Understanding Atmospheric Differential Refraction and Spectrometer Slit Rotation

1. Atmospheric Differential Refraction and Spectrometer Slit Rotation S. Donnell, Kiowa Creek Observatory AAVSO, Colorado Springs Astronomical Society donnellsp@gmail.com

2. Atmospheric Differential Refraction • Atmospheric differential refraction is the wavelength-dependent refraction as light passes through the atmosphere. • Blue is refracted more than red. • Refraction is perpendicular to the horizon – an image of a star is actually a small spectrum in the vertical direction – blue on top and red on the bottom.

3. • Atmospheric differential refraction, or dispersion, varies with zenith angle and wavelength. Zenith Angle (deg) 0.0 29.6 48.2 60.0 66.4 70.5 73.4 75.5 77.2 78.2 Sec Z 1.00 1.15 1.50 2.00 2.50 3.00 3.50 4.00 4.50 4.90 • Dispersion is zero at the zenith and increases with increasing zenith angle. Sec(z) >>1 Sec(z) > 1 Sec z ~ air mass • The difference in dispersion in blue and red also increases with increasing zenith angle.

4. Effect of Slit Orientation • The amount of dispersion is comparable to the width of a slit in a typical spectrometer • A slit oriented perpendicular to the horizon will pass most or all of the light across the spectrum. • A slit oriented horizontal to the horizon will have a decrease in intensity in the blue and red ends. • Mostly affects slit spectrometers with broad spectral ranges and narrow slits.

5. • Dispersion results in a decrease in intensity of light passing through the slit on the blue and red ends of the spectrum. Sec Z = 1.0 Sec Z = 1.15 Sec Z = 1.50 Sec Z = 2.0 Sec Z = 2.5 Sec Z = 3.0 Sec Z = 3.5 Sec Z = 4.0 1 -9% at sec z= 2.0 0.9 -18% at sec z = 1.5 0.8 0.7 -33% at sec z = 4.9 • This reduction in intensity increases with increasing zenith angle. Fraction of INtensity Through SLit -38% at sec z = 2.0 0.6 0.5 Zenith Angle (deg) 0.0 29.6 48.2 60.0 66.4 70.5 73.4 75.5 77.2 78.2 Sec Z 1.00 1.15 1.50 2.00 2.50 3.00 3.50 4.00 4.50 4.90 0.4 • Reduction in intensity also means a decrease in signal to noise. 0.3 Slit Parallel to Horizon 0.2 Sec z ~ air mass -90% at sec z = 4.9 0.1 • Relative and absolute flux calibration is compromised. 0 350 400 450 500 550 600 650 700 750 800 Wavelength (nm)

6. Effect on Calibration • Relative Flux Calibration • Also known as atmosphere/instrument response correction. • No effect if both the target spectrum and reference star spectrum are obtained with the slit perpendicular to the horizon for both objects. • If the reference star spectrum was obtained at one slit angle and the target star another, then there will be a difference in the blue/red response that will remain after the atmosphere/instrument response correction has been applied to the target star. • This will generally be the case for any fixed slit orientation. • Absolute Flux Calibration • Same issue as with relative flux calibration.

7. Setting the Slit Parallel to RA/Hour Circles • Typical fixed slit orientation. • Slit is oriented perpendicular to the horizon when facing south. • Slit remains fixed thereafter - parallel to the hour circles and in the direction of the pole. Zenith Parallactic Angle Horizon

8. Parallactic Angle by Declination and Hour Angle Latitude 40 deg N Hour Angle Dec (deg) 0 1 2 3 4 5 6 7 8 9 10 11 12 0.0 0.0 0.0 0.0 0.0 85 80 70 60 50 40 30 20 10 0 -10 -20 -30 -40 180.0 180.0 180.0 180.0 180.0 Zenith 163.8 162.2 157.4 148.2 127.8 85.2 46.7 29.5 21.5 17.1 14.6 13.0 12.1 11.6 147.7 144.7 136.5 123.5 103.9 80.2 59.6 45.4 36.5 30.8 27.1 24.7 23.3 22.6 131.8 127.9 118.1 105.3 90.2 75.1 62.2 52.3 45.1 40.1 36.7 34.5 33.2 32.8 116.1 111.8 101.9 90.9 79.8 69.6 61.2 54.5 49.5 45.9 43.5 42.1 41.6 100.8 96.4 87.4 78.6 70.6 63.7 58.3 54.1 51.0 49.0 47.9 85.8 81.7 74.0 67.2 61.7 57.3 54.0 51.7 50.4 50.0 71.1 67.5 61.2 56.3 52.6 50.0 48.4 47.8 56.6 53.6 48.8 45.4 43.2 41.9 42.3 40.0 36.6 34.4 33.2 28.1 26.6 24.4 23.1 22.6 14.0 13.3 12.2 11.6 11.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

