Musings on $\pi$ Day

1. The Ubiquity of $\pi$      or, Life, the Universe, and Everything: A Simple Statement of Fact

$\pi$ is everywhere you look. It is even the case that there is $\pi$ in the sky. We need $\pi$ in order to live and function. These three observations are fundamental to the way our universe is put together.

2. The Value of $\pi$      or, How Much is that Round Thing in the Window?

(with apologies to Patti Page) So, for fun, let’s calculate $\pi$ using Ramanujan’s famous infinite series formula, and check the error against a clever, arbitrary-precision algorithm for $\pi$, based on the Chudnovsky brothers’ improvement on Ramanujan’s series approximation, and which is correct to as many digits as we care to specify. While we’re at it, we’ll include just a straight-up implementation of the Chudnovsky brothers’ series approximation, too.

Ramanujan’s formula (see also here, and here):

\begin{equation}
\dfrac{1}{\pi} = \dfrac{2\sqrt 2}{9801}
\sum_{n=0}^{\infty} \dfrac{\left(4 n\right)!}{\left(n!\right)^4}
\dfrac{1103 + 26390\,n}{396^{4n}} \label{eq:ram}
\end{equation}

As mentioned here, the Chudnovsky brothers derived a Ramanujan-like formula that converges considerably faster(!) than Ramanujan’s original:

\begin{equation}
\dfrac{1}{\pi} = \dfrac{1}{53360\sqrt{640320}}
\sum_{n=0}^{\infty} \left(-1\right)^n
\dfrac{\left(6 n\right)!}{\left(n!\right)^3\left(3 n\right)!}
\dfrac{13591409 + 545140134\,n}{640320^{3n}} \label{eq:chud}
\end{equation}

We can take advantage of Python’s decimal module for exact arithmetic to as many digits of precision as we might want in calculating each term of the series. Doing so, we find the following errors after each successive iteration of the two series (note the exponents!):

Ramanujan   Chudnovsky
n Rpi(n)-pi Cpi(n)-pi
-- ---------- -----------
0 7.642E-8 -5.903E-14
1 6.395E-16 3.078E-28
2 5.682E-24 -1.721E-42
3 5.239E-32 1E-56
4 4.944E-40 -5.959E-71
5 4.741E-48 3.609E-85
6 4.599E-56 -2.212E-99
7 4.5E-64 1.368E-113
8 4.433E-72 -8.515E-128
9 4.391E-80 5.331E-142
10 4.37E-88 -3.353E-156
11 4.364E-96 2.117E-170
12 4.372E-104 -1.341E-184
13 4.393E-112 8.513E-199
14 4.424E-120 -5.42E-213

As we can see, Ramanujan’s formula, eq. \eqref{eq:ram}, gives eight orders of improvement (i.e., eight more digits of accuracy) per successive iteration, while the Chudnovsky formula, eq. \eqref{eq:chud}, yields fourteen orders of precision per iteration!

To illustrate, after fifteen Chudnovsky series terms, the difference between the series approximation and the actual value of $\pi$ is:

-0.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
00000 00000 00000 00000 00000 00000 00000 00000 00000 00000
00000 00000 00542...

Even just the first Chudnovsky term by itself (or just the first two Ramanujan terms) gives $\pi$ to almost machine precision ($2^{-52}\approx 2.22\!\times\!10^{-16}$) on a 64-bit computer.

For another perspective (thanks for the idea, Daniel Greenspan), let’s calculate (roughly!) the total number of atoms in the universe. As you might imagine, this will be a big number. We’ll break it down into two parts.

First, how many stars are in the universe? This is a number we can estimate from observations of galaxies and the amount of light that they emit. Essentially, since we can determine distances to galaxies, we just add it all up. Modern astronomical estimates for the equivalent number of solar-mass stars in our universe, based on the amount of light we detect coupled with the distances to the objects (galaxies) emitting that light, all come in at around

\begin{equation}
N_{stars} \approx 2\!\times\!10^{23} \label{eq:Nstars}
\end{equation}

This is equivalent to the mass of the visible universe divided by the mass of the Sun.

The amount of baryonic—that is, visible, or what we think of as “normal”—matter in the universe is only a small fraction of the total mass of the universe. Our universe, based on several different kinds of observations, is $68.3\%$ dark energy, $26.8\%$ dark matter, and $4.9\%$ ordinary matter. But that’s another story. We’ll just stick to the ordinary matter that we can detect via the electromagnetic radiation it emits.

Second, how many atoms are in a star the mass of our Sun? Now, the Sun has a measured mass of $M_{\odot} = 1.9884\!\times\!10^{30}$kg() and is composed of about $74.9\%$ hydrogen and $23.8\%$ helium by mass(). For this exercise, we will assume that the mass contributions of electrons and the other elements besides hydrogen and helium are negligible. The mass of a proton is $1.00784$amu, and the mass of a helium nucleus is $4.002602$amu. One amu (atomic mass unit) is $1.66053904\!\times\!10^{-27}$kg. The approximate number of atoms in the Sun, $N_{\odot}$, is then

\begin{equation}
N_{\odot} \approx
\dfrac{0.749 M_{\odot}}{1.00784\mathrm{amu}} +
\dfrac{0.238 M_{\odot}}{4.002602\mathrm{amu}}
\approx 9.6 \!\times\!10^{56} \mathrm{atoms} \label{eq:Nsuns}
\end{equation}

Hence, combining \eqref{eq:Nstars} and \eqref{eq:Nsuns}, the number of atoms in the universe, $N_{universe}$, is, roughly,

\begin{equation}
N_{universe} \approx N_{stars}\cdot N_{\odot}
\approx 1.9\!\times\!10^{80} \mathrm{atoms}
\end{equation}

Notice from the table above that the error of the Chudnovsky series after only the first six terms is about one part in $2.8\!\times\!10^{84}$—a number that is several orders of magnitude larger than the number of atoms in the entire universe!

