DDA 2015 – MMRs and the Origins of Extrasolar Orbital Architectures

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Konstantin Batygin (CalTech)

Abstract

The early stages of dynamical evolution of planetary systems are often shaped by dissipative processes that drive orbital migration. In multi-planet systems, convergent amassing of orbits inevitably leads to encounters with rational period ratios, which may result in establishment of mean motion resonances. The success or failure of resonant capture yields exceedingly different subsequent evolutions, and thus plays a central role in determining the ensuing orbital architecture of planetary systems. In this talk, we will show how an integrable Hamiltonian formalism for planetary resonances that allows both secondary bodies to have finite masses and eccentricities, can be used to construct a comprehensive theory for resonant capture. Employing the developed analytical model, we shall examine the origins of the dominantly non-resonant orbital distribution of sub-Jovian extrasolar planets, and demonstrate that the commonly observed extrasolar orbital structure can be understood if planet pairs encounter mean motion commensurabilities on slightly eccentric (e ~ 0.02) orbits. Accordingly, we speculate that resonant capture among low-mass planets is typically rendered unsuccessful due to subtle axial asymmetries inherent to the global structure of protoplanetary disks.

Notes

  • SeeMécaniqueCéleste, Laplace 1805!
    • But origins not really understood until Roy & Ovenden 1954, Goldreich 1964 (MNRAS)
  • Disk-satellite interactions (Goldreich & Tremaine)
  • But what about more than one planet?
  • All tend to migrateinward then lock intoMMRs (Pierens 2013 A&A)
    • $\Rightarrow$ numerical models predict MMR lock
    • BUT only ~15% of observed planet pairs are in resonance
  • The real Hamiltonian (planet-planet interactions) is actually probably a mess.
    • See Poincare’s book, vol. 2(!)
    • Define a canonical rotation that gives an integral of the motion (“generalized reducing transformation” –Poincare)
      • Basically, a generalized Tisserand parameter
    • Batygin & Morbidelli 2013(?)
  • An analytical theory for resonant capture: unrestricted ETB problem.
    • Batygin 2015 (MNRAS, submitted)
    • Capture prob. only depends on total mass of the planets, NOT the mass ratio
      • phase space area occupied by planet is small
    • Kepler sample: critical eccentricity is ~0.02 — very small!
      • Larger than this, capture fails
      • Matches observed value!
    • Explains Jupiter-Saturn MMR lock
    • Perhaps slight deviations from axial symmetry in protoplanetary disks are responsible for the orbital architecture we observe today.

DDA 2015 – Consolidating and Crushing Exoplanet Systems

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Kathryn Volk (U. British Columbia)

Abstract

Kepler revealed the common existence of tightly-packed planetary systems around solar-type stars, existing entirely on orbits with periods shorter than ~200 days. Those systems must have survived for the ages of their host stars (~5 Gyr), so their formation mechanism must provide inter-planet spacings that permit long-term stability. If one postulates that most planetary systems form with tightly-packed inner planets, their current absence in some systems could be explained by the collisional destruction of the inner system after a period of meta-stability. The signatures of such intense collisional environments may have been observed around stars in the form of rapidly varying debris disks; in these observed disks, collisional products are being disposed of via drag down onto the star or grinding to the nearly instantaneous dust blow-out limit. We use the orbital spacings and planet masses of the observed Kepler multi-planet systems to investigate the stability and long-term behavior of the systems. We find that many of our Kepler system analogs are unstable on 100 Myr timescales, even for initially small eccentricities (0-0.05); the instability timescales in these systems are distributed such that equal fractions of the systems experience planetary collisions in each decade in time. We discuss the likely outcomes of collisions in these systems based on the typical collision speeds from our numerical integrations and what implications this has for interpreting the observed Kepler multi-planet systems. The possible implications for our Solar System are discussed in a companion abstract (Gladman and Volk).

