# Can a milky way stars radial velocity be measured to within 1.5 m/s?

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I've read that using techniques like temperature controlled Iodine cells in spectrometers allows for +/- 1.5 m/s velocimeter resolution. My understanding is this is equivalent to observing a 0.00003 Angstrom shift at 6000A wavelength. Or in other words, measuring a spectrum shift 1/1000th of a pixel on a spectrometer CCD array. Apparently we can do even better by removing the atmosphere in spectrometers and making the whole system from non thermally expanding materials. Wow, this seems incredible!

My Question: Can we actually measure a stars speed to 1.5 m/s precision. Is there some weird solar effect like flares / rotation / expansion/contraction of atmosphere that would stop us from measuring this accurately?

To ask my question in a slightly different way: I've done a 3D simulation of an observatory on Earth observing Alpha Centauri redshift. Due to the spin and tilted orbit of the earth, we can see fluctuations in the redshift. I've drawn tiny error bars of +/-1.5 m/s (or +/- 0.00003 Angstrom shift) to show the size of the velocimeter measurement error. The error bars are absolutely tiny compared to the the effect produced by the spin of the Earth! Would we actually be able to reproduce the theoretical curve really accurately with an expensive velocimeter?

Image 1. Redshift curve

Image 2. Zoomed in Redshift curve - You can just make out the minuscule error bars.

Another way of asking this question is, consider an ideal scenario. No spin or orbit of the Earth and the star is moving radially away from us and also isn't spinning. Would a single velocimeter measurement tell us the star's speed to within about 1.5 m/s? Would something like rhythmic expansion and contraction of a stars atmosphere cause a false measurement?

Thanks for any help,

John

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I don't know the answer to your question but if I was trying to find out the limits of what's possible I would look up the details of the measurements made by the gaia satellite.  Certainly accuracy  will depend on how bright the star is etc.

The other area to explore would be the planet hunting spectrographs.

Regards Andrew

Edited by andrew s
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Some 25 years ago  I had the privelige of working for Birmingham University Physics and Astronomy dept under the late Professor George Isaak, who, for the record was one of nature's natural gentlemen. The BISON group did a lot of the early work looking at oscillations of sound waves on the sun, from which models of the solar interior could be derived. We made spectrometers that focussed on a single atomic absorption line ( usually Potassium at 770 nm) using potassium vapour as a local atomic reference. The idea is to isolate 2 wings (called red and blue) as areas in the absorption line either side of the minimum. the spectral line in a star is broadened due to thermal motion so the stellar absorption line has a very distinct width and shape. Your local reference sits at typically 100C and has a much smaller width, so can be used a a probe to look at the details of the wider stellar line.

If the relative velocity was zero, the red and blue wings would be of equal intensity being halfway up the line profile. If you define an intensity ratio of (R-B)/(R+B) this gives a function which is independent of light intensity and scales with the relative velocity. When the stellar line shifts due to the doppler effect, one wing will approach the line minimum while the other will will move up towards the continuum, thus the ratio changes.

How do you look at the 2 wings? Some elements have spectral lines that split into two components in the presence of a magnetic field....you put your potassium cell between the poles of a strong magnet and it is possible to look simultaneously at the red and blue wings using polarised light. It's called the Zeeman Effect.

If you Fast Fourier the data and average the results over weeks and months we could see relative motions of millimetres per second in the sun since the photon statistics are really good, and about a metre per second in 1st magnitude stellar velocities. To do this we cast our own mirrors about 1-2  metres in diameter from araldite spun in a dish using the centrifugal force to produce the parabola shape. Not good enough to image with, but plenty good enough to pass starlight down an optical fibre to a spectrometer.

Basically the more data the better...the precision you achieve generally goes with the square root of the number of observations.

Don't confuse absolute accuracy with the ability so see small changes...you are right in that getting an accurate answer to the true radial velocity means knowing a lot of other numbers to the same precision...the moon's effect, gravitational redshift in the case of the sun, the earth's orbital velocity and barycentric motion.

At the time the group had the world's most accurate radial velocity for Arcturus and one or two other stars.