3. A Short Introduction to Astrophysics     or, Is there Really $\pi$ in the Sky?     or, Here are the Footnotes

You might not have seen this coming. But here it is, wherein we demonstrate that, indeed, there is $\pi$ in the sky.

(†) We can determine the mass of the Sun by measuring the motions of the planets and asteroids in our Solar System, and then using Newton’s Law of Gravity. As Kepler discovered from Tycho Brahe’s meticulous observations, and Newton proved mathematically after he invented calculus and then turned his attention to the Moon’s motion, the orbital period $P$ of a body of mass $m$ and its mean distance $a$ from the Sun with mass $M_{\odot}$ are related by

\begin{equation}
P^2 = \dfrac{4 {\pi}^2}{G\left(M_{\odot}+m\right)} a^3
\end{equation}

Look at that: $\pi$ is in this equation that describes what we see in the sky.

(‡) We can determine the relative abundances of the elements that make up the Sun (and almost any star) by measuring, with a spectroscope, the amount of radiation absorbed by those elements in the atmosphere of the Sun (called the photosphere). Every element has its own discrete spectral signature in the form of absorption lines at specific sets of wavelengths. The amount of radiation absorbed by an element, relative to the other elements present, and in combination with the measured temperature, luminosity, and mass of a star, tells us what fraction of the star’s photosphere consists of that element. (We also need to know the distance to the star, but that’s a long story.)

Stars are, roughly speaking (i.e., ignoring the radiation absorbed by the elements in their photospheres), black body radiators. This means we can relate their luminosity (total radiated energy per unit time) to their radius $R$ and their effective surface temperature, $T_{eff}$. Simply put, the luminosity is the surface area of the star ($4\pi R^2$) times the amount of radiation emitted per unit surface area of the star:

\begin{equation}
L = 4\pi R^2 \sigma T_{eff}^4 \label{eq:L}
\end{equation}

where $\sigma = \dfrac{2\pi^5 k^4}{15 c^2 h^3}$ is the Stefan-Boltzmann constant, $k=1.38064852\!\times\!10^{−23}$ Joules per degree Kelvin ($J\cdot K^{-1}$) is the Boltzmann constant, $c$ is the speed of light in vacuum, and $h=6.62607015\!\times\!10^{−34} J\cdot s$ is the Planck constant from quantum mechanics. Eq. \eqref{eq:L} is a consequence of the physics of black body radiation.

Look at that: $\pi$ is integral to these relations that describe what we see in the sky, too.

(Don’t ask about the quantum mechanics connection. You can go down that rabbit hole by following the provided links. Quantum mechanics hurts my head.)

4. Full Disclosure      or, So This is Where That Came From

The Python code that produces the Ramanujan and Chudnovsky results (table and plot) is:


import decimal
from decimal import Decimal as D
from utils import mutils
from utils import mplot as plt

prec = 300  # Set the number of digits of precision
# for calculations.
decimal.getcontext().prec = prec

def dfac(n):
""" Arbitrary digits factorial. """
m = D('1')
for k in range(1,n+1):
m *= k
return m

def Rpi(n):
"""
Calculate pi using n iterations of Ramanujan's
formula.
"""
s = D('0')
for k in range(n+1):
facterm = dfac(4*k)/dfac(k)**4
num = D('1103') + D('26390')*k
den = D('396')**(4*k)
s += facterm*num/den
s *= D('8').sqrt()/D('9801')
return 1/s

def Cpi(n):
"""
Calculate pi using n iterations of the Chudnovsky
brothers' Ramanujan-like formula.
"""
s = D('0')
for k in range(n+1):
facterm = dfac(6*k)/(dfac(k)**3*dfac(3*k))
num = D('13591409') + D('545140134')*k
den = D('640320')**(3*k)
s += D('-1')**k*facterm*num/den
s *= D('1')/(D('53360')*D('640320').sqrt())
return 1/s

# Print a table of the error of n iterations
# of Ramanujan's formula.
print('     Ramanujan   Chudnovsky')
print(' n   Rpi(n)-pi    Cpi(n)-pi')
print('--  ----------  -----------')
fmt = '{:2d}  {:>10s}  {:>11s}'
c4  = decimal.Context(prec=4)
rerrs = []
cerrs = []
for n in range(15):
exact_pi = D(mutils.pi_chudnovsky(prec))
errR = Rpi(n) - exact_pi*D('1e-{:d}'.format(prec))
errC = Cpi(n) - exact_pi*D('1e-{:d}'.format(prec))
normerrR = errR.normalize(c4)
normerrC = errC.normalize(c4)
print(fmt.format(n, str(normerrR), str(normerrC)))
rerrs.append(float(errR))
cerrs.append(float(errC))