Notes

  • Architectures of close-in (closely packed) planetary systems (from Kepler)
  • Fabrycky 2014
  • ~5-10% ofFGK field stars
    • These systems must be stable on Gyr timescales
  • Are all stars formed tightly packed?
  • Modeled 13 such Kepler systems
    • Preserved $a$ and masses, orbital angles randomized
    • Allowed $e_0$ to vary $0 < e_0 < 0.05$
    • Sudden onset of instability in 11 of these 13 after tens to ~100 Myr
      • [why is she surprised?]
    • These eccentricities are in range of observed values
    • Decay rates consistent with e.g. Holman & Wisdom (1992 AJ)
  • Why sudden onset?
    • History is very sensitive to ICs [duh]
    • Consolidation (low-speed collisions) vs. Destruction (high-speed collisions)
    • First collision is often near the accretion/erosion boundary — i.e., low-speed
    • Masses in 4-5 planet systems tend to be lower, while individual masses in ~3-planet systems are higher: mergers?
    • Tracked collision speeds during integrations.
    • Second collision often goes into erosion regime (i.e., high-speed)
      • Observing debris should be rare (but see Meng et al. 2012)
    • Ergodicity allows large variety of outcomes
  • $\Rightarrow$ tightly packed systems could be ubiquitous initially
    • Young stars should show higher fraction
  • The remaining ~95% should be 0-2 planet systems

DDA 2015 – Capture into Mean-Motion Resonances for Exoplanetary Systems

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Exoplanet Theory II

Maryame El Moutamid (Cornell)

Abstract

Many bodies in the Solar System and some exo-planets are close to or captured in Mean Motion Resonances (MMR). Capture into such resonances has been investigated by many authors. Indeed, the Hamiltonian equations of motion in presence of migration are given by Sicardy and Dubois Cel. Mech. & Dyn. Astron., 86, 321-350 (2003). Fleming and Hamilton, Icarus 148, 479-493 (2000), studied the problem in a less generic context. In these two papers, the authors studied the problem of 1:1 corotation (Lagrange points L4 and L5), rather than m+1:m corotations (El Moutamid et al, Cel. Mech. & Dyn. Astron, 118, 235-252 (2014)). We will present a generic way to analyze details of a successful (or not) capture in the case of an oblate (or not) central body in the context of Restricted Three Body Problem (RTBP) and a more General Three Body Problem in the context of known statistics for captured exoplanets (candidates) observed by Kepler.

Notes

  • Captures partial near MMR (Fabrycky et al. 2012)
  • No generic study on coupling between associated resonances (ERTB vs. general TB)
  • 1) simple model,2DoF — $(m+1) n’ \approx m n$
    • splitting the corotation and Lindblad resonances (by $J_2 \neq 0$)
    • Lindblad: vary $e$
    • corotation: pendular motion (conserves $e$)
    • plot: $J_c – J_L$ vs. $\phi_C$
  • 2) general case
    • can define a constant of motion: $J_{c,relat} = \frac{A^2 \xi}{m} – \frac{A’^2 e’}{m+1} – ?? = const.$
  • Add dissipationforMMR capture
    • ratio: potential barrier of one vs. other body
    • plot: potential energy vs. critical angle of corotation
    • probability of capture: very very small

DDA 2015 – Obliquity Evolution of Earth-Like Exoplanets in Systems with Large Inclinations

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Russell Deitrick (U. Washington)

