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.

DDA 2015 – The onset of dynamical instability and chaos in navigation satellite orbits

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.

Aaron Jay Rosengren (IFAC-CNR)


Orbital resonances are ubiquitous in the Solar System and are harbingers for the onset of dynamical instability and chaos. It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Here we will show that the same underlying physical mechanism, the overlapping of secular resonances, responsible for the eventual destabilization of Mercury and recently proposed to explain the orbital architecture of extrasolar planetary systems (Lithwick Y., Wu Y., 2014, PNAS; Batygin K., Morbidelli A., Holman M.J., 2015, ApJ) is at the heart of the orbital instabilities of seemingly more mundane celestial bodies—the Earth’s navigation satellites. We will demonstrate that the occurrence and nature of the secular resonances driving these dynamics depend chiefly on one aspect of the Moon’s perturbed motion, the regression of the line of nodes. This talk will present analytical models that accurately reflect the true nature of the resonant interactions, and will show how chaotic diffusion is mediated by the web-like structure of secular resonances. We will also present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance overlap criterion. The obtained results have remarkable practical applications for space debris mitigation and for satellite technology, and are both of essential dynamical and theoretical importance, with broad implications for planetary science.


  • Motivation: space debris problem
    • Active debris removal is becoming necessary
    • New: exploit resonant orbits to obtain relatively stable graveyards or highly unstable disposal orbits
  • Resonance overlap & chaos
    • asteroid belt resonances: cf. DeMeo & Carry 2014 (Nature Rev)
    • What is resonant structure of cislunar space?
      • actually less well known than resonant structure of asteroid belt
    • Cislunar resonant phenomena:
      • tesseral resonances
      • MMRs
      • lunisolar semi-secular resonances (sun-synchronous, evection resonance)
      • secular resonances (crit. inclination, Kxxxx resonance)
    • Navsat orbits (European) are unstable!
      • Chao 2000, Jenkin & Gick 2002, Chao & Gick 2004
      • Also: interference from sats in disposal orbits
    • Ref: Mercury’s orbit and secular chaos
  • Harmonic analysis of Lunar perturbations
    • Tesseral and lunisolar semi-secular resonances cannot be the cause of orbital instabilities observed in numerical surveys
    • Role of secular resonances in producing chaos
      • simplifications:
        • 2nd order in ratio of semimajor axes
        • short periodic terms of disturbing function can be averaged out
      • resonance: $\dot{\psi} = (2-2p) \dot{\omega} + m \dot{\Omega} \pm s\dot{\Omega}_2 \approx 0$
    • chaotic diffusion (~250 yr)
      • Daquin et al. CMDA (in prep)
      • Chirikov res overlap criterion
      • chaotic web
        • plot: $e$ vs $i$
      • FLI stability maps
        • heat map: $e$ vs $i$
        • too many dimensions $\rightarrow$ far from understood