fig = plt.figure(figsize=(8.2, 5))
xlab = ['$\mathrm{number\ of\ series\ terms}\ n$', 12]
ylab = ['$\mid f(n) - \pi \mid$', 12]
pt = ['$\mathrm{\pi\ series\ approximation\ error}$',
14]
labs = [['$f(n) = \mathrm{Ramanujan}$', 10],
'$f(n) = \mathrm{Chudnovsky}$']
xticks = np.arange(15)
yticks = np.array([0.1**k for k in range(0, 240, 30)])
ylim = (1e-220, 1e-1)
plt.lineplot([np.array(rerrs), abs(np.array(cerrs))],
np.arange(15), ['k-', 'r-'], [1, 1],
ylim=ylim, logy=True, xlab=xlab,
ylab=ylab, xticks=xticks, yticks=yticks,
doxticks='bottom', doyticklabels='both',
dolegend=True, labels=labs, plottitle=pt)
fname = (os.environ['PYTHONPATH'] +
'/misc/Ramanujan pi.jpg')
plt.savefig(fname, dpi=300)


Moonlit Snowscape

The Moon illuminates a snowy scene (my back yard) in the pre-dawn darkness. This is an 80 second exposure at 35mm f/3.2 and ISO 125. The yellow-gold color on the background trees is from low-pressure sodium streetlights on the next street over. Click on the image to enlarge; right-click to open the full-resolution version in a new tab.

Twilight Dome Fun

Here are a few images from goofing around yesterday evening with the NOFS 61-inch telescope, dome, and twilight. Images were taken with a Canon G3 X at ISO 125 on 2019-02-07 MST. Click on an image to enlarge; right-click to open the full-resolution version in a new tab. Ordering is reverse-chronological because I like the 7:02pm image best.

All images © 2019 Marc A. Murison: CC-BY-NC-ND

When I am at my desk, preparing for tonight’s observing. And it is evening.

Notes to self, part 437.

1. When I am at my desk, preparing for tonight’s observing.
1. And it is evening.
2. If an email arrives from the satellite tracking app, you could open it.
1. Be aware that this alert is for tonight.
2. You did bring clothing for the weather, right?
1. Not that it matters. You don’t pay attention to these things.
2. Maybe you should.
3. Come to think of it, you do recall thinking, this morning, that you could get away with not paying attention today, since you figured you’d be inside anyway.
1. Running late, you were in a hurry.
2. And you are lazy, when possible: it makes life more efficient.
3. You bring your digital camera with you to the Observatory, because you never know what will demand photos on any given day.
1. Or night.
2. Mountain weather dances, flits, pirouettes.
1. Cloud formations tend to be awesome.
2. Atmospheric effects abound.
3. Evanescent.
4. Most, even in such a wondrous, sky-dance land, never look up.
1. Is the mundaneness of our daily routines so important? That we must concentrate our gaze, glazed, on the mud of our feet?
2. This is a great sadness.
4. According to the alert, the International Space Station is due to pass overhead.
1. Tonight.
2. It is an especially good pass:
1. For once, its path will track straight overhead.
2. For once, it will largely miss the Earth’s shadow.
1. This means the ISS will be a bright beacon from nearly horizon to horizon.
2. This means it must be nearly either a north-to-south or a south-to-north pass. Ah, spatial geometry.
3. For once, this good fortune is not tied to a predawn pass.
1. You do not function well in the predawn hours.
3. To compensate, Murphy’s Law will demand its due.
1. It always does.
1. This is consistent with observation.
2. You hypothesize that this is a conservation law.
3. Murison’s Corollary: When fortuitous good things happen, the balance of the Universe must be restored.
1. Count on it.
5. Fire up the satellite ephemeris program you wrote.
1. Fetch the latest orbital elements from space-track.org.
2. Create plots of azimuth and height above the horizon.
3. Check that your observing window matches the alert’s prediction.
6. Glance at the outside temperature: +12°F.
1. You are surprised.
2. But then you remember this morning, and your decision to leave the coat, the scarf, the gloves, behind.
3. Tell yourself: that’s okay, this should be quick, it’s not that cold.
7. Grab the camera and head outside ten minutes early.
1. Always start early. Things go wrong.
2. Rats: you didn’t bring a tripod.
1. Hand-held video recording it is, then.
2. You are secretly a little relieved at not being able to try anything fancy.
1. Even though nobody else is here, it feels like a secret.
2. Can we really keep secrets from ourselves?
8. The door locks behind you: click.
1. Memory trigger.
2. Check your pocket for keys. After it locks behind you.
3. This strikes you as humorous.
9. Find a good spot: the middle of the small parking lot.
1. Unobstructed view north, west, and south.
2. The main telescope dome, three stories high, with a halo of Flagstaff light pollution, swallows the eastern sky.
3. The satellite is on a south-to-north path tonight.
4. Yes, this is perfect.
10. The southwest wind is brisk.
11. Unpack and check your camera.
1. Breathe. Go slow. Be methodical. Think.
2. Everything functions as expected.
3. You don’t expect this. What will be the yin to this yang?
12. +12°F is cold.
13. Bare hands in +12°F will quickly go numb.
1. Forty-five seconds to a minute, tops.
2. You will marvel at the pain, though you cannot feel anything.
3. Configure and start your camera before this happens.
14. Check your watch: seven minutes to go.
1. This, too, is unexpected.
2. Try not to think about your body heat rapidly fleeing with the wind, that thief.
1. Your warm, warm, cozy, comfortable body heat.
2. Via your hands, and neck, and head, and feet.
3. When did these jeans become so thin?
3. Seven minutes is an eternity.
1. When there is nothing to do but not think about how uncomfortable it is.
2. When standing exposed in the wind.
3. When it is +12°F.
15. Keep your eyes on the view through the camera.
1. Is that it, there, low in the southwest?
2. Look up, blink-flick distorting tears, and verify with your eyes: yes, there it is.
1. Right on time.
2. In the right place.
3. Glorious.
16. Follow it slowly up, and over, and down to the northeast, where it softly slips into shadow before reaching the treeline. The five-minute pass passes quickly.
1. Now you cannot feel your feet.
17. It is done.
1. Note the satisfaction in your gut: good data acquired, it says.
2. Bask in that warmth as you lean down to pack up.
3. And then your circumstances impinge.
18. Fifteen minutes is a surprisingly long time when it’s +12°F out.
1. And you’re wearing only a t-shirt and light jacket.
2. And Birkies.
19. If you can’t feel anything with the stumps at the ends of your arms, there will be consequences.
1. You won’t be able to turn off or stow your camera.
2. It will be surprisingly hard, and hence take a surprisingly large number of tries, and hence take a surprisingly long time, to get your key into the door lock and scurry back inside, to your office.
3. Where it is not +12°F.