Abstract

In order to properly assess the potential for habitability and prioritize target selection for the characterization of exoplanets, we need to understand the limits of orbital and rotational dynamics. Large satellites may be rare and very difficult to detect. Consequently, it is necessary to quantify the likelihood of a planet’s having extreme obliquity cycles in the absence of a moon and to model the potential impact on the planet’s climate. We explore the obliquity evolution of (1) known exoplanet systems that could contain Earth-like planets in the habitable zone and (2) hypothetical planets in mutually inclined, chaotic resonant configurations that experience some of the most extreme orbital evolution possible. We use a secular obliquity model coupled to either an N-body models or a 4 order secular orbital model. We find that in some known systems, planets’ obliquity variations are small and unlikely to have a major effect on climate, unless undetected planets are present. Systems with three or more planets are significantly more dynamically rich, with planets that undergo obliquity changes of ~10° over 50,000 years and >30° over a few million years. In resonant configurations, Earth-like exoplanets can undergo dramatic and chaotic evolution in eccentricity and inclination while remaining stable for over 10 Gyr. In configurations in which eccentricities and inclinations stay below ~0.1 and~10°, respectively, obliquities oscillate quasi-periodically with amplitudes similar to the non-resonant, three-planet configurations. In more dynamically active configurations, in which eccentricities and inclinations evolve to e > 0.3 and i > 15°, obliquities can extend from ~0° to well past 90°. In extreme cases eccentricities can reach >0.9999 and inclinations >179.9 degrees, driving precession rates in excess of degrees per year. However, these planets can graze or impact the stellar surface and are probably not habitable.

Notes

  • $\upsilon$Andromedae c and d
    • obliquity oscillations
  • Model description
    • Barnes, Deitrick et al. 2015
    • Using the secular disturbing function (Murray & Dermott) and a secular obliquity model (Kinoshita 1975, 1977)
    • HD190360
      • obliquity varies w large amplitude in a “strip” in $\Delta i_0$ – $e_0$ plane — WTH?
      • two planets interacting (an Earth and a super-Jupiter) … somehow
      • Inside the “strip”, a commensurabilitylibrates
        • $(\varpi’ – \varpi) – (\Omega + p_A)$
        • outside the “strip”: no libration
      • Analogous to a compound pendulum
  • Summary
    • Non-coplanar systems in MMR exhibit long-lived chaos.
    • These systems can be formed by scattering.
    • Possible way to form misaligned hot Jupiters.
    • Earth-like planets in these systems can also have chaotic obliquity variations.

DDA 2015 – Loners, Groupies, and Long-term Eccentricity Behavior – Insights from Secular Theory

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Christa Van Laerhoven (CITA)

Abstract

Considering the secular dynamics of multi-planet systems provides substantial insight into the interactions between planets in those systems. Secular interactions are those that don’t involve knowing where a planet is along its orbit, and they dominate when planets are not involved in mean motion resonances. These interactions exchange angular momentum among the planets, evolving their eccentricities and inclinations. To second order in the planets’ eccentricities and inclinations, the eccentricity and inclination perturbations are decoupled. Given the right variable choice, the relevant differential equations are linear and thus the eccentricity and inclination behaviors can be described as a sum of eigenmodes. Since the underlying structure of the secular eigenmodes can be calculated using only the planets’ masses and semi-major axes, one can elucidate the eccentricity and inclination behavior of planets in exoplanet systems even without knowing the planets’ current eccentricities and inclinations. I have calculated both the eccentricity and inclination secular eigenmodes for the population of known multi-planet systems whose planets have well determined masses and periods. Using this catalog of secular character, I will discuss the prevalence of dynamically grouped planets (‘groupies’) versus dynamically uncoupled planets (‘loners’) and how this relates to the exoplanets ‘long-term eccentricity and inclination behavior. I will also touch on the distribution of the secular eigenfreqiencies.

Notes

  • Secular character of multi-planet system
  • planet-planet interactions
  • only need masses and semimajor axes (not eccentricity, not inclination) to set secular structure
  • two-planet system: two eccentricityeigenmodes
    • $h = e \sin \varpi$, $k = e \cos \varpi$ plot: $e$ is a vector
    • each $e$ vector is the sum of two eigenvectors
  • 3-planet system: “groupie”-ness and loners
    • groupies:
      • $e$ highly variable
      • $\varpi$ precession not uniform
    • loners:
      • $e$ does not vary by much
      • $\varpi$ precesses steadily
  • Kepler-11
    • outer planet is a loner — does not interact with others
    • 5 inner planets are groupies — interact strongly with each other
  • Summary: most planets are “groupies”, “loners” are rare.

