of the orbital plane and the pericentre, which is broken by the additional perturbations from the other planets. For a detailed analysis of this stabilisation mechanism, see this paper.

While the inclination shown in the above figure reaches a maximum of around 120 degrees, this is dependent on the unknown orbital parameters of the companion star. The authors therefore integrate 500 systems with different binary orbits, to determine statistically the distribution of planetary inclinations that would result. The results are shown below:

Distribution of planetary inclinations

Distribution of planetary inclinations, summed over all possible perturber orbits. The triangles show the true inclinations, while the circles show the observable values, projected onto the plane of the sky. Most configurations result in very large inclinations for the planetary system, some being retrograde (greater than 90 degrees). From Kaib, Raymond & Duncan (2011).

This paper therefore makes a startling prediction: the planetary system is overwhelmingly likely to be misaligned with its host star. This is in contrast to our own Solar System, where all the planets’ orbits are within a few degrees of the Sun’s equator. For 55 Cancri, the median value is around 60 degrees, and the planets can often be driven onto retrograde orbits, going backwards relative to their star, as shown above.

Since planet e transits the star, it is possible to determine the inclination observationally, by means of the Rossiter-Mclaughlin effect. Thus, the authors’ prediction that the system is misaligned with the host star will be able to be tested in the near future.


Chaos II

Last week I described the background to my recent paper, which investigates how planets destabilise bodies on nearby orbits, and the implications for attempting to characterise planets by studying how they interact with debris discs. Here I’ll go into a little more detail on what we actually did.

We were attempting to find out under what conditions orbits of small bodies (henceforth “particles”–they can be comets or asteroids) near planets are unstable. Since instability can often take a long time to become manifest (as an extreme example, there is a change that the planets of the inner Solar System will become unstable in several billion years), we chose as a proxy whether or not the orbits were chaotic, which can be measured on shorter timescales; in our case, we used around 10,000 orbits. It is fairly easy to tell, by plotting the evolution of orbit elements as a function of time, whether or not an orbit is chaotic:

Regular and Chaotic orbits

The evolution of orbital eccentricity as a function of time (measured by the number of conjunctions with the planet) for particles on a regular orbit (top) and a chaotic orbit (bottom).

The argument then is that the chaotic orbits are unstable because they are free to wander through a large set of values, rather than being restricted in the way that non-chaotic, or regular, orbits are. They may then come close enough to the planet to collide with it, or experience a very strong perturbation that flings them out of the system or into the star. This is not strictly true, but is a good approximation.

There are basically three relevant parameters in this problem: the ratio of the masses of the planet and the star, the difference between the orbital semi-major axes of the planet and the particle, and the eccentricity of the particle’s orbit. The work of Wisdom (1980) showed that, as the planet mass is increased, orbits at greater semi-major axes become unstable. We then investigated the role of eccentricity. We followed the orbits of many thousands of particles, and produced plots such as these:

The chaotic zone

The chaotic zone, as a function of planet:star mass ratio mu, eccentricity e, and semi-major axis ratio epsilon=(a-a_pl)/a_pl.* In each plot, the planet lies to the left. White regions are populated with chaotic orbits; black with regular.

Here we show, for a fixed planet mass, where orbits with different semi-major axes (along the x-axis) and eccentricities (along the y-axis) are regular or chaotic. At each grid cell we followed the orbits of 100 particles; in white cells, all 100 were chaotic, while in black cells, none were. The planet lies to the left of the plot, so we see that orbits closer to the planet are chaotic whereas orbits further away are not. The vertical line on the plot shows the result for the extent of the chaotic zone derived by Wisdom in 1980. At low eccentricities, it underestimates the extent slightly, but on the whole does a good job. However, we see that for particles at higher eccentricities the chaotic zone extends considerably beyond this.

