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This answer to Why does the Gaia space telescope have two main mirrors says:

According to the GAIA FAQs which does an excellent job: http://www.cosmos.esa.int/web/gaia/faqs:

Why is there an angle of 106.5 degrees between Gaia's 2 telescopes?

The choice of the so-called basic angle of GAIA was a non-trivial one. On the one hand, it should be of order 90 degrees to allow simultaneous measurements of stars separated by large angles on the sky. On the other hand, it should not be a harmonic ratio of a 360-degree circle (e.g., 60 deg, 90 deg, or 120 deg). Taking these considerations into account, acceptable ranges for the basic angle are 96.8 +/- 0.1 deg, 99.4 +/- 0.1 deg, 100.5 +/- 0.1 deg, 105.3 +/- 0.1 deg, 106.5 +/- 0.1 deg, 109.3 +/- 0.1 deg, 109.9 +/- 0.1 deg, etc. Accommodation aspects identified during industrial studies subsequently favoured 106.5 deg as the value finally adopted for Gaia.

I don't understand any of that.

Questions:

  1. "On the one hand, it should be of order 90 degrees to allow simultaneous measurements of stars separated by large angles on the sky." Well any large angle is a large angle, why should it be 90?
  2. "should not be a harmonic ratio of a 360-degree circle" I can imagine for instrumental reasons, but why are "acceptable ranges for the basic angle are 96.8 +/- 0.1 deg, 99.4 +/- 0.1 deg, 100.5 +/- 0.1 deg, 105.3 +/- 0.1 deg, 106.5 +/- 0.1 deg, 109.3 +/- 0.1 deg, 109.9 +/- 0.1 deg, etc." and what does "etc." mean here? Is there a pattern I'm not seeing?

Any angle with an uncertainty of +/- 0.1 deg can be reached with some rational number with a denominator of 1800 or less, that's as far as I've gotten.