DDA 2015 – Using Populations of Gas Giants to Probe the Dynamics of Planet Formation

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

Session: Exoplanet Theory I

Ruth Murray-Clay (UC Santa Barbara) (invited)

Abstract

[none]

Notes

  • How do giant planets and brown dwarfs form?
  • Architecture of Solar System is atypical.
  • Lots of gas giants at large distances, small distances (“hot Jupiters”), but not much in between a la Solar System. Why?
  • SS: rocky planets (~1 AU), gas giants (~5-10 AU), ice giants (~20-30 AU)
  • Theory: cannot predict numbers, but can predict patters in system architectures and statistical populations
  • How to get companions to stars: 1) turbulent fragmentation, 2) grav. instability, 3) core accretion
  • HR8799: testbed for planet formation theories
    • 4 Jupiter-mass planets
    • turbulent frag.? No: system is not hierarchichal
    • grav. inst.?
      • iffy – minimum fragment distance problems (but could have migrated)
      • Timing – collapse must occur at end of infall or a binary star results
    • core accretion?
      • dynamical (growth) timescale is too long ($t_{grow} > t_{infall}$)
      • $t_{grow} > t_{disk}$
      • cross section regimes — all problematic:
        • physical cross section
        • grav focusing
        • gas drag capture
  • Make gas useful.
    • no gas: particles can orbit inside core Hill radius
    • gas: “wind shear”
      • binary capture
      • particle capture can occur out to Hill radius
      • growth time at 70 AU can be short enough to nucleate an atmosphere
      • turbulent gas: okay
    • accretion cross sections increase by up to $10^4$
  • Gemini Planet Imager could confirm this theory.
  • Metal-rich stars hostmorehotJupiters and highly eccentric planets: signature of planet-planet interactions? Why?
    • Scattering?
    • Secular chaos?
    • Perhaps those systems form many Jupiters.
  • Are the solar system analogs orbiting low metallicity stars?

The Printer and I: A Tale of Spinning Fans, Diseased Hearts, and the Tragedy that is Life

psprinter-overview
[Click to embiggen.]

This (see photo) is how I spent my afternoon and evening, today. I have a conference to attend next week and must present a poster paper on some recent research results. Because I know by now that both Old Man Murphy and Loki the Trickster always lie in wait, snickering — I hear you, you bastards — I go to check the large-format printer. It is a Beast, and it turns electrons into poster papers. I flip the power switch, and it makes a horrible noise, won’t boot up, freezes, then whines plaintively, “call HP … call HP … please, won’t you call HP ….” Not very encouraging. Screw you, Loki — thou art a Puck.

As with all things computer that misbehave, I keep trying the same thing over and over, hoping for a different result, though I know full well that no different result will … um … result. Indeed, no dice. Run around the building and check with everybody: nobody knows what’s wrong or what happened. Yeah, sure.

What to do? Go find some screwdrivers, of course. The horrible noise emanates from somewhere around the power supply. Sort of. It’s buried in the guts of the Beast, so it’s hard to tell from the outside. It is a place to start, anyway. I roll up the sleeves of my robe, pick up a Holy Implement of Torx, and get to work …

Several hours later, I finally have figured out, cuss word by cuss word (proper ordering is important), how to get past all the barriers cleverly designed by Evil HP Engineers to make rational disassembly near-impossible. (Ever disassemble a laptop computer, down to the bare metal? This is harder, I kid you not.) Sixty screws later (I count them, twice), I get to the power supply fan. The heart of the Beast is diseased, despoiled. It is not turning quite right, and the motor shaft wiggles a little. It is not supposed to wiggle. Even a little. Culprit apprehended at last? Perhaps. Fortunately, it’s just a cheap $8 cooling fan you can pick up at any Radio Shack.

But Radio Shack does not exist anymore. When did that happen?

We have come round to this place again: what to do? Rummage around in the junk spare parts room, of course. It is a glorious room, beloved of tinkerers on staff. Bingo: six salvaged computer power supplies, just lying there on a shelf, calling to me. No, seven! But I am wise to their siren song. One after another, a closer look reveals frightening ugliness — mostly in the form of caked-on dust and dirt and grime. Their hearts spin, but they are Unclean and Decrepit. Sigh … last one: yay, Cleanliness! The Blessed One, Savior of the Beast, is found.