9. • Flux loss at 400 nm (relative to 550 nm) for a slit parallel to RA/hour angle circles • Zero flux loss at zenith since dispersion is zero. • Larger hour angles and lower declinations contribute to increasing air mass with a resulting flux loss

10. • Flux loss at 750 nm (relative to 550 nm) for a slit parallel to RA/hour angle circles • Losses are less than on the blue end, but still a concern for larger hour angles and lower declinations

11. The Miracle of Slit Rotation • By rotating the slit for both target and reference stars to be perpendicular to the horizon you: 1. Minimize or eliminate entirely the loss of flux and signal to noise at the blue and red ends of the spectrum. 2. Allow for the construction of a better atmosphere/instrument response curve by ensuring that the slit orientation for both target and reference is at the same parallactic angle. 3. Generate a more accurate absolute flux calibration. Zenith Horizon

12. Rotating the Slit • Ideally, the slit should be rotated continuously throughout the exposure - generally not practical for amateur setups. • A compromise solution is to set the slit orientation to be perpendicular to the horizon at the exposure mid time, minimizing slit loss. • A manual rotator between the spectrometer and scope allows for rotation of the spectrometer while keeping it firmly attached to the scope. • About $55 from Agena Astro. Blue Fireball Camera Angle Adjustor / Rotator

13. Orienting the Slit to Zero Parallactic Angle Bubble Level • Attach a bubble level to your spectrometer. • Easy to determine when your slit is perpendicular to the horizon. • No need to calculate and dial-in parallactic angles.

14. Method for Setting the Slit Orientation • Goal is for the slit to perpendicular to the horizon at the exposure mid- time. • Prior to the target or reference star exposure start time, move the scope to the west and rotate the spectrometer so that the slit is perpendicular to the horizon (i.e, the bubble is centered in the level). • Move the scope back to the target/reference star and wait until the indicated exposure start time. • If all goes well, the slit will be in the correct orientation perpendicular to the horizon at the exposure mid time. Offset = Delta Time * Earth rotation rate Example: Delta Time = 21:17 – 21:02 = 15 min Earth rotation rate ~ 0.25 deg/min Offset = 15 * 0.25 = 3.75 deg = 3 deg 45 min

15. Additional Comments • Applies to broad spectrum spectrometers like the Alpy 600 • Slitless setups need not worry. • High resolution spectrometers mostly unaffected due to their narrow spectral range. • A properly rotated slit will still result in some reduction in intensity at the red and blue wavelengths over the exposure time. • Especially true for long exposures. • Minimize by setting slit angle for mid-exposure time • Difference in exposure times between target and reference will introduce some difference in the blue/red flux even if the slit is oriented properly. • For very long exposures, setting the slit parallel to the horizon may be a better option (Szokoly, 2005)

16. Slit Loss Calculator Slit Loss Calculator Version 1.0, Jan 2024 Slit Orientation Slit Parallel to RA/Hour Circles Object Parameters Object ID J2000 RA J2000 Dec Observing Location Longitude Latitude Alrescha 02:02:03 hh:mm:ss 02:45:50 +/-dd:mm:ss 255.0000 degE 40.0000 deg Spectrometer Configuration Scope Focal Length Slit Width Slit Length Exposure Parameters Date Exposure Mid Time Exposure Duration 1768 mm 25 microns 250.0 microns 1/1/2024 UTC 0:00 UTC 270 sec Wavelength Range Blue Wavelength Red Wavelength Seeing and Atmosphere Conditions Seeing Atmospheric Pressure Temperature Water Vapor Pressure 400 nm 700 nm 2.5 arcsec 600 mm Hg 7 C 8 mm Hg Input the conditions specific to your equipment and target star to determine the slit loss for any of three slit orientations: 1. Perpendicular to the horizon (ideal), 2. Parallel to RA/hour circles, and 3. Parallel to the celestial equator.