It is still, however, the same basic mechanism at work. Recall that the chaos here is driven by the overlap of mean motion resonances, and that these resonances have a width that grows with the eccentricity of the particle. We derived an improved condition for the overlap of these resonances, accounting for their increasing width, and the results are plotted as red lines. These match the edge of the chaotic zone at higher eccentricities extremely well, over 5 orders of magnitude of planet mass! The width of the chaotic zone changes from 1.3\mu^{2/7} to 1.8e^{1/5}\mu^{1/5}. It now includes the eccentricity dependence, and the mass dependence changes slightly. Our new result works for higher eccentricities while Wisdom’s is valid for lower.

So we see there are two regimes: for low eccentricities the chaotic zone width is given by the classical Wisdom result. However, for larger eccentricities, the chaotic zone can be significantly larger. These eccentricities need not be very large: the boundary separating the regimes is around 0.01 for a planet of Jupiter’s mass (\mu=10^{-3}). Since objects such as Pluto in the Kuiper Belt can have eccentricities significantly in excess of this, this is potentially important for understanding the interactions of such bodies with planets.

Disc edge profiles of HR8799

Calculated profiles of the inner edge of the debris disc of HR 8799. The density of particles is plotted as a function of semi-major axis. Solid black lines show the results using our new criterion for the edge of the chaotic zone. Red lines show the results from the Wisdom criterion, ignoring (vertical) or including (sloping) the smearing-out effects of eccentricity. Lines are shown for planet masses of 2, 4, 6, 8 and 10 times Jupiter's, increasing from left to right.

To show the importance of this effect, we took a real system. The young star HR 8799 is orbited by at least four planets and two planetesimal discs. The outermost planet is around 68AU from the star, while the disc’s inner edge is not exactly known but estimated at 90AU. We compared the expected shape of the edge of the debris disc, if it is made up of eccentric particles, using our new result (black lines in the above plot) to those using the Wisdom result (red lines; the sloping ones include the smearing-out effect of the particles’ eccentricities while the vertical do not). Since the planet mass is not known with certainty, we computed the profiles for a range of planet masses, from twice to ten times Jupiter’s. Notice how the new result requires a smaller planet mass to attain the same degree of clearing as the Wisdom result.

The masses of the planets are very uncertain, because they must be derived from theoretical models of their interior structure and cooling history. Independent limits on the masses, as are provided by dynamical studies such as this, are very valuable. The hope is that, if the disc edge and planetary separation are known, the mass of the outermost planet can be estimated from the size of the hole it has cleared, in the same way as in the Fomalhaut system. At the moment, this is not possible, because of the uncertainties both on the planet’s orbit and the location of the planetesimals. However, we revealed an additional complication: the size of the clearing for a given mass, or the planet’s mass for a given clearing, depends on the eccentricities of the particles in the disc. Only very detailed images of the disc’s edge can fully determine the planet’s mass. It is to be hoped that future observations will provide this.

*Does anyone know how to get LaTeX into figure captions? If I put the BBcode in it gets really messed up…

Chaos I

Today we return to celestial mechanics, and I’d like to discuss the background to a paper I’ve written with my Ph.D. supervisor Mark Wyatt that has just been accepted to the Monthly Notices of the Royal Astronomical Society. In it we investigated one way in which a planet can destabilise nearby bodies in the same planetary system, and the implications of this for estimating the masses of planets in extra-Solar systems. Before describing our work I need to say a little about both the astronomical and mathematical background.


It is clear in our Solar System that planets clear asteroids and comets from orbits that are too close to them. For example, there are no stable populations of bodies (with the exception of the Trojans) between the orbits of Jupiter and Neptune, while inside Jupiter’s orbit and beyond Neptune’s there are stable populations — the Asteroid and Kuiper Belts. While the proximate cause of bodies being removed from the unstable regions is by coming very close to a planet and being scattered onto a very different orbit, the region of space over which this is effective is smaller than the cleared region. Some other dynamical mechanism is at work to move some particles onto orbits where they will encounter a planet, but without moving others.

