absolute flux calibration

Basic concept

0) measure area of known aperture, at CNS
1) install source ~ Modulated narrowband at around 800 nm, LED or laser diode. Lack of coherence for LED is an advantage. Monitor wavefront brightness at input of aperture. 
     question- use a broadband source that has same shape as Vega in that spectral region?  
2) put known aperture in front of solar cell, behind collimator. Need to watch out for back-reflections off collimator. This puts parallel light into the aperture of telescope and onto solar cell.
     this establishes flux of N photons per sec through known area, so 
3) Move solar cell out of the way, and let collimated light impinge on telescope. This establishes narrowband sensitivity of telescope. 
4) Use slitless dispersed imaging mode- determine relative filter transmission with in-beam and out-of-beam difference. Use 4-inch CBP to get relative instrumental throughput. Use airmass dependence to regress to zero atmosphere. 
5) Peg absolute through
5) point telescope with known aperture at Vega/Sirius, and regress to top of atmosphere. This establishes received flux.  

Whole thing is basically a big beam expander. Imagine 1 mrad divergence from a collimated laser, 1mm beam. If we expand it to 4 inch diameter then angular demagnification is factor of 100, and divergence is 10 microradians or about 2 arcsec, good match to diffraction limit of 4 inch aperture. 

Diffraction-limited aperture would give a PSF that is independent of seeing. 

For 800 nm:

Best Thor labs AR coating at 800 nm:

So B type coating is 0.25% or 2 parts per thousand. Any double-bounce contamination would be at 5E-6 level. 

Thor labs does make 3 inch plano-convex lenses.

Calibration Logic

If we use a single-frequency laser diode at 808nm, we get monochromatic flux calibration at a single wavelength. That gives us photoelectons per photons per square cm per sec at calibration wavelength. 

Still need to figure out monochromatic to Spectral photon density connection

We can observe bright sources through 4 inch aperture, and then shift to fainter ones that we can also observe through full aperture.

collection area ratio is (100mm/1.2m)^2  which corresponds to magnitude difference of 2.5 log_10(0.1/1.2)^2 = 5.4 magnitudes. Exposure time ratio is 144. So 1 sec exposure for full aperture is two minutes of exposure on small aperture. 

Sirius spectrum from HST CALSPEC FLUX STANDARDS: SIRIUS (AND VEGA), R. C. Bohlin. Sirius A is V=-1.4, has a faint companion. 

So region at 808 nm is very clean

Sirius data from CalSpec: Sirius.csv

I have no idea why there is a glitch in the continuum value at 8200A. 

Sirius is at (J2000)

Atmosphere:

Collimator:

Bought 100mm dia -150mm focal length concave and convex lenses (uncoated) for zoom collimator.  For a pair of back-to-back lenses with equal and opposite focal lengths, effective focal length depends on their axial separation. We want an effective focal length of around 1 km or 1e5mm. That can barely be accomplished, with lenses touching each other. Edmund optics sells mounts that will hold these, Stock #15-869. 

Filter

These folks make LIDAR filters. 808nm x 10nm wide 100mm diameter, $1250: https://opticalfiltershop.com/product-category/specialty-filters/lidar/ 

wavelengths w 10nm width: 532, 660, 808, 830, 840, 850, 940nm. 

Sources, and surface brightness uniformity:

For a collimating lens and a disc source, surface brightness goes as cos(phi)^4. If we want part-per-thousand surface brightness uniformity, then Cos(phi)^4 > 0.999. That means cos(phi) > 0.9997, so half-angle subtended by collimator can't exceed 1.28 degrees. For sin(theta)=theta approximation, collimator subtends 2.5 degrees, that's f/23. So for a 6 inch collimator we need a focal length of 138 inches, or a focal length of 3.25 meters. 

So actually if we're 10 meters away, that suffices for surface brightness uniformity. 

Edmund sells a diode laser at 808 nm with 150 mW for $3000. Divergence is a few mrad. So at 1 km beam spot is a few meters. Fresnel length at 1 km is sqrt(800nm*1000m) = 28mm. A larger receive aperture will help suppress scintillation. 

From Edmund Optics site: 

Thor labs single frequency laser diodes, no driver included: https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=7573 

Fluxes:

A zeroth magnitude A0 star delivers around 10,000 photons/sec/nm/cm^2 at top of atmosphere, at 500 nm. If we have a 3 inch dia aperture, that's an area of 45 cm^2. If we have a dispersed system with, say, 1 nm per pixel then we expect about 450,000 photons per sec per pixel. Even at 25% overall efficiency, that works OK. Even a 2 inch diameter optic would work OK. Those filters are not hard to get. 790-810 nm has 95% atmos throughput. There are laser diodes and LEDs at 808. Vega has declination +38, Sirius has declination -16. 