Camera: Canon G3 X. Video processed using kdenlive.

Moved to here.

Preparations

The evening looks promising.

Transparent air, crystalline blue—emblematic of Flagstaff even in summer—has soothed my soul since childhood. “I can see for miles and miles…” spins in my head, unbidden, as I walk a short patch of worn asphalt, dull gray and pitted from winter’s attacks. Dark green Ponderosa forest broods to the horizon, turning black as the light dims and the usefulness of my retinal color sensors fades. Thin, dry air is a poor thermal insulator, so it chills rapidly after sunset. I zip my jacket.

Ritual scan of the sky, projecting ahead several hours: gauging the night’s weather and observing conditions is an habitual game. I occasionally misjudge, but not tonight. The door clacks shut behind me. I know my keys are in my pocket, but I check anyway. I aim toward the chipped, institutional-turquoise railing in front of the dome. Cirrus lie low in the southwest, painted grunge by twilight and distance (thirty, forty miles). These will likely keep to their remove and not interfere. I pretend to decree it so.

As I shuffle southwest, my face parts the breeze. My felt hat stays on my head, unassisted. I’ve no need to glance at our rooftop weather station’s wind vane or anemometer. It should be a good night, the air clinging to the forest laid out before me as it flows, laminar and unturbulent, lifting with the ridge upon which we root and gliding smoothly overhead. Trudy, our night observer, should get one arcsecond seeing at the 61-inch telescope, perched on its massive concrete pedestal three stories up. It is the world’s most precise star measuring engine. Down here at the parking lot, the air will be more agitated. I’ll see two arcseconds, maybe a bit less, at the 51-inch telescope which squats inside a dome off the edge of the cooling asphalt. The dome resembles R2D2 from Star Wars.

“Seeing” is astronomer jargon for what our roiling atmosphere does to starlight, pushing and shoving it, forcing it to wiggle erratically in random directions as cells of turbulent air, refractive indexes varying slightly from their neighbors, scurry across our line of sight. These pockets of air, fleeing distant large-scale atmospheric pressure gradients, attest to forces at work beyond our tiny purview. This is why stars twinkle.

Baleful blood-red Scorpion heart, Antares, sits low in the south. The orb flashes sharp red and green and yellow and blue (if you stare carefully), dancing. Astronomers hate that. Twinkling harbingers fuzzed, mushy, corpulent images. Spica is higher in the sky, its hard, white-blue light passing through less of our atmosphere. It holds fairly steady, only an occasional flicker. I look higher. Orangey Arcturus, one of my favorite stars, stares unblinking, steady, solid. Some part of my brain registers that stars higher than about forty degrees above the horizon will be sharp tonight. I notice muscles relaxing, a growing anxiety over data quality now dissipates. Mona Lisa smile: in this clear air, the night will be dark and the Milky Way will billow, almost flocculent, and span the entire vault of the sky. I will remember later to emerge and gape at this wonder until my neck aches. Da Vinci would understand. models fitted to a star’s intensity profile (click to enlarge)

Scientists quantify. Astronomers’ measure of seeing—our means of taking the guesswork out of comparing one night to another—is the size of a star’s disk at the focal point of a telescope (that is, on the sensor hanging off its butt end). The width of a circle drawn half-way down from the central, brightest point of the disk that is a star image to its edge as it merges into the sky background is that measure. We call it “full width, half max”, or FWHM.

We measure angles with telescopes—immense, expensive protractors. This star is so many fractions of a degree from that star. A sixtieth of a degree is an arcminute. Your eyes can resolve details down to about one arcminute, or slightly less. A sixtieth of an arcminute is an arcsecond. An arcsecond is a very small angle: the apparent size of a U.S. quarter, 3.1 miles away. (The 61-inch telescope can measure angles to within one thousandth of an arcsecond.) “Good” seeing is when the FWHM of a star image is one arcsecond or less—a useful cultural agreement. Three arcseconds is bad. Five is horrendous, and the stars are dancing madly, taunting and useless, all the way to the zenith.