The Asteroid Belt, its outer edge truncated by Jupiter. Image credit: Wikipedia

Similar clearings are seen in the debris discs around other stars. These debris discs are made of the dust formed in collisions between asteroids or comets, and there are many that have been imaged at a variety of wavelengths (see here for a gallery). Many are seen to have holes in the centre; a good example is that around Fomalhaut, which has a very sharp inner edge at around 130AU. Fomalhaut also hosts a planet which has been detected with the Hubble Space Telescope, as seen in the Figure below. The planet’s existence and general properties had been predicted by Alice Quillen (2007), who calculated what the nature of the planet must be in order to account for the shape of the disc edge. The planet was discovered in 2008 by Paul Kalas et al., at a location in very good agreement with the theoretical predictions. (This method of predicting an unknown planet based on the orbits of known bodies has a venerable tradition going back to the discovery of Neptune in the 19th Century.) Indeed, the mass of the planet as estimated by the amount of light received from it is very uncertain, and currently the best estimate of the planet’s mass is from the dynamical models (Chiang et al., 2009).

The planet and disc of Fomalhaut, as seen by HST. The star's light was blocked with a coronograph, allowing the much fainter planet and disc to be seen. The planet's orbit lies just interior to the disc, and orbital motion over a two-year period can be seen. Image credit: Kalas et al., Science, 322, 1345.

When material is destabilised and scattered by the planet, it may suffer one of three ultimate fates: collision with the star, collision with the planet, or ejection from the system. This leads to another scenario in which whether material can be destabilised is important: destabilised material colliding with the star has been invoked as an explanation for the unusual compositions of some White Dwarf atmospheres. White Dwarfs are the remnants of low-mass stars at the end of their lives, and their high density (around the mass of the Sun in a volume the size of the Earth) gives them a strong gravitational field at their surface. This gravitational field should act to separate the elements in the atmosphere, so instead of different species being mixed together, as is the case in Earth’s atmosphere, the heavier should sink to the bottom, as sand does in water. Hence, only the lightest elements, hydrogen or helium, should be seen. However, some White Dwarfs show spectroscopic evidence of “pollution” by heavier elements (called metals by astronomers), for which there must be an ongoing source to replace those that are sinking. A leading hypothesis is that they originate from destabilised asteroids or comets that have collided with the star.

We might expect orbits to be less stable around White Dwarfs than around their progenitors. This is becasue, during the giant phases of a star’s evolution that precede the white dwarf phase, the star can lose a significant quantity of its mass—around half, for a star such as the Sun. Therefore, the planets around White Dwarfs are more massive, relative to the star, than when they were around the progenitor. Since most dynamical effects depend on the ratio of planetary to stellar mass, these effects will become stronger as the star loses mass. Hence, unstable regions around planets would be expected to grow. This was something we explored in a paper earlier this year, finding that the amount of material scattered from typical planet–disc systems could broadly acccount for the amount of metal pollution in White Dwarf atmospheres of various ages. If this is the correct explanation, then it is telling us about the long-term fate of planets and asteroids as their stars age and die — including our own.


NB: See this post for definitions of orbital elements.

Although there are several ways by which one or more planets might destabilise other bodies, the one with the widest applicability is the overlap of mean motion resonances. These resonances occur when two bodies’ orbital periods are in a simple integer ratio, such as 2:1. From Kepler’s Law, P^3\propto a^2, this occurs at a ratio of semi-major axes of around 1.6. So if one planet is located at 1AU, a planet at 1.6AU would be at the 2:1 resonance. Resonances with Neptune in the Kuiper Belt are shown schematically here:

Resonances in the Kuiper Belt. Note that the resonances become more closely spaced the closer they are to Neptune. Image Credit: Wikipedia

Now, although the actual period ratio only occurs at one specific semi-major axis ratio, the resonance also affects nearby orbits. The elements of these orbits oscillate, with a maximum amplitude of oscillation that depends on the mass of the planet and the particular resonance being considered. In particular, the more massive the planet, the more powerful the resonance, and the greater the range of semi-major axes affected by it. If a resonance acts alone on an orbit, the result is a regular oscillation of the orbital elements.