It believes it has been bestowed a new chance at life. I wish I could be happy for it. Little does it know its fate. Surely it deserves to be told of its pending doom? Yet that would crush its new-found hopes. You are perverse and cruel, you Fates! I do not have the heart to tell it.

True to my calling as Lord High Tinkerer, I pick up the Holy Implement of Torx and sacrifice the Blessed One upon the Ancient Altar of Gorthung (a fifty-year-old, government-issue desk, solid and heavy as a tank, with an ice-cold slate top). I flay its body and cut out its heart. I know no mercy.

Fan in bloody hand (a blood blister acquired some time during printer pieces-parts separation has popped), I trundle down the hill to the electronics lab. There, a colleague — the Wizard of Wire, Lord of Circuit — performs minor surgery. Lo, and behold! Upon application of the Lightning of Zoltar (a 12-volt power supply), the heart of the Blessed One lives again, spinning round and round in a most pleasing whir. Back up the hill.

That dreaded niggle squatting in the back of my mind finds a crack and blossoms. It dawns on me: now I have to put it all back together. Sixty screws. I realize I am tired. I’ll never remember where they all go. Come back tomorrow with freshly caffeinated veins? Pffft. Such is for wusses, unbecoming of a Tinkerer. So, since the operation of my memory — even on a good day — resembles most closely that of a sieve, I have little choice but to re-figure out how to take apart the Beast but in reverse. I am reminded of Ginger Rogers. I miss Ann Richards and her rapier wit. Today is not a good day.

Another hour passes by. I wave hi. We do that a lot, Time and I. My finger leaks on the table; I wipe it. And also on the housing of the reassembled printer power supply. I look at the smear, and I do not wipe it. I have left my mark upon this Beast, I think to myself. I shall not remove it. It will be buried amidst your guts; only you and I will ever know it is there. This token of my toil is enough, I decide. I move on.

At last, it is back together, despite all the King’s men staying home, watching TV. I do not want to plug it in. I’m sure you understand. Don’t you? Even so, I still roll the Beast back to its lair. I reattach its stiff black tail. I notice it is dirty, the cord, this conduit of the Lightning of Zoltar.

We have arrived at the moment of truth: I flip the switch. And wait. As with a pot of water that has yet to boil, it is best not to stare at a booting computer, especially one as slow and dumb as the Beast’s. I stare anyway. I wave hi to passing Time again, then it whirs with a pleasing sound. And dies. And tells me to call HP.

Naturally, I turn it off, wait ten seconds (capacitors can be slow to bleed, you know), and then turn it on again. Maybe something different will happen this time.

Null Hypothesis

I write

Into the void.

Transient, ephemeral, aperiodic,

Sharp.

 

Null hypothesis. By assumption

Not significant. You are

Alone. Free. Leap

Into the ether, the abyss —

Into the void.

 

math test

Here’s how to get MathJax up and running for your blog: part I, part II. The three tests below are text lifted from elsewhere.

Test 1

Consider first what we shall call the direct geometry case, in which we use only the zenith angle $z$ and bypass the geocentric angle $\theta$. The length of side $\overline{CM}$ follows from the right triangle $\widehat{CMP}$:

$$\begin{equation}\begin{array}[b]{ccl}\left(R+H\right)^{2} & = & \left(D\sin z\right)^{2}+\left(R+h+D\cos z\right)^{2}\\ \\ & = & D^{2}+\left(R+h\right)^{2}+2\left(R+h\right)D\cos z\end{array}\label{eq:R+H-test}\end{equation}$$

or

\begin{equation}D^{2}+2\left(R+h\right)D\cos z-\left[\left(R+H\right)^{2}-\left(R+h\right)^{2}\right]=0\label{eq:D eqn-test}\end{equation}