uhoh
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    The down vote is for the question being too challenging? Too mathematical? Not every question can be "How many stars are there?" – uhoh Apr 04 '21 at 21:41
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    My answer on April 1st would have been: "The bond angle of liquid water is 106.1 ± 1.8°, so using an angle inside the errorbars of this for GAIA sounded cool." That's obviously not the answer. I also do not see any pattern (yet) in the series of possible angle choice. Could it have something to do with the position of the maxima of spherical harmonics? Something like "we want want to compress the data of our measured view as spherical harmonics with highest possible resolution"? – B--rian Apr 06 '21 at 14:39
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    @B--rian speaking of the H2O bond angle: How to find the angle between Mickey Mouse's ears as specified in an official trademark? – uhoh Apr 06 '21 at 15:39
  • There are 106 CCD's on Gaia and the telescopes have a 106.5 degrees angle... Is that co-incidental? If you divide 360 by 106.5, the result has 7 decimal places, the same is true for most numbers... 360 doesn't really exist on GAIA's rotation because it drifts by 1' at the end, so what kind of harmonic resonance of 360 degrees is ESA referring to? 1 degrees is alot, after two turns, the field of view has changed completely. – bandybabboon Apr 08 '21 at 13:41
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    @aliential the concern about harmonics and systematic errors may be related to heating from the sun causing mechanical deformation in the "optical bench". Since the rotational axis is inclined there is a constantly rotating thermal gradient across the huge structure, and since it has quite a non-uniform distribution of objects and reflectivities, this could result in higher harmonics in the thermal expansion/contraction and torques within the structure. It's quite an engineering wonder, there's a bit about that in this answer. – uhoh Apr 08 '21 at 15:14
  • @aliential had you already considered all of that when you wrote "The second statement is wrong:"? – uhoh Apr 08 '21 at 15:15
  • @uhoh thermal expansion and harmonic resonance are typically not associated very often, and when they are related, the wording should be quite specific in notions of dilation coeficients and degrees kelvin... I couldn't find any google scholar studies of "harmonic resonance & thermal expansion" ... The context is stated as "harmonic ratio of a 360-degree circle", That's typically from rotation ... The silicon carbide opitcal bench is a quasi-octagonal torus. Is there is CO2 in the torus to aid thermal management? How many kelvin and microns of deviations can happent to the rig? – bandybabboon Apr 08 '21 at 18:44
  • @aliential I'm asking the question here, you've chosen to answer :-) – uhoh Apr 08 '21 at 21:49
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    @aliential From The Gaia mission: “4.2.3. Periodic basic angle variations... These fluctuations are two orders of magnitude larger than expected based on pre-launch calculations and show a strong asymmetry between both telescopes; the line of sight of the preceding telescope fluctuates within a ±1000 µas range, while the other line-of-sight range is only ±200 µas. These fluctuations show a modulation with the 6-h spacecraft rotation” The effect is huge! It's likely that harmonics of this are what the weird basic angle was chosen to avoid. – uhoh Apr 08 '21 at 22:38
  • @aliential see also GAIA payload module mechanical development and The HSOB GAIA: a cryogenic high stability cesic optical bench for missions requiring sub-nanometric optical stability – uhoh Apr 08 '21 at 22:39
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    @uhoh, thanks, interesting study. 4.2.3, That's a third of a millionth of a degree, if the focal length is 35m, (219m circumference) a degree is 610mm, and the focal error is perhaps 0.0002mm. It's surprisng that ESA measured the rig using lasers prior to takeoff, and the field warp is 100 times greater than expected. It's unreasonable to compare the harmonic ratio of a circle (120,90,60) and the straight edge rig? i.e. physical modes of a drum https://www.researchgate.net/figure/The-ratio-fr-between-the-frequency-of-the-corresponding-overtone-and-that-of-the_fig10_7974151 – bandybabboon Apr 09 '21 at 08:37
  • Also, the main warp happens every 6 hours, that's like banging a 3 meter metal gong every 6 hours and worrying it will warp like the tacoma narrows bridge? I am way off track! So rather than thermal based harmonic ratio, perhaps for vibrations during takeoff? that could indeed cause a build-up of waves to damage the ceramic. The angles would be very specific to the quasi octagon torus and not at all 120/90/60. I think the ESA editorial perhaps over-summarized a scientists quote there? ESA is multilingual, si perhaps the translator became confused. – bandybabboon Apr 09 '21 at 08:52
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    From what I can tell, it has basically nothing to do with details of the instrumentation. It's more to do with the mathematics of trying to convert an integer number of scan observations into (preliminary) coordinates along a great circle on the celestial sphere (the term of art seems to be "great circle reductions"). They had exactly the same issue with the Hipparcos satellite even though the satellite and the detectors were very different (the chosen angle in that case was 58 degrees, so there's evidently some dependence on details of the spacecraft, scan mode, etc.). – Peter Erwin Apr 10 '21 at 00:24
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    @PeterErwin On the 'great circle reduction' in the data analysis for the astronomic satellite Hipparcos Searching for "torques" takes us to Table 6.2 which mentions up to 100 harmonics for solar torques on the spacecraft attitude, then lengthy discussions after that. I'll read further but if you had a strong $n$th harmonic in solar torque there may be some basic angles 360/$m$ that could be impacted? – uhoh Apr 10 '21 at 00:40
  • @PeterErwin I can't figure out what the relationship between $n$ and $m$ could be yet, my coffee is just kicking in and this seems hard :-) Thank you for the tip, searching "Hipparcos great circle reductions" brings up quite a lot! – uhoh Apr 10 '21 at 00:47
  • @uhoh No, I don't think it has anything to do with torques. It's a mathematical aspect of the data reduction. – Peter Erwin Apr 10 '21 at 02:19
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    I'll see if I can come up with some kind of hand-waving summary, even though it won't be a proper answer because I don't understand all the details. – Peter Erwin Apr 10 '21 at 02:27

2 Answers2

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I did not find anything conclusive on the web, and I am yet waiting to receive feedback from ESA's contact page. In a German speaking IT forum there is something a bit less vague than the FAQ cited in the question:

The reason for the 106.5 degrees is not easy to see. It has to do with the fact that we are (also) doing an astrometric solution on a great circle and for this all angles 360x(n/m) [n, m = small integers] have to be avoided because otherwise one would have a congruence state that would hinder the solution. Values other than 106.5 degrees would also be possible. 90 degrees would be ideal for parallax determination, but unfortunately 90=360x(1/4).