However, since the resonances have a width, and bunch up more closely the closer you are to the planet, there comes a point where the resonances overlap. When this happens, the evolution of an orbit becomes chaotic and unpredictable—crudely, instead of being confined to one resonance, you can imagine the particle being passed amongst many since their regions of oscillation overlap. This overlap of resonances driving chaotic behaviour was described by Chirikov (1979) and applied to the stability of orbits in the Solar System’s Asteroid Belt by Wisdom (1980).

It is useful to have a simple formula to give the boundary of the chaotic region for any planet mass. This Wisdom derived assuming low particle eccentricities, finding \delta a = 1.3 a \mu^{2/7}, where \delta a is the width of the cleared chaotic zone and \mu is the ratio of planet mass to stellar mass. I.e., more massive planets clear out wider regions around their orbit, which is what one intuitively expects. This result is now used to estimate the mass of planets truncating debris discs, since if the planet and disc location are known, one can solve for \mu.

In our previous paper, we had used the Wisdom result to estimate the amount of material that would destabilised as the star loses mass (this changes \mu and so changes the width of the chaotic zone), and compared it to more accurate numerical results using both integrations of the full equations of motion, and a simpler numerical investigation using a dynamical map (described in Duncan, Quinn & Tremaine 1989). We considered a disc in which particles had moderate eccentricities, up to 0.1, since in our Solar System the Kuiper Belt bodies have eccentricities in this range. We found that the chaotic zone width increases with increasing eccentricity—this is also seen in Wisdom’s original paper, although Wisdom’s analytical result was only valid for low eccentricities. In our latest paper, we followed up these investigations, deriving a simple formula for the chaotic zone width at non-zero eccentricity.

In the next post, I’ll describe what exactly we did in our new paper, and the implications for the interactions of planets with planetesimal discs. Concisely, we found that if particles are on eccentric orbits, the width of the chaotic zone is no longer given by the Wisdom (1980) formula, and thus any attempt to estimate a planet mass in the way described above will give an erroneous result.

Scientific prose

Having recently moved country, with all the attendant difficulties imposed by the language barrier, I have begun to think a little about communication, how we inform one another of our thoughts and make ourselves understood. As a scientist I am particularly interested in the style of writing in scientific prose, since this at the same time needs both to convey very complex, novel ideas, and to be understood by the international scientific community, many of whose members do not have Engish as a first language. I also feel that, ideally, a good scientific paper should be at least somewhat understandable to the interested layperson, particularly in areas of science such as medicine where personal, corporate and state decisions will be based on published research. I do confess, however, that when writing my papers I write for scientists and not the general public!

Personally I can find reading scientific papers, even within my own field, more challeging than many other styles of English. Why is this? I can think of at least four explanations:

  1. Unfamiliarity of language. If I am informed that “the slithy toves did gyre and gimbol in the wabe1“, I don’t have a clue what any of those words mean. (Although, curiously, I can identify their parts of speech…)
  2. Unfamiliarity of concepts. I know what the words “closed”, “metric” and “space” mean in everyday parlance, but it takes a bit of effort to recollect from my undergraduate days what exactly a “closed metric space” is.
  3. Complexity of language. Scientific papers typically contain many long complex and/or compound sentences such as this: “We show that when mass loss is slow, systems of two planets that are marginally stable can become unstable to close encounters, while for three planets the timescale for close approaches decreases significantly with increasing mass ratio.” (Debes & Sigurdsson, 2002) Here a short main clause “we show” introduces a huge subordinate clause 35 words long, which itself is composed of three nested levels of clauses.
  4. Complexity of concepts. The above quotation is describing the effects of a star losing mass (as happens before it becomes a white dwarf) on any orbiting planets. Whether these planets are stable depends on their masses relative to the star’s, which increase as the star loses mass. Systems of two and three planets behave somewhat differently, but in both cases the question is whether, or on what timescale, the planets will approach each other closely. This complex set of phenomena, and the causal relationships between them, are described by the authors in the quoted sentence.