The whole project was a triumph of cost-effective science. We built a world-wide network of remote observatories for less than a million quid....

Have a look....

I can recommend the following book, with which I have no financial interest (although author Prof Chaplin is an excellent chap..it was his araldite mirrors that did the stellar work!)

I have no idea what the current state of the art is.

Edited by rl
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Good post @rl I had forgotten about helio and stellar seismology.  I love the idea of spin cast araldite mirrors.

Regards Andrew

Edited by andrew s
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Omg, incredible story rl! Had no idea a single line could get that accuracy, I was considering spectrometer technology because I assumed you needed to average multiple lines to measure small shifts. That answers some questions I had. Also, wasn't sure if repeated long term measurements would be low enough noise for my purposes, sounds like it could actually work. I never considered that imperfect diy mirrors could be used for a spectrometer, that's good to know.

I'm all about absolute accuracy. Haven't factored in tidal bulges from the moon yet (+ Sun & Jupiter tides), hard to model exactly but should be easy to measure. Barycentric motion of earth was simple with de405 ephemerides from nasa. Wgs84 geoid and sidereal time for calculating our observing station motion as it rotates around earths axis. Arbitrary precision numerical vector calculations make the math for 3D relative motion redshifts straightforward, just tell the computer the star data, observer data, date/time and let the computer calculate and display all the predictions. This is getting exciting, might have an idea that could actually work out.

Edited by Physics_dude
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Basically it's all down to using a local atomic reference, which is not subject to drifts due to ageing or temperature..the atoms just work with absolute consistency for much the same reasons that the best clocks are atomic in principle. The basic principle of operation is pretty much immune to small slow temperature variations. A standard grating is subject to all sorts of environmental effects and such spectrometers are usually housed in air-conditioned rooms with exquisite temperature control, fixed down to vibration-damped benches. Yes, I believe you can get somewhere near the same sort of accuracy if you are careful enough....I believe the first exoplanets were found this way..but it is harder. The atomic based spectrometer is not perfect..it still has issues with the analogue electronics drifting with temperature  and optics shifting, but the level of effort required to fix the secondary drift problems is less. Sure, there are ways of messing it up if you build one yourself; there is a learning curve.  And you only get data on one line..it's a one trick pony. Horses for courses...

Our spectrometers were mounted on a standard equatorial mount open to the weather and still gave very stable results in spite of getting tilted as the mount moved. They worked even better given the same environment as a precision echelle.

You need to have a good look at the photon statistics..most of the starlight is thrown away using an atomic based spectrometer since youre only using the 2 wings of one line. Not a problem on the sun and 1st magnitude stars with a big mirror, but if you're looking at 12th magnitude stars with a 6" mirror you might need a very long integration time just to get a decent signal-to-noise ratio.

A grating makes more efficient use of the light available. And you have the option of calibrating with an atomic standard at regular intervals. The point I'm making is that no one technique is the best in all observing situations. Sorry if that's obvious..!

If you've got a genuine university project have a look at all the measurement technology out there today...my information is 25 years out of date. There is a lot to consider in order to get the most precise measurement given the constraints of observing time, development costs and operational convenience/ reliability.

Edited by rl
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Not currently associated with any university. Trying to learn the astronomy terminology, science, technology and current knowledge with papers, forums, calculations, physics theory etc.

Did some calculations on photon statistics, took a magnitude 1 star, 1 square metre collecting mirror, used a 2000x1 linear CCD detector with wavelength dependent efficiency and calculated the electron-hole generation rate per pixel to see how the signal compares to the noise:

Looks like we have a total photon rate of 20 billions photons / s hitting the detector, and about 14 billion electrons / s produced by the CCD detector.

For individual CCD pixels the electron rate is as low as ~4 million electrons per second at ~1000 nm wavelength. The read noise for a typical CCD is about ~10 electrons / read for each pixel and dark current is about 25 electrons / s per pixel.

Seems the Poisson photon distribution of the signal dominates the other errors at these relatively high photon counts. A 10 minute scan gives a signal to noise ratio of ~35000 on the worst pixel, compared to ~11000 for a 1 minute scan.