This said, why not simply choosing 91° as $\gcd(360,91)=1$ (with $\gcd$ being the greatest common divisor)?

Maybe it is really only a numerology of the engineers, just like the Mars parachute Morse message. An argument for that would be that Gaia also has also exactly 106 CCD sensors. Or it is simply a logistical issue on how to mount things on the spacecraft. I am really curious what the official ESA statement will be.

B--rian
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    Okay so 106.5 is 360 times (71/240), so perhaps 71 is considered a "not small" integer. How about the others? This is a question about rational numbers and the largeness of integers, so an answer will be based on rational numbers and the largeness of integers as well. But you also want to be comfortably far away from a small integer rational number, the smaller it is, the farther you want to be away from it. 90 and 120 degrees are 1/4 and 1/3 and systematic errors those low harmonics may be strong. Not by coincidence our magic angle is almost equally far away from both of them. – uhoh Apr 07 '21 at 18:26
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    @uhoh I guess you are asking about the other possible values in your question, right? – B--rian Apr 07 '21 at 18:32
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    "but why are 'acceptable ranges for the basic angle are 96.8 +/- 0.1 deg, 99.4 +/- 0.1 deg, 100.5 +/- 0.1 deg, 105.3 +/- 0.1 deg, 106.5 +/- 0.1 deg, 109.3 +/- 0.1 deg, 109.9 +/- 0.1 deg, etc.'" – uhoh Apr 07 '21 at 18:39
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    The 0.1 deg error bars irritate me - it could be mechanical limitations which limit us to 0.1 deg steps. I am so curious what ESA will answer, if they answer. – B--rian Apr 07 '21 at 18:52
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    The focal plane detector is huge, of order 0.5 meters wide a focal length of 35 m the samples both star fields (of the two primaries) over a rotation of about .8 degrees, so not sure what the 0.1 degrees is from. Unrelated but just asked in Space SE: What causes GAIA rotational axis to precess the way it does? How exactly is this accomplished? – uhoh Apr 07 '21 at 21:25
  • I considered floating point accuracy on the processor with millions of images but on 32-bit from my experience it would be trivial. I think that at 106 is a pretty number: 1 symbolises a rocket taking off vertically, 0 symbolises a map of the sphere transected obliquely by the bars of the Milky Way, and 6, we know what 6 is! Based on some years of physical modelling synthesis, the harmonic ratio story is trivial for the given physical model. It's not a mission Critical page so I think it's just been over summarised misleadingly, it's a public relations document. – bandybabboon Apr 10 '21 at 00:38
  • Isn't ~70 about the minimum number of times Gaia is/was expected to image a region of space? Maybe it was a choice based on the number of repeat observations expected? – Rob Apr 11 '21 at 08:39
  • No answer from ESA by the way :-/ – B--rian Apr 20 '21 at 09:40
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EDIT- The Frame of GAIA is not symmetrical so it's not related to wave resonance. It can be related to the 360' rotations of the cameras:

1/ 90 degrees gives the most precise angle for triangulations of normal vectors and vector magnitude on a cartesian graph. You can try it out by programming 3D mesh, else by glancing sideways travelling on a bike: glancing at 90'C, you have the highest measurement of the movement of X relative to Y compared to looking at 45'. If you have binoculars on the bike helmet (i.e. a narrow field of vision) you will find that 90 degrees views gives the best relative difference.