Inside the dome, chill seeps through my clothes as I wait for dome shutters and mirror covers to open the telescope’s eye to the heavens. As the liquid nitrogen tank satiates the camera dewar in pulsing spurts, a valve trips and vents excess pressure; the hiss is painful. I escape into the side room and toggle switches, powering various devices. The air compressor initiates a new aural assault. I plug my ears. Why did it choose now, I think. Several of us conjecture that the 51-inch telescope is inhabited by gremlins, not so much malevolent as impish, irritating. Maybe they are leprechauns. Back in the dome, motors stop and the nitrogen tank has finished its rhythmic regurgitation of cold. Pulling on insulated blue gloves meant for such things, I disconnect and stow the thick umbilical hose, its business end caked with ice, thin sheets of condensing air flowing to the dome floor. The drive motors wake and hum, a happy sound, as I feed them power. Everything inside this dome is thirsty.

We are ready for the night.

Prolegomenon

You recognize as a youngster that science, and music, and literature and writing—creative wonders—draw you along comfortable invisible force lines. But not opera. Overbearing, embarrassing falsetto vibrato is just wrong. As your joints grow creaky and more of your pate warms to the Sun, you know that this is a misperception. You stumble upon more of these, as you notice yourself more often assigning past vigorous feats of physical prowess to the unimportant pursuits of the unimportant young. You ponder these, your various misperceptions. And your misperceptions of misperceptions. Recursion tickles you.

$$\dfrac{\mathrm{d}^2\overrightarrow{r}}{\mathrm{d}\theta^2}+2\widehat{z}\times\dfrac{\mathrm{d}\overrightarrow{r}}{\mathrm{d}\theta}+{\left(\widehat{z}\cdot\overrightarrow{r}\right)}\widehat{z}=\frac{1}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}\overrightarrow{\nabla}\mathrm{\Omega}$$

You realize in the shower one day that your—and others’—universal cognitive foibles smacking into observable reality are an irresistible rabbit hole, wondrously vast and an endless source of material to contemplate. Like a particle in the three-body problem of celestial mechanics, your orbit is a tangled meandering, variously lured into the sphere of influence of first one and then the other of those two massive attractors, science and the creative urge. This resonates, and you realize a re-appreciation of past love.

$$\mathrm{\Omega}=\frac{1}{2}r^{2}+U=\frac{1}{2}r^{2}+\frac{{1-\mathrm{\mu}}}{r_{1}}+\frac{\mathrm{\mu}}{r_{2}}$$

Thus: what shall you write? Unuseful question. The world is big. Where shall you intend your aim? Better. Get thee to the shower!, your ever-reliable Delphic font of nearly every good idea.§ You love nature, and science—especially astronomy and math—and the scientific way of thinking, which come to you with joy and not pain. (This cannot be weird, surely—friends’ and society’s protestations notwithstanding.) The chasm awaits.

$$r_{1}=\sqrt{{{\left(x+\mathrm{\mu}\right)}^{2}+y^{2}+z^{2}}}\hspace{2.222222em}r_{2}=\sqrt{{{\left(x-1+\mathrm{\mu}\right)}^{2}+y^{2}+z^{2}}}$$

On a whim you schlep to a National Association of Science Writers conference, where you are isolated and small, sole introvert amidst a mind-bruising cacophony. Drilling through your crushing discomfort, you meet Roy Peter Clark’s Writing Tools: 50 Essential Strategies for Every Writer (you buy three copies), you hear Jonathan Coulton sing his wistful nerd anthem, “Code Monkey” (you buy three CDs), and a merciful soul tells you to read Lewis Thomas’s classic medley of essays, The Lives of a Cell: Notes of a Biology Watcher (why is there no Kindle version?). This is it. A trigger, an unlatching: your dormant writing compulsion awakens.

Astronomy with math. True stories, precisely told. A worthwhile target.

$$v^{2}-\frac{{2\mathrm{\Omega}}}{{1+e_{p}\mathrm{cos}\mathrm{\theta}}}+z^{2}+C+2\int\frac{{e_{p}\mathrm{sin}\mathrm{\theta}}}{{\left(1+e_{p}\mathrm{cos}\mathrm{\theta}\right)}^{2}}\mathrm{\Omega}\hspace{0.222222em}d\mathrm{\theta}=0$$

Halfway through college, you end the pleasant agony and decide astronomy over music. Seemingly by crazy random utterly naive inevitability, you become a professional astronomer. As your mop grows thinner and your knuckles grow larger, you realize the apparent randomicity is a misperception.

The equations, if you are wondering, tell how a massless particle moves in the combined gravitational fields of two massive objects in orbit about each other.¤ Think, for example, Sun–Jupiter–spacecraft. In astronomy, we call this the restricted three-body problem. It is astonishingly complex.

§ Perhaps only Death is a greater surety—though, surely, only by a little.

¤ For completeness:

$$\mathrm{\mu}=\frac{m_{2}}{{m_{1}+m_{2}}},\hspace{2.2em}r=\sqrt{x^2+y^2+z^2}$$

and

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

Flagstaff Skywheel 2015-11-07

Dark skies are a treasure, a part of our culture, a part of who we are as humans that we must preserve. Due to some enlightened and forward thinking in the late 1980s, the outdoor lighting code implemented in Flagstaff has thus far kept light pollution from completely overrunning our beautiful natural skies.

From my back yard, 2.5 miles from the downtown commercial business center (click the thumbnail at right), I can see stars as faint as about magnitude 5.5 on a clear, Moonless night. In the video, North is towards the upper left corner. On the left side (NE), you can see that the sky background is noticeably brighter than toward the SW at right. The center of downtown Flagstaff is toward the  NE.