Attenuator

We will need a precision attenuator. Options include pulse duty cycle, polarization, glass reflection, aperture. 

Same-path version:

Entire prism array moves vertically, changes from no attenuation to 4 bounces. Can put monitor diode in lower leg and measure 2-bounce attenuation. 

Quad-bounce off NBK-7 transmits 2.5E-6 but SF11 has a much higher index and quad bounce has more reflection (around twice a much) so attenuation would be 2.4E-5. That's a better choice. SF11 index at 080 nm is 1.7643

Another glass for anamorphic prism pairs is N-KZFS8  with index of 1.7059 at 808nm.

Beam pickoff plates from Thor Labs have one side AR coated, one not. That's useful for this application. https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=913

Anamorphic prisms are useful as well, we'd need 4 of them:

This last version has the advantage of wide dynamic range in attenuation. This can all sit in front of the beam expander, in collimated beam so spherical aberration is not an issue. 

We want to use absorptive filters, which are various kinds of Schott glass. Tech note on optical quality: tie-28_bubbles_and_inclusions_us.pdf

Presumably expanding the beam to full filter diameter would be good, to average over inclusions and bubbles. Schott NG9 filter glass spec sheet: SCHOTT-Neutral-density-NG9-2020-e.pdf

instrumentation notes

dispersion at input to span a few arcminutes from 800 nm to 1000 nm means an angular dispersion of 2 arcmin/200 nm = 0.01 arcmin/nm or 1.666E-4 deg/nm. In first order, with sin(q)=q approximation, lambda=d * theta, so d=lambda/theta. 
d is spacing between lines in meters. That's 1/d = 3E-6 rad/nm, or d=1e-9/3e-6 = 3.33e-4m = 0.33 mm, which corresponds to a line spacing of 3 lines/mm. We might as well use a prism? Here's a tool:

https://lightmachinery.com/optical-design-center/more-optical-design-tools/prism-designer/ 

LW-3-3050-UV from CVIlaser optics has 3 degree wedge. We can counter-rotate two of them to tune dispersion as an adjustable objective prism for 3 inch aperture. Made from fused silica. 

Using two of them lets us tune from 0 to 6 degrees of wedge angle. Use Thor labs SM3 slip ring and 3 inch lens mount to do the adjustment by hand. 

This will work best with alt-az mount telescope and rotator, otherwise atmospheric refraction rotates.

Alignment:

We need to have the aperture be aligned normal to the telescope boresight. So: 
- place aperture on front end
- point telescope at zenith
- make aperture horizontal

That will put aperture normal to pointing boresight. How well do we need to do? For projected aperture to be within 1e-4 of full area, we need cos(theta)>0.9999 so theta to within 0.8 degrees. Actually, now that I think about it as long as image of artificial source and celestial source land on same pixels even that doesn't matter. 

To Do (March 14 2022 update)

August 24 2024. 

Good prospect of an 8-inch refractor being available for this, in Colorado! It's f/12 so focal length is 2438mm. Plate scale for 10 micron pixels is 0.84 arcsec per 10 micron pixel. That's good. 

Plan would be retroreflector on adjacent peak. Telescope can point below the horizon out the slit. Simple objective lens means mirror won't fall out!

Put calibrated photodiode behind narrowband filter. That plus AM modulation should select out calibration light with low background. 

808 nm is a good place to operate. 

Downloaded the CalSpec model and STIS data for Alpha Lyra = Vega. 

SITS data have:

WAVELENGTH (A) ,FLUX,STATERROR,SYSERROR,FWHM,DATAQUAL,TOTEXP,Photons_per_nm_per_sec_per_m2

8081.912598, 1.0793e-09, 3.537e-13, 1.0793e-11, 9.769531, 1, 2.4, 43.91163361251095

calspec model has: (wavelength (A), flux, continuum)

8088.563417266212, 1.0856448e-09, 1.0856448e-09

Flux is F-lambda which is Watts per nm per sq meter. 

check on photon conversion: 

say energy flux density is 1E-9 W/nm/m^2. 

A photon at 800 nm has energy E=hc/lambda = 1.5 eV = 2.46E-19 Joules

One photon per second carries 2.46E-13 microWatts or 2.46E-19 Watts. 