with solution

\begin{equation}\begin{array}[b]{ccl}D & = & -\left(R+h\right)\cos z\pm\sqrt{\left(R+h\right)^{2}\cos^{2}z+\left[\left(R+H\right)^{2}-\left(R+h\right)^{2}\right]}\\ \\& = & \left(R+h\right)\left(\sqrt{\cos^{2}z+\dfrac{\left(R+H\right)^{2}-\left(R+h\right)^{2}}{\left(R+h\right)^{2}}}-\cos z\right)\end{array}\label{eq:D soln quadratic ugly-test}\end{equation}

where the geometry of the problem requires the positive root. For convenience, define

\begin{equation}\epsilon\equiv\dfrac{H}{R}\quad\mathrm{and}\quad\xi\equiv\dfrac{h}{R}\label{eq:eps and xsi defs-test}\end{equation}

Then we can write eq. \eqref{eq:D soln quadratic ugly-test} as

\begin{equation}D=\left(R+h\right)\left(\sqrt{\cos^{2}z+\left(\dfrac{1+\epsilon}{1+\xi}\right)^{2}-1}-\cos z\right)\label{eq:D soln quadratic-test}\end{equation}

Eq. \eqref{eq:D soln quadratic-test} has the disadvantage of subtraction of two nearly equal numbers.

Test 2

We would like to know what is the radius $\bar{r}$ of the center of mass

of a grid cell of inner radius $r_{1}$ and outer radius $r_{2}$. In polar coordinates $\left(r,\theta\right)$ an infinitesimal area element is $dA=r\,dr\,d\theta$, so

\begin{equation}\bar{r}=\frac{1}{\Delta A}\intop_{0}^{\Delta\theta}\intop_{r_{1}}^{r_{2}}r\,dA=\frac{1}{\Delta A}\intop_{0}^{\Delta\theta}\intop_{r_{1}}^{r_{2}}r^{2}dr\,d\theta\label{eq: area-weighted r integral-test}\end{equation}

where $\Delta A=\frac{\Delta\theta}{2\pi}\cdot\pi\left(r_{2}^{2}-r_{1}^{2}\right)$.

Thus,

\begin{equation}\Delta A=\frac{\Delta\theta}{2}\left(r_{2}^{2}-r_{1}^{2}\right)\label{eq: cell area-test}\end{equation}

and

\begin{equation}\bar{r}=\frac{1}{3}\frac{\Delta\theta}{\Delta A}\left(r_{2}^{3}-r_{1}^{3}\right)=\frac{2}{3}\frac{r_{2}^{2}+r_{1}r_{2}+r_{1}^{2}}{r_{1}+r_{2}}\label{eq: area-weighted r-test}\end{equation}

[…]

Thus, we have the bootstrapping scheme

\begin{equation}\begin{array}{rclcrcl}\bar{r}_{0} & = & \dfrac{2}{3\Delta^{2}}\left(r_{2,0}^{3}-r_{1,0}^{3}\right) & & r_{2,0} & = & \sqrt{r_{1,0}^{2}+\Delta^{2}}\\& \vdots & & & & \vdots\\\bar{r}_{k} & = & \dfrac{2}{3\Delta^{2}}\left(r_{2,\,k}^{3}-r_{2,\,k-1}^{3}\right) & & r_{2,\,k} & = & \sqrt{r_{2,\,k-1}^{2}+\Delta^{2}}\\& \vdots & & & & \vdots\\\bar{r}_{N_{r}-1} & = & \dfrac{2}{3\Delta^{2}}\left(r_{2,\,N_{r}-1}^{3}-r_{2,\,N_{r}-2}^{3}\right) & & r_{2,\,N_{r}-1} & = & \sqrt{r_{2,\,N_{r}-2}^{2}+\Delta^{2}}\end{array}\label{eq: bootstrap scheme}\end{equation}

where, again, we start with $r_{1,0}=r_{min}$ .