2/ The second statement is incomplete, missing context: On the other hand, it should not be a harmonic ratio of a 360-degree circle (e.g., 60 deg, 90 deg, or 120 deg). Taking these considerations into account, acceptable ranges for the basic angle are 96.8 +/- 0.1 deg, 99.4 +/- 0.1 deg, 100.5 +/- 0.1 deg, 105.3 +/- 0.1 deg, 106.5 +/- 0.1 deg, 109.3 +/- 0.1 deg, 109.9 +/- 0.1 deg, etc. . The suggested values don't impart notable harmonic ratios in the same way that phi rotations give the best coverage of light for a plant:

  • The CCD time is set to 4.4 seconds and can be changed to 4.47, rendering angular resonance NEARLY TRIVIAL.
  • The rotation time is 1 degrees every minute, one rotation every 6 hours (4009 photographs at 4.4 seconds apiece), which can be changed to i.e. 6:20.
  • GAIA has already rotated sideways by ~1 degrees every time the cameras have scanned round once, and the aperture of the CCD is 1.7 x 0.6 degrees, The resonance shifts mostly depend on those figures.
  • The sweep has to also keep the solar panels towards the sun and the sun away from the CCDs and to provide a very constant thermal environment. some very decisive constraints compared to rotation resonance.
  • The satellite frame that holds the equipment isn't regular or symmetrical, so any vibrations and movements that travel through the frame will have irregular vibration modes at angles other than 90'C

Here's a video of the complexity of the supposed "360 degrees".

https://www.gaia.ac.uk/science/parallax/scan

bandybabboon
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    I'm not so confident that all this "ESA is wrong" is really true. In Stack Exchange it's important to support assertions in answers using authoritative sources or at least going through the argument and the math in some details. Comparing GAIA to a plant isn't helpful, and writing "NEARLY TRIVIAL" and "IS A WRONG STATEMENT" in ALLCAPS is a little dramatic especially for an answer so thin on substance. This currently reads like an opinion and a little bit like a rant, can you elaborate on these points? Thanks! – uhoh Apr 08 '21 at 12:02
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    This isn't correct. There certainly are small ranges of angles which are possible and have seen a plot showing the expected errors vs basic angle that shows the peaks and troughs and that the troughs occupy narrow ranges of angles. – ProfRob Apr 08 '21 at 12:05
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    I don't think the CCD exposure time has anything to do with it. The important factors are the time interval between when a star appears on the detector and the length of time it is on the detector. – ProfRob Apr 08 '21 at 12:08
  • The satellite rotates at most by 0.06 degrees for every image capture (4.4 seconds). Do you mean that it's important for the 106.5 angle or for the exposure time? What kind of resonance were you referring to? vibrations in the steel? I'm interpreting the suggested resonance issue as timing the camera wrong relative to position so as to miss some views. – bandybabboon Apr 08 '21 at 13:06
  • @uhoh Sorry i had edited the ALLCAPS to bold and forgot to press enter. I program 3D models of platonic solids a lot using quaternions and Euclidian space, and digital signal processing of sine waves with specific resonances, so I am interpreting angles and degrees from that training: I didn't compare GAIA to a plant i compared the 106'5 degrees "harmonic ratio" to the harmonic ratio of Phi, a mathematical constant used by plants. My point is: harmonic resonance is a function of time. it's tweakable using time differences. – bandybabboon Apr 08 '21 at 13:24
  • From The Gaia mission: “4.2.3. Periodic basic angle variations... These fluctuations are two orders of magnitude larger than expected based on pre-launch calculations and show a strong asymmetry between both telescopes; the line of sight of the preceding telescope fluctuates within a ±1000 µas range, while the other line-of-sight range is only ±200 µas. These fluctuations show a modulation with the 6-h spacecraft rotation” The effect is huge! It's likely that harmonics of this are what the weird basic angle was chosen to avoid. – uhoh Apr 08 '21 at 22:35
  • When you use the ' symbol do you mean degrees or minutes of an angle? – Fred Mar 07 '22 at 14:36
  • Degrees because the question is on the degrees metric. – bandybabboon Mar 07 '22 at 14:45