That's an impressive number of photons! Should definitely be possible to collect high signal, low noise data on the brighter stars. Sounds like the difficulties in a velocimeter measurements of bright stars are in stable, accurate measurements of the shifts, rather than the photon count.

CCD vs two photodiodes around an absorption line vs other technology is a good question. Every velocimeter I'm reading about seems to be using CCD's. I suppose an iodine cell in front of a CCD guarantees a consistent, real time calibrated spectra, something I need. InGaAs arrays look pretty interesting, really good spectral response at higher wavelengths.

Actually feeling more optimistic after seeing the high photon rates, signals are really high when measuring a bright star and a fair sized reflecting mirror. Looks like it would cost \$\$\$ for proper equipment.

My summary of velocimeter problems and solutions:

- Thermal effects on spectrometer optics mounts: I2 cell should deal with slow thermal variations, Invar/zerodur material could be used for optics mounts to minimize thermal expansion. Air conditioner to minimize background temp fluctuations.

- Thermal effects on CCD sensor: Could carefully control CCD temperature + use a CCD temp sensor and micro-controller for real time corrections to data

- Other drifting effects of electronics: utilize optical chopper with calibrated light source to ensure all pixels have up-to-date calibration data on response, dark current, noise etc.

- detector SNR: measure bright stars only and use a big mirror / lens area.

- Vibrations: Make spectrometer + scope heavy and vibration isolated from ground. Shield from wind (observatory).

- I2 cell aging / variations: Use really pure I2 gas, quality quartz container. Tightly control temperature and pressure of cell for stable calibration reference.

- Tracking (slight star tracking error could cause a false redshift/blueshift? I think?) Will need to investigate this problem, may be important? Might need to orientate CCD perpendicular to star motion? Might need a really accurate and smooth tracking system?

- 2 wings around a line problem: I think computational measurements of shifting spectra now allow for every peak and absorption line along a spectra to be measured statistically whether they are sharp or really broad? I don't think we have to throw most photons away anymore? I'm trying to find example software to understand this. Am I an idiot?

Any major stuff I've got wrong or missed?

Edited by Physics_dude
fixed my noise analysis math
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On 24/01/2021 at 03:50, Physics_dude said:

My Question: Can we actually measure a stars speed to 1.5 m/s precision. Is there some weird solar effect like flares / rotation / expansion/contraction of atmosphere that would stop us from measuring this accurately?

To ask my question in a slightly different way: I've done a 3D simulation of an observatory on Earth observing Alpha Centauri redshift. Due to the spin and tilted orbit of the earth, we can see fluctuations in the redshift. I've drawn tiny error bars of +/-1.5 m/s (or +/- 0.00003 Angstrom shift) to show the size of the velocimeter measurement error. The error bars are absolutely tiny compared to the the effect produced by the spin of the Earth! Would we actually be able to reproduce the theoretical curve really accurately with an expensive velocimeter?

The answer is yes it is done to this precision by professionals for measuring the wobble due to exoplanets. As you say you do have to make corrections for many factors and chose your star as the stability of  the star puts a lower limit on the measurement. (Fortunately In the search for planets capable of supporting life  the stars are likely to be stable, like our sun for life to evolve).

You might be interested in this measurement by Christian Buil which describes in detail how it can be done even by an amateur to a 1 sigma precision of 5m/s

Cheers

Robin

Edited by robin_astro
typo
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16 minutes ago, robin_astro said:

.....and chose your star as the stability of  the star puts a lower limit on the measurement.

As an example  here are some measurement I made of the variations in the radial velocity of red supergiant Deneb due to non radial pulsations which are several km/s and as far as is known are chaotic in nature, compared with the much more stable main sequence star Vega.

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[update]

Did more calculations on photon statistics, actually the photon count matters more than I thought! Even with a 1 square metre collecting mirror! A 14 minute scan produces a photon error equal to the photon change on a pixel due to the spectra shifting across a sharp iodine peak from a 1.5m/s velocity change. Since there are multiple iodine peaks to measure the shift from, I suppose a ~5 minute scan with my proposed setup would be sufficient to resolve 1.5 m/s changes with a reasonably small error from photon statistics.  Thanks rl for pointing this out! I definitely underestimated the photon statistical error.