Test 3

Now, $-\widehat{z}\times{\left(\widehat{z}\times\overrightarrow{r}\right)}=\overrightarrow{r}-{\left(\widehat{z}\cdot\overrightarrow{r}\right)}\widehat{z}$, so

\begin{equation}\overrightarrow{r}^{\prime\prime}+2\widehat{z}\times\overrightarrow{r}^{\prime}=\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}{\left(\overrightarrow{r}+\overrightarrow{\nabla}U\right)}-{\left(\widehat{z}\cdot\overrightarrow{r}\right)}\widehat{z}\label{}\end{equation}

Define a new effective potential

\begin{equation}\mathrm{\Omega}=\frac{1}{2}r^{2}+U=\frac{1}{2}r^{2}+\frac{{1-\mathrm{\mu}}}{r_{1}}+\frac{\mathrm{\mu}}{r_{2}}\label{EQUATION.5d0b51dc-3a17-4d57-95ed-8e8768257778}\end{equation}

where

\begin{equation}r_{1}=\sqrt{{{\left(x+\mathrm{\mu}\right)}^{2}+y^{2}+z^{2}}}\hspace{2em}r_{2}=\sqrt{{{\left(x-1+\mathrm{\mu}\right)}^{2}+y^{2}+z^{2}}}\label{EQUATION.10d1bacb-a0cf-4bdc-8b6d-c72d845b975b}\end{equation}

Then we find the satisfying result

\begin{equation}\overrightarrow{r}^{\prime\prime}+2\widehat{z}\times\overrightarrow{r}^{\prime}+{\left(\widehat{z}\cdot\overrightarrow{r}\right)}\widehat{z}=\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}\overrightarrow{\nabla}\mathrm{\Omega}\label{EQUATION.7aeaeb03-1226-46ab-815a-4b28e71a84a5}\end{equation}

The individual components of \eqref{EQUATION.7aeaeb03-1226-46ab-815a-4b28e71a84a5} are

\begin{equation}\begin{aligned}x^{\prime\prime}-2y^{\prime} & =\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}\frac{{\partial\mathrm{\Omega}}}{{\partial x}}\\y^{\prime\prime}+2x^{\prime} & =\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}\frac{{\partial\mathrm{\Omega}}}{{\partial y}}\\z^{\prime\prime}+z\hspace{0.9em} & =\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}\frac{{\partial\mathrm{\Omega}}}{{\partial z}}\end{aligned}\label{}\end{equation}

where

\begin{equation}\begin{array}{rcl}\overrightarrow{\nabla}\mathrm{\Omega} & = & \left[\begin{matrix}x-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}\left(x+\mathrm{\mu}\right)-\dfrac{\mathrm{\mu}}{r_{2}^{3}}\left(x-1+\mathrm{\mu}\right)\\y\left(1-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}-\dfrac{\mathrm{\mu}}{r_{2}^{3}}\right)\\z\left(1-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}-\dfrac{\mathrm{\mu}}{r_{2}^{3}}\right)\end{matrix}\right]\\ \\& = & \left(1-\dfrac{1-\mathrm{\mu}}{r_{1}^{3}}-\dfrac{\mu}{r_{2}^{3}}\right)\overrightarrow{r}-\mathrm{\mu}\left(1-\mathrm{\mu}\right)\left(\dfrac{1}{r_{1}^{3}}-\dfrac{1}{r_{2}^{3}}\right)\widehat{x}\end{array}\label{}\end{equation}

How I Do MathJax II. Example

To render equations in a WordPress blog, you have several options. The most aesthetically pleasing is MathJax. An earlier post tells you how to install MathJax for your WordPress site. This second post shows a few pointers by way of an example (you’ll probably want to view the page source, then search for “For example”). Here are a few more usage examples.

How to Do Math in a Blog Post

If you’ve installed MathJax in your site, then in a blog post you can trigger the loading of MathJax by putting the shortcode at the top of your post. It will not show up in your readers’ browsers.

That’s it! You can write your post now.

What I usually do, if the document has a lot of equations, is to compose the post in the quasi-WYSIWYG LaTeX editor, LyX. You can, of course, use whatever writing tool you like. When you’re happy with how your article looks, then copy the text to the clipboard. (With LyX, open up the source pane (View→Source Pane) and select the text.) Paste to your WordPress post editor.