This is 3.25 hours of the sky wheeling by in my Flagstaff back yard. Famous objects that appear: the Andromeda Galaxy (passes straight overhead), the Double Cluster in Perseus (left of Andromeda Galaxy), the Pleiades (towards the end, at the bottom), and Capella (towards the end, bright star at left).

Camera: Canon G3 X, 30 seconds per “video” frame (15-second exposures).

Zodiacal Light West of Flagstaff, Feb. 2015 U.S. Naval Observatory – Flagstaff Station (click to enlarge)

The zodiacal light at 7:51 pm (MST) on February 10, 2015, as seen from the west parking lot of the U.S. Naval Observatory near Flagstaff. If you’re wondering where the Observatory is, it’s about five miles west of downtown (Google maps link).

Below are two versions of a stack of eight 30-second exposures taken with a ZWO ASI120MM camera mounted on a camera tripod. This was 1h 47m after sunset (6:04 pm), and 21 minutes after the end of astronomical twilight (7:30 pm). You can see several naked-eye astronomical wonders, which are marked on the annotated version: Zodiacal light from NOFS, 2015-02-11, with annotations (click to enlarge)

DDA 2015 – Constraints on Titan’s rotation from Cassini mission radar data

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.

Bruce Bills (JPL)

Abstract

We present results of a new analysis of the rotational kinematics of Titan, as constrained by Cassini radar data, extending over the entire currently available set of flyby encounters. Our analysis provides a good constraint on the current orientation of the spin pole, but does not have sufficient accuracy and duration to clearly see the expected spin pole precession. In contrast, we do clearly see temporal variations in the spin rate, which are driven by gravitational torques which attempt to keep the prime meridian oriented toward Saturn.

Titan is a synchronous rotator. At lowest order, that means that the rotational and orbital motions are synchronized. At the level of accuracy required to fit the Cassini radar data, we can see that synchronous rotation and uniform rotation are not quite the same thing. Our best fibng model has a fixed pole, and a rotation rate which varies with time, so as to keep Titan’s prime meridian oriented towards Saturn, as the orbit varies.

A gravitational torque on the tri-axial figure of Titan attempts to keep the axis of least inertia oriented toward Saturn. The main effect is to synchronize the orbit and rotation periods, as seen in inertial space. The response of the rotation angle, to periodic changes in orbital mean longitude, is modeled as a damped, forced harmonic oscillator. This acts as a low-pass filter. The rotation angle accurately tracks orbital variations at periods longer than the free libration period, but is unable to follow higher frequency variations.

The mean longitude of Titan’s orbit varies on a wide range of time scales. The largest variations are at Saturn’s orbital period (29.46 years), and are due to solar torques. There are also variations at periods of 640 and 5800 days, due to resonant interaction with Hyperion.

For a rigid body, with moments of inertia estimated from observed gravity, the free libration period for Titan would be 850 days. The best fit to the radar data is obtained with a libration period of 645 days, and a damping time of 1000 years.

The principal deviation of Titan’s rotation from uniform angular rate, as seen in the Cassini radar data, is a periodic signal resonantly forced by Hyperion.

Notes

• Titan:
• hard to see surface
• Cassini’s radar intended for mapping surface
• didn’t get much by way of repeat observations (“tie points”), which are needed to constrain rotation
• most data near poles — not terribly helpful
• Rotation model from tie-point observations
• Stiles et al. 2008: 50 tie points over 2.8 yr
• Now: 2602 tie points over 10 yr
• solve for 3 params (RA & DEC of spin pole, angular rate)
• $P = 15.94547727 \pm 6.03 \times 10^{-7}$ d
• spin pole precession
• gravity model: ~250 yr
• not clearly seen in data
• spin rate variations
• seen in data
• dynamical model
• assume Titan in synch. rotation
• gravity torque
• dissipation
• $\rightarrow$ libration period ~850 d
• Hyperion has nontrivial influence
• fit: libration period = 645.4 d, damping time = 430 yr, rotation period slightly changed

DDA 2015 – Recent Formation of Saturnian Moons: Constraints from Their Cratering Records

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: Moon Formation and Dynamics I

Henry C. (Luke) Dones (SWRI)

Abstract

Charnoz et al. (2010) proposed that Saturn’s small “ring moons” out to Janus and Epimetheus consist of ring material that viscously spread beyond the Roche limit and coagulated into moonlets. The moonlets evolve outward due to the torques they exert at resonances in the rings. More massive moonlets migrate faster; orbits can cross and bodies can merge, resulting in a steep trend of mass vs. distance from the planet. Canup (2010) theorized that Saturn’s rings are primordial and originated when a differentiated, Titan-like moon migrated inward when the planet was still surrounded by a gas disk. The satellite’s icy shell could have been tidally stripped, and would have given rise to today’s rings and the mid-sized moons out to Tethys. Charnoz et al. (2011) investigated the formation of satellites out to Rhea from a spreading massive ring, and Crida and Charnoz (2012) extended this scenario to other planets. Once the mid-sized moons recede far from the rings, tidal interaction with the planet determines the rate at which the satellites migrate. Charnoz et al. (2011) found that Mimas would have formed about 1 billion years more recently than Rhea. The cratering records of these moons (Kirchoff and Schenk 2010; Robbins et al. 2015) provide a test of this scenario. If the mid-sized moons are primordial, most of their craters were created through hypervelocity impacts by ecliptic comets from the Kuiper Belt/Scattered Disk (Zahnle et al. 2003; Dones et al. 2009). In the Charnoz et al. scenario, the oldest craters on the moons would result from low-speed accretionary impacts. We thank the Cassini Data Analysis program for support.