Ah yes, Christian Buil has very nice stuff! I browsed that page days ago but skipped over the error analysis because I didn't understand it. Now things make more sense. His telescope is much smaller than in my calculations and it's a 5.5 magnitude star, that's about 1000x less photons than in my calculations. This'll be a fun exercise to use my new photon statistics math to calculate Christians experimental errors.

Oh nice graph robin_astro! Do you happen to have details on the measurements? Telescope size? Scan time? Pixel array? Wavelength calibration method? Trying to connect as much theory, data and experiment that I can. Learning so much stuff at the moment, thanks heaps guys! hmmm, "non radial pulsations", will have to look into that.

Edited by Physics_dude
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1 minute ago, Physics_dude said:

His telescope is much smaller than in my calculations and it's a 5.5 magnitude star, that's about 1000x less photons than in my calculations. This'll be a fun exercise to use my new photon statistics math to calculate Christians experimental errors.

For bright targets I suspect you will find the systematic errors are much larger than the photon statistics. In Christian's case he believes they are fibre noise and spectrograph thermal stability. In the professional case ISTR is the stability of the star which makes sub m/s precision difficult

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I think the current state of the art is the ESPRESSO spectrograph on the VLT which was specified for 10cm/s precision and recently got down to 30cm/s when measuring Proxima Centauri-b and were able to detect activity due to star spots

Robin

Edited by robin_astro
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Oh wow, I see how people are getting away with smaller scopes. I just learnt what an echelle spectrograph is!

What a clever idea! I no longer need to waste 99% of the pixels on the array. This technology can improve the spectroscopic and velocimeter resolution by quite a bit! So that's how Christian did so well with his tiny 0.28m diameter scope, very clever.

Edited by Physics_dude
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6 minutes ago, Physics_dude said:

I just learnt what an echelle spectrograph is

Yep it gives you a double hit. You use all the photons and cross correlation over a wide wavelength range gives you incredible precision (again helped by the choice of star, G/K/M dwarf stars have a lot of lines)

Edited by robin_astro
carification
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This is what I got looking at two lines with my now retired homemade temperature controlled echelle spectrograph.

Regards Andrew

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3 hours ago, Physics_dude said:

Oh nice graph robin_astro! Do you happen to have details on the measurements? Telescope size? Scan time? Pixel array? Wavelength calibration method?

This was with a LHIRES III telescope mounted slit spectrograph at ~0.4A resolution. It is a notoriously unstable instrument so I superimposed precise wavelength markers on the star spectrum with a calibration lamp. I talk about it briefly here at 18:36 min

Ultimately though slit spectrographs are not so good for precise  radial velocity measurement where you are trying to measure the centroids of lines accuracy because the shape of the line can change significantly depending on where exactly you place the star relative to the central line of the slit. Fibre fed spectrographs are better in this respect because the fibre scrambles the profile. (As well as the big advantage of mounting them off the telescope in a temperature controlled environment)  These (and the echelle advantages) are the key  things that allowed Christian to get about 2 orders of magnitude more precision than here. He is also an extremely skilled observer !

Cheers

Robin

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Fair enough, sounds like fibre is the way to go for high precision RV. My understanding of 3 main problems solved by fibre:

- non-fibre telescope mounted spectrometers can bend/move slightly when rotated during star tracking, giving false redshift/blueshift/other problems.

- non-fibre systems are more exposed to the elements, temp, vibrations, rain, wind etc that can give false redshifts.

- small deviations from perfect alignment of a star and telescope line of sight can cause false redshift/blueshift for high precision RV measurements in non fibre systems.

Fibre systems lose a lot of light on the air-fibre coupling, Christian's setup measured only ~5 photons / s per ccd pixel due to dim starlight, light losses and high ccd/nm density. It seems cross correlation computations can handle photon noise really well.

That espresso spectrometer looks nice. I'm trying to find out how to download the fits data used by that paper (EDIT: ok got hold of some different espresso data, this should be fun)

Edited by Physics_dude

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