Clearly, to a large degree 1 and 2 are interdependent. Wovon man nicht sprechen kann, darüber muss man schweigen. We must have the necessary vocabulary to begin to discuss anything, for if we do not have words to describe something, how can we discuss it? On the other hand, scientific language tends to take familiar words and give them a very precise meaning. For example, one cannot properly understand a closed metric space simply by referring to the everyday meanings of the three component words. Increasing technical vocabulary and concepts go hand in hand when studying any field, sport as much as science, and together present one barrier to understanding a text. This is unavoidable, since it is not possible in any paper to explain all the technical terms used: that is a matter for the textbooks! It is, however, a reasonable assumption that anyone (scientist or layperson) interested in reading a paper will have some knowledge of the requisite background, and it is common to introduce less well-known terms. In my field, for example, it is customary to inform readers that a “debris disc” is an extra-Solar analogue of our Solar System’s Asteroid and Kuiper Belts: any professional astronomer, or layperson with an interest in astronomy, will understand that.

What I’ve been wondering is whether 3 is so necessarily dependent on 4; and, if so, whether that is a barrier to, or maybe rather an aid to, understanding. I.e., does the semantic complexity of scientific concepts—complex chains of causation, conditionals and counterfactuals, caveats etc.—necessarily entail a concomitant syntactic complexity when they are expressed in natural language? And if so, is this a hindrance to understanding them, or would lots of shorter, simpler sentences be harder to understand than one equivalent long one? To give an example, could the sentence from Debes & Sigurdsson quoted above: “We show that when mass loss is slow, systems of two planets that are marginally stable can become unstable to close encounters, while for three planets the timescale for close approaches decreases significantly with increasing mass ratio.” be rewritten something like this: “Some systems of two planets are marginally stable. We show that slow mass loss can render them unstable to close encounters. Systems of three planets have a timescale for close approaches to occur. We show that slow mass loss decreases this timescale significantly.” This is more verbose (43 words against 37), as there is some repetition (e.g., of “we show”), and breaking up the sentences means more pronouns had to be introduced. Question to you, readers: Do you find the original sentence, or my rewritten sentences, easier to understand?

While pondering that, it’s worth comparing scientific prose to other factual English prose written to inform at a relatively high level. To do this, I compared two texts: First, the abstract and introduction from the Debes & Sigurdsson paper from which the above sentence was taken; this had the triple advantage of being (1) close to hand, (2) related to my research, and (3) despite my complaints about the difficulty of scientific writing, actually a very well-written paper. Second, the opening paragraphs of “The History of Madrid” chapter from my Dorling Kindersley travel guide “Madrid”; this has the advantage of needing to introduce new vocabulary itself, in the form of Spanish words, persons etc., in the same manner as the introduction to a scientific paper.

The simplest measure of linguistic complexity might be to simply count the number of words in the sentences. This I did for each text, removing sentences containing semi- or full-colons (as they could just as easily be parsed as compound sentences or strings o simple sentences). For the scientific paper, I ignored any references in parentheses such as “…similar to the Solar system (Duncan & Lissauer 1998).”, but counted references essential to the meaning of a sentence such as “Duncan & Lissauer (1997) … found that…”, in this case, as 4 words.

There were 32 sentences in the Debes & Sigurdsson text, and 31 in the Dorling Kindersley. The former had an average length of 30.2 +/- 8.9 words, the latter 21.9 +/- 7.4. While the sentences in the scientific text are longer, with such a range of lengths the difference is probably not statistically significant, and I will need more samples of texts to say definitively whether the scientific paper’s sentences are longer. I should also draw from more than one source, since there might be great variations of individual style between authors.

On the other hand, just because a sentence is long, does that make it complicated? The sentence “The big red car, the small green car, and the huge black bus stopped at the traffic lights”, containing 18 words, I would say is less complicated than “The car which was big and red, and the bus indicating to turn left, stopped when they wanted”, despite having the same number of words. The former is a simple sentence with only one main clause, the latter a complex sentence with one main and two subordinate clauses, as well as a participial phrase. To attempt to measure this sort of complexity, I counted the number of finite verbs (those that can stand alone in a clause or sentence) in each sentence. This gives the average number of clauses per sentence, although it misses out constructions such as participial phrases, infinitive phrases, gerunds and the like. The results were: Debes & Sigurdsson, 2.25 +/- 1.14 finite verbs per sentence; Dorling Kindersley, 1.94 +/- 1.06 finite verbs per sentence. Again, we see a hint that the scientific text is indeed more complicated.