References
Canup, R. M. (2010). Nature 468, 943
Charnoz, S.; Salmon, J., Crida, A. (2010). Nature 465, 752
Charnoz, S., et al. (2011). Icarus 216, 535
Crida, A.; Charnoz, S. (2012). Science 338, 1196
Dones, L., et al. (2009). In Saturn from Cassini-Huygens, p. 613
Kirchoff, M. R.; Schenk, P. (2010). Icarus 206, 485
Robbins, S. J.; Bierhaus, E. B.; Dones, L. (2015). Lunar and Planetary Science Conference 46, abstract 1654
(http://www.hou.usra.edu/meetings/lpsc2015/eposter/1654.pdf)
Zahnle, K.; Schenk, P.; Levison, H.; Dones, L. (2003). Icarus 163, 263

Notes

• Can cratering records constrain moon ages?
• see http://space.jpl.nasa.gov
• small inner moons (and Mimas) interact strongly with rings — the so-called “ring moons”
• migrated from outer edge of rings ~100 Myr
• regular moons (Mimas-Iapetus) are (assumed?) primordial
• transition is abrupt where tidal forces prevent formation
• formation of moons from spreading rings:Charnoz et al. 2010,Canup 2010,Charnoz et al. 2011,Crida &Charnoz 2012
• ring spreads viscously
• outside Roche limit, formation
• Lainey et al. 2012: dissipation stronger than thought
• decreases timescale considerably
• Impact rates
• $R_{moon} = R_J \dfrac{R_S}{R_J} \dfrac{R_{moon}}{R_S}$
• Crater scaling: diameter vs. velocity
• impacts/$10^9$ yr: Mimas 8.5, Rhea 48
• Mimas & Rhea counts: Robbins et al. 2015 (LPSC)
• plot: #craters larger than D vs. D
• Mimas: saturated up to $D \sim 20-45$ km
• Rhea: saturated up to $D \sim 25$ km
• Summary
• Mimas: Craters are near saturation for diameters < 20 km
• Rhea: saturation < 25 km
• Ages may be underestimated

DDA 2015 – On the Spin-axis Dynamics of the Earth

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.

Gongjie Li (Harvard) (Duncombe award winner)

Abstract

The variation of a planet’s obliquity is influenced by the existence of satellites with a high mass ratio. For instance, the Earth’s obliquity is stabilized by the Moon, and would undergo chaotic variations in the Moon’s absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically-forced climate change is the rate of chaotic diffusion. Accordingly, here we reexamine the spin-axis evolution of a Moonless Earth within the context of a simplified perturbative framework. We present analytical estimates of the characteristic Lyapunov coefficient as well as the chaotic diffusion rate and demonstrate that even in absence of the Moon, the stochastic change in the Earth’s obliquity is sufficiently slow to not preclude long-term habitability. Our calculations are consistent with published numerical experiments and illustrate the putative system’s underlying dynamical structure in a simple and intuitive manner. In addition, we examine if at any point in the Earth’s evolutionary history, the obliquity varied significantly. We find that even though the orbital perturbations were different in the past, the system nevertheless avoided resonant encounters throughout its evolution. This indicates that the Earth obtained its current obliquity during the formation of the Moon.

Notes

• Obliquity $\cos \epsilon$ affects climate
• Mars obliquity variations caused collapse of Martian atmosphere
• Obliquity variations of a Moonless Earth
• without Moon, $\epsilon$ is chaotic (Laskar et al. 1993)
• geostrophic winds
• but N-body sims: $\epsilon$ constrained to $\epsilon \lesssim 45$ deg — why?
• Sun and planetary torques: spin precession rate, inclination variation
• model as superposition of linear modes
• resonance overlap: two connected chaotic zones — Laskar 1993, Morby 2000, Laskar 1996
• average over primary resonances $\rightarrow$ secondary resonances
• overlap of secondary resonances creates the chaotic bridge (Chirikov 1979)
• Results
• regular at $\ge 85$ deg
• less chaotic in bridge
• analytic and numerical are consistent
• Li & Batygin 2014a
• diffusion timescale 10 Myr in primary chaotic zones, 2 Gyr in the bridge
• Pre-late heavy bombardment evolution of Earth’s obliquity
• Li & Batygin 2014b
• solar system starts more compact (Nice model)
• study evolution of mode freqs and effects on Earth’s inclination
• also, Moon was closer
• two freqs match prior to LHB only if $\epsilon \ge 85$ deg
• $\therefore$ Earth’s obliquity arose during the formation of the Moon

DDA 2015 – Recent dynamical evolution of Mimas and Enceladus

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: Moon Formation and Dynamics III

Maja Cuk (SETI Institute)