You now have to make one change to the pasted text: remove the line breaks inside AMS environments

\begin{...} ... \end{...}

For example,

\begin{equation}
\begin{array}[b]{ccl}
D & = & -\left(R+h\right)\cos z\pm\sqrt{\left(R+h\right)^{2}\cos^{2}z+\left[\left(R+H\right)^{2}-\left(R+h\right)^{2}\right]}\\
\\
& = & \left(R+h\right)\left(\sqrt{\cos^{2}z+\dfrac{\left(R+H\right)^{2}-\left(R+h\right)^{2}}{\left(R+h\right)^{2}}}-\cos z\right)
\end{array}\label{eq:D soln quadratic ugly}
\end{equation}

becomes

\begin{equation}\begin{array}[b]{ccl}D & = & -\left(R+h\right)\cos z\pm\sqrt{\left(R+h\right)^{2}\cos^{2}z+\left[\left(R+H\right)^{2}-\left(R+h\right)^{2}\right]}\\\\& = & \left(R+h\right)\left(\sqrt{\cos^{2}z+\dfrac{\left(R+H\right)^{2}-\left(R+h\right)^{2}}{\left(R+h\right)^{2}}} \cos z\right)\end{array}\label{eq:D soln quadratic ugly-how}\end{equation}

Here’s how to refer to the above equation. Write, for example,

eq. \eqref{eq:D soln quadratic ugly}

which renders as eq. \eqref{eq:D soln quadratic ugly-how}.

How I Do MathJax I. Installation

I use equations. To enable equations in a WordPress blog, there are several options. The most comprehensive—and aesthetically pleasing—is to use MathJax. This post tells you how to install MathJax for your WordPress site. A second post has a few pointers. Here are a few usage examples.

1. Edit default.js

I do not use the MathJax CDN since occasionally their site has problems. When that happens, your math stops working and your pages containing math become ugly. So I download MathJax to my WordPress install. Rather than futz with <script> tags in my site’s header, I edit the default configuration file to my liking. Thus:

  • Download the latest version of MathJax: go to https://github.com/mathjax/MathJax/, click on Releases, and download the latest version.
  • Unpack the archive file to your hard drive.
  • Edit default.js in the config directory. My preferences:
    • You’ll probably want to add to your extensions, something like:
      extensions: ["tex2jax.js", "TeX/AMSsymbols.js", "TeX/AMSmath.js"]
    • Scroll down and set messageStyle to your liking (I changed mine to messageStyle: "simple").
    • Scroll down to menuSettings and change these to your liking (I set zoom: "Hover").
    • In the tex2jax section that immediately follows:
      • Under inlineMath uncomment the line with inline delimiters ['$','$']. This enables normal LaTeX inline delimiters. You’ll have to escape actual dollar signs with \\\$.
      • processEscapes: true
      • preview: "[math]"
    • Scroll down to the TeX section.
      • Under equationNumbers, set autoNumber: "AMS".
      • Fiddle with whatever else there catches your fancy.
    • Fiddle with whatever else catches your fancy.
  • Finally, upload your entire MathJax directory to your WordPress site, something like http://yourdomain/mathjax/.

2. Get the WordPress plugin.

Next, get the MathJax-LaTeX plugin and set the settings. The easiest way is to go to your blog administration Dashboard→Plugins→Add New, and type mathjax in the search box. My plugin settings (Dashboard→Settings→MathJax-LaTeX) are

  • Force Load = unchecked
  • Default [latex] syntax attribute = inline (this seems to have no effect with my configuration)
  • Use wp-latex syntax? = unchecked
  • Use MathJax CDN Service? = unchecked
  • Custom MathJax location? = http://yourdomain/mathjax/MathJax.js
  • MathJax Configuration = default

Do not forget to click the Save Changes button!

image lightbox test

 

M37-g15d053.020.SDSSR_.c.stack-159.fits

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