Although this needs to be confirmed with a larger study, this suggests that scientific prose may be more syntactically complex than other informative factual writing. Whether this is a hindrance to understanding, and whether it is practical to simplify our language, are of course further questions to explore…

Oh, and I just read through the draft of this, and Damn! do I write some long sentences!

Edit (25/09/11 17:46 UT): A t-test tells me that the difference in mean sentence length between the two texts is significantly different (p=0.00016, about three and a half sigma) but not the difference in number of verbs (p=0.26).

1OK, I know this isn’t science, but it’s a good example. Try this from last week’s Nature: “In contrast with previous assumptions, we report here that the nascent antizyme polypeptide is the relevant polyamine sensor that operates in cis to negatively regulate upstream RFS on the polysomes…”


Well, I’ve now got most of my hassle of moving jobs, country, language etc. sorted out now. (OK, the last one was a lie; I still can’t communicate. I just about managed to get some throat sweets this morning though.) Unfortunately, to my great distress, my favourite and most useful textbook, Solar System Dynamics by Murray & Dermott, is MIA after the move! While it awaits recovery or replacement (I’d rather it were recovery–it’s got 4 years’ worth of marginalia inside!) here are some more photos of Madrid…

I was surprised to learn on moving here how modern the city is. As I’d done some reading up on Spanish history before I came, I had learned that it was relatively unimportant in the Mediæval period, only being adopted as the seat of the monarchy during the reign of Felipe II (him of Spanish Armada fame) in the late 16th Century. So there are precious few Mediæval buildings left. An example is San Jerónimo el Real, built in the early 16th Century but extensively restored in the 19th.

San Jerónimo el Real

The church of San Jerónimo el Real, close to the Prado museum.

However, what surprised me more was really how few of the important buildings date to the Siglo de Oro, the “Golden Age” of the 16th to 17th Centuries. An example is Plaza Mayor, laid down at the turn of the 17th Century, albeit with some restoration following later fires &c.

Plaza Mayor

Plaza Mayor, surrounded by tapas bars. Rather pricey, but very tasty!

Many of the most famous buildings are rather modern: The Palacio Real, for example, built in the mid 18th Century…

Palacio Real

The South facade of the Palacio Real.

…and the Prado, built at the end of the 18th Century:

The Prado

The North facade of the Prado.

In my naïveté I had imagined that this time period, which geopolitically would be regarded as the time of the decline of the Spanish Empire, would have engendered less civic architecture, but that is not the case indeed! Development continued throughout the 19th Century, including buildings such as the Biblioteca Nacional:

Biblioteca Nacional

Biblioteca Nacional, the National Library. To make this post tangentially related to celestial mechanics, the figure seated at front right is King Alfonso "The Wise" of Castille, who ordered the compilation of the Alphonsine Tables of planetary positions, used in Europe throughout the Middle Ages and into the Renaissance.

The 20th Century saw transformation of Madrid’s skyline with the introduction of skyscrapers, one of the first being the Telefonica building:


The Telefónica building on the Gran Vía.

To one used to the more modest architecture of Cambridge, all the tall buildings here felt somewhat overbearing at first. I am, however, grateful for the shade they provide from the Sun here (note sparsity of clouds in these pictures!)

What I did on my holidays

If there hasn’t been much activity here recently, that’s because I’ve been:

  1. On holiday, visiting my family in Sheffield, a city rich in industrial heritage

    Bessemer Converter at Kelham Island Industrial Museum, Sheffield

    and surrounded by lovely countryside

    Hope Valley, Derbyshire, seen from Mam Tor. The nearby village is Castleton; the Hope Valley Cement Works are behind it, and Winnats Pass is to the right.