Abstract

Mimas and Enceladus are the smallest and innermost mid-sized icy moons of Saturn. They are each caught in a 2:1 orbital resonance with an outer, larger moon: Mimas with Tethys, Enceladus with Dione. This is where the similarities end. Mimas is heavily cratered and appears geologically inactive, while Enceladus has a young surface and high tidal heat flow. Large free eccentricity of Mimas implies low tidal dissipation, while Enceladus appears very dissipative, likely due to an internal ocean. Their resonances are very different too. Mimas is caught in a 4:2 inclination type resonance with Tethys which involves inclinations of both moons. Enceladus is in a 2:1 resonance with Dione which affects only Enceladus’s eccentricity. The well-known controversy over the heat flow of Enceladus can be solved by invoking a faster tidal evolution rate than previously expected (Lainey et al. 2012), but other mysteries remain. It has been long known that Mimas has very low probability of being captured into the present resonance, assuming that the large resonant libration amplitude reflects sizable pre-capture inclination of Mimas. Furthermore, Enceladus seems to have avoided capture into a number of sub-resonances that should have preceded the present one. An order of magnitude increase in the rate of tidal evolution does not solve these problems. It may be the time to reconsider the dominance of tides in the establishment of these resonances, especially if the moons themselves may be relatively young. An even faster orbital evolution due to ring/disk torques can help avoid capture into smaller resonances. Additionally, past interaction of Mimas with Janus and Epimetheus produce some of the peculiarities of Mimas’ current orbit. At the meeting I will present numerical integrations that confirm the the existence of these problems, and demonstrate the proposed solutions.

Notes

• tidal rates $\dfrac{1}{a}\dfrac{d a}{d t}$: Mimas = 59, Enceladus = 23
• numerical integrations — brute force
• artificial migration
• slow
• the trouble with Mimas
• Mimas and Tethys in inclination-type 4:2 MMR
• inclination of both moons affected by the resonance
• libration amp. of resonance is large, ~100 deg $\rightarrow$ primordial Mimas inclination — doesn’t work
• eccentricity of Tethys has complex effects
• Mimas-Tethys evolution rate: $\dfrac{da_{moon}}{dt} \propto \dfrac{R^5_{planet}}{a^{3/2}}$
• introduce ad hoc ring torques — artificial torque on Prometheus
• gives Tethys resonance a kick
• $\therefore$ don’t take Mimas-Tethys resonance too seriously
• …more ad hoc games…
• rings-Janus-Mimas-Enceladus-Dione system evolution is very complex

DDA 2015 – On the in situ formation of Pluto’s small satellites

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.

Man Yin Woo (University of Hong Kong)

Abstract

The formation of Pluto’s small satellites – Styx, Nix, Keberos and Hydra remains a mystery. Their orbits are nearly circular (eccentricity $e = 0.0055$ or less) and near resonances and coplanar with respect to Charon. One scenario suggests that they all formed close to their current locations from a disk of debris, which was ejected from the Charon-forming impact. We test the validity of this scenario by performing N-body simulations with Pluto-Charon evolving tidally from an initial orbit at a few Pluto radii. The small satellites are modeled as test particles with initial orbital distances within the range of the current small satellites and damped to their coldest orbits by collisional damping. It is found that if Charon is formed from a debris disk and has low initial eccentricity, all test particles survive to the end of the tidal evolution, but there is no preference for resonances and the test particles’ final $e$ is typically > 0.01. If Charon is formed in the preferred intact capture scenario and has initial orbital eccentricity ~ 0.2, the outcome depends on the relative rate of tidal dissipation in Charon and Pluto, $A$. If $A$ is large and Charon’s orbit circularizes quickly, a significant fraction of the test particles survives outside resonances with $e \gtrsim 0.01$. Others are ejected by resonance or survive in resonance with very large $e$ (> 0.1). If $A$ is small and Charon’s orbit remains eccentric throughout most of the tidal evolution, most of the test particles are ejected. The test particles that survive have $e \gtrsim 0.01$, including some with $e \gt 0.1$. None of the above cases results in test particles with sufficiently low final $e$.

This work is supported in part by Hong Kong RGC grant HKU 7030/11P.

Notes

• Pluto satellite system
• 5 known
• Charon dominant
• all nearly coplanar
• all nearly circular
• all near MMR with Charon
• Brozovic et al. 2015
• Formation scenarios
• forced resonant migration
• Ward & Canup 2006
• Nix & Hydra formed in same giant impact that formed Charon
• ruled out by Lithwick & Wu
• multi-resonance capture
• unlikely (Cheng et al. 2014)
• collisional capture of planetesimals
• Lithwick & Wu 2008, Dos Santos et al. 2012
• ruled out: capture time << collisional timescale, also Walsh & Levison 2015
• in situ formation
• Kenyon & Bromley 2014
• giant impact produced debris ring
• problem: outward tidal evolution of Charon
• Solving the migration problem
• forced eccentricity — Leung and Lee 2013
• for $e_C = 0.24$, $e_f \sim 0.01$ to $0.02$ for test particles (small moons)
• integrate two tidal models, constant $\Delta t$ and constant $Q$
• For constant $\Delta t$, no preference for resonances and $e \gt 0.01$
• Conclusion: it is unlikely that all the small satellites formed close to their current position

DDA 2015 – Rotational and interior models for Enceladus II

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.

Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

Notes

• Libration measurement
• 3D reconstruction of coords of a network of control point (fiducial satellite surface points — e.g. craters)
• most of Enceladus’s orbit was covered
• Thomas et al. 2015
• minimize RMS residual $\rightarrow 0.120 \pm 0.014$ deg
• Solid models
• core plus two-layer in hydro.equilib. plus south polar sea
• measured libration amplitude rules this out
• decoupled shell from the core (indep.librations)
• consistent with observed libration amplitude if shell thickness 21-26 km and ocean thickness 26-31 km
• Gravity data
• suggests a local mass anomaly — interpreted as ocean thicker under south pole