Vega Calspec files, from https://archive.stsci.edu/hlsps/reference-atlases/cdbs/current_calspec/ 

alpha_lyr_stis_011.fits
alpha_lyr_mod_004.fits

converted to photons per nm per sq meter per sec:

CalSpecVegaModelPhotonSED.csv
STISVegaPhotonSED.csv

Jupyter notebook:
CalspecToCsv.ipynb

So we get around 44 photons per sec per sq cm per nm. 
In a 10 nm bandwidth and 50mm dia aperture, we would collect 44*10*20.26 = 8,900 photons per second. 

scintillation expectation: 

from 

Potential narrowband filters:

science motivation:

stellar atmospheres

axions

cosmology:
]  arXiv:2101.05897 [pdf, other]

Model-independent measurement of the Hubble Constant and the absolute magnitude of Type Ia Supernovae
Jian-Chen Zhang, Kang Jiao, Tong-Jie Zhang
Comments: 10 pages, 2 figures
Subjects: Cosmology and Nongalactic Astrophysics (astro-ph.CO)

In this work, we propose a cosmological model-independent and non-local method to constrain the Hubble Constant H0. Inspired by the quasi cosmological model-independent and H0-free properties of the `shifted' Hubble diagram of HII galaxies (HIIGx) defined by Wei et al. (2016), we joint analyze it with the parametric type Ia supernova (SN Ia) Hubble diagram (e.g. the joint-lightcurves-analysis sample, JLA) and get a Bayesian Inference of Hubble constant, H0=71±20 km s−1 Mpc−1. Although with large uncertainty, we find that H0 is only strongly degenerate with the B-band absolute magnitude (MB) of SN Ia but almost independent on other nuisance parameters. Therefore the accuracy can be simultaneously improved by a tight constraint of MB through a cosmological and H0 independent way. This method can be extended further to get more-literally non-local results of H0 by using other Hubble diagrams at higher redshifts.

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Energy balance in stars is a probe of fundamental physics, depending on stellar atmosphere models. 

Apparent Luminosity scales as Mbol/R2. dL/L = 2 dR/R. 

Absolute flux scale is a systematic error in this. For a population of N well-understood stars at distances known to sigma_i, apparent flux scales uncertainty goes as <sigma_i>/sqrt(N). 

Use as an example the volume-limited WD sample provided by The Gaia DR2 halo white dwarf population: the luminosity function, mass distribution and its star formation history

Santiago Torres1,2, et al

Gaia distances via parallax are key here. 

The single DA white dwarf star in their sample has details at 

http://vizier.u-strasbg.fr/viz-bin/VizieR-S?Gaia%20DR2%205142197118950177280 

Parallax to this object is 12.95 mas, for a (crude) distance of 77 pc. The parallax uncertainty is given as 0.15 mas. That gives a one sigma distance range of 78.12 to 76.34, or a fractional uncertainly in distance of 1.2%. That corresponds to fractional flux uncertainty of 2.4%. So after ten stars we become flux-accuracy systematics dominated. 

So from that perspective absolute flux calibration is useful. 

And in fact there is a range of luminosities where the emission is dominated by axions:

Mass-radius relation means we could use surface gravity to determine radius. We could look for consistence between distance, temperature, luminosity, and radius. 

The limitation is going to be radius determination, which has to come from log-g.

If axion emission provides an additional energy emission path, then would the star be cooler than expected? Not clear how it would affect the photosphere. 

THis star has asterseismology with period shift that constrains axion mass:
http://vizier.u-strasbg.fr/viz-bin/VizieR-S?Gaia%20DR2%20798566915774818176 

Parallax DR3 to G117-B15A is 17.386 mas with uncertainty of (!) 0.08 so distance is known to half a percent! Gmag is 15th. 

The most precise stellar clock is R548=ZZ Ceti. It’s 14th magnitude. 

http://vizier.u-strasbg.fr/viz-bin/VizieR-S?Gaia%20DR2%202457759374023232768 

Parallax is 30.52 mas and sigma_parallax is 0.033 so distance is known to one ppt!!

This paper: An independent limit on the axion mass from the

variable white dwarf star R548

Claims to know luminosity to 6% at 1 sigma, radius to 3.5% and that seems to dominate luminosity uncertainty. 

References:

Axion limits from WD luminosity fcn:

https://iopscience.iop.org/article/10.1088/1475-7516/2014/10/069/meta 

Parallax to Sirius (a binary) is 379 mas, distance is known to way better than 1%. Sirius is a fundamental star with Teff and log-g known, also its radius is known from interferometers. 

axion vs. photon emission vs. luminosity. 

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absolute calibration references