  2. Preparing for and undergoing my Ph.D. viva in Cambridge, where I said farewell to my beautiful office building

    The Observatory Building, Institute of Astronomy, Cambridge

  3. Attending a meeting in the Spanish Sierra

    Sierra de Guadarrama, just north of Madrid

  4. And settling into my new job in Madrid

    The picturesque view from my new office window!

Back to Celestial Mechanisation soon!

Anatomy of an orbit

Since I’m going to be discussing orbital dynamics fairly extensively on this ‘blog, I thought I’d put together a post describing how orbits are described, since the terminology can be fairly confusing, even to professional astronomers.

First, recall that particles orbiting under an inverse-square force such as gravity follow orbits which are conic sections — circles, ellipses, parabolae or hyperbolae. Particles which are bound to the body they are orbiting follow circular or elliptical orbits, while unbound particles follow parabolic or hyperbolic orbits. I’ll confine my attention to bound orbits here, since most objects of interest (planets, moons, asteroids…) are on bound orbits.

Such orbits are confined to a single plane. Restricting attention to this plane, an elliptical orbit is described algebraically by r=\frac{a(1-e^2)}{1+e\cos f}, while geometrically it looks like this:

Eccentric orbit

An orbit with an eccentricity of 0.3. The orbiting particle is located at the filled symbol, having moved through an angle f.

The orbit has one parameter describing its size, the semi-major axis a, and one describing its shape, the eccentricity e. The higher the eccentricity, the less circular the orbit is. The particle’s location on the orbit, measured from the point of closest approach to the central body, is measured by the angle f, the true anomaly. This point of closest approach is located at a distance  q=a(1-e) from the central body. It is known variously as the pericentre, periapse (pl. periapses) or periapsis (pl. periapsides). For orbits around specific bodies, it may be called the perigee (Earth), perihelion (Sun), etc.

Note that the symbols marking the orbit in the above Figure are not evenly spaced. They in fact represent the particle’s position at equal intervals in time. The particle moves fastest at periapsis; this is a simple consequence of angular momentum conservation.

While this description of an orbit is quite simple, it is often necessary to describe an orbit with respect to a different reference frame. For example, when observing an extra-solar planetary system it is convenient to align the reference frame with the line of sight and the plane of the sky. The orientation of the orbital plane with respect to the reference plane is then given by two angles, the inclination I and the longitude of ascending node \Omega. The former describes the angle between the x-y plane of the reference frame and the orbital plane, while the latter describes the line of intersection of the two planes (this is known as the line of nodes). These angles are shown below:

Orbital angles

We also see the familiar angle f, the true anomaly, making its appearance. One final angle is needed to completely specify the orientation of the orbit: this is the argument of periapsis \omega, which describes which way the periapsis points within the orbital plane.

To summarize, there are now six parameters, called orbital elements, describing the particle’s position in space:

  • I, inclination, describing the angle between a reference plane and the orbital plane
  • \Omega, longitude of ascending node, describing the orientation of the intersection of these two planes
  • \omega, argument of periapsis, describing the direction the periapsis points relative to the line of nodes.
  • f, true anomaly, describing the particle’s position on its orbit relative to the periapsis.
  • a, semi-major axis, the average distance of the particle from the central body
  • e, eccentricity, a measure of how non-circular the orbit is

If the particle only experiences an inverse square force from the central body, then all these apart from the true anomaly are constant, and the orbit remains unchanged in time. However, when more bodies are introduced, they can cause all the orbital elements to change. The goal of celestial mechanics is to determine what these changes are, which is a very difficult problem for which there is no general solution.

Finally, there are alternative orbital elements. It is common to use the longitude of periapsis \varpi=\omega+\Omega to express the direction of periapsis relative to the reference direction, and the true longitude \theta=f+\omega+\Omega to express the particle’s position on its orbit relative to this direction. Remember:

  • Longitudes are measured relative to a fixed reference direction
  • Arguments are measured relative to the line of nodes
  • Anomalies are measured relative to the periapsis