17. Slit Loss Calculator Results Object ID RA Dec UTC Date Reference Wavelength Slit Orientation Alrescha 02:02:03 hh:mm:ss 02:45:50 +/-dd:mm:ss 1/1/2024 550 nm Slit Parallel to RA/Hour Circles Start Mid End Time Azimuth Elevation (unrefracted) Zenith Angle Local Mean Sidereal Time Hour Angle Parallactic Angle Flux Loss at 400 nm Flux Loss at 700 nm 23:57:45 129.49 40.59 49.41 23.64 -2.39 -36.29 -6.34% -1.11% 00:00:00 130.09 40.92 49.08 23.68 -2.36 -35.92 -6.09% -1.06% 00:02:15 UTC 130.70 deg 41.25 deg 48.75 deg 23.71 hrs -2.32 hrs -35.55 deg -5.85% -1.02%

18. Example: Target and Reference Star with Slit Parallel to RA/Hour Circles

19. Results From Slit Loss Calculator for Target and Reference Stars Object ID RA Dec UTC Date Reference Wavelength Slit Orientation 13-Nu Aqr 21:09:35 hh:mm:ss -11:22:18 +/-dd:mm:ss 11/5/2023 550 nm Slit Perpendicular to CE Object ID RA Dec UTC Date Reference Wavelength Slit Orientation HD 174567 18:49:44 hh:mm:ss 31:37:45 +/-dd:mm:ss 11/5/2023 550 nm Slit Perpendicular to CE Start Mid End Start Mid End Time Azimuth Elevation (unrefracted) Zenith Angle Local Mean Sidereal Time Hour Angle Parallactic Angle Intensity Loss at 400 nm Intenisty Loss at 700 nm 03:20:45 217.06 30.72 59.28 23.29 2.13 28.09 -8.27% -1.46% 03:23:00 217.62 30.46 59.54 23.32 2.16 28.49 -8.65% -1.53% 03:25:15 UTC 218.19 deg 30.19 deg 59.81 deg 23.36 hrs 2.20 hrs 28.89 deg -9.04% -1.60% Time Azimuth Elevation (unrefracted) Zenith Angle Local Mean Sidereal Time Hour Angle Parallactic Angle Intensity Loss at 400 nm Intenisty Loss at 700 nm 03:27:30 284.27 35.14 54.86 23.40 4.57 60.68 -19.14% -3.53% 03:40:00 285.87 32.82 57.18 23.61 4.78 59.93 -22.09% -4.13% 03:52:30 UTC 287.46 deg 30.52 deg 59.48 deg 23.82 hrs 4.99 hrs 59.12 deg -25.52% -4.84% Both stars show losses at 400 nm and 700 nm Difference in slit loss is ~13% for 400 nm and ~2.6% for 700 nm

20. Effect on Spectral Profile -13% Black – Spectrum Calibrated with No Slit Losses Red – Spectrum Calibrated with Slit Losses

21. Summary • Wavelength-dependent atmospheric refraction (dispersion) causes starlight to be differentially refracted in a direction perpendicular to the horizon. • A slit oriented in any direction other than the perpendicular results in a significant loss of intensity at the blue and red ends of the spectrum. • This is a problem primarily for spectrometers like the Alpy 600 that produce a spectrum across the visual range. • This loss of intensity adversely affects the quality of the atmosphere/instrument response correction as well as absolute flux calibration • This problem is easily mitigated by rotating the slit to the optimal angle for both the target and reference stars.

22. References • Alexei Filippenko (1982), The Importance of Atmospheric Differential Refraction in Spectrophotometry, https://articles.adsabs.harvard.edu/pdf/1982PASP...94..715F • Ref: Szokoly, 2005, Optimal slit orientation for long multi-object spectroscopic exposures • https://arxiv.org/abs/astro-ph/0506688

23. Questions?

24. Parallactic Angle • The parallactic angle is the angle between the great circle through a celestial object and the zenith, and the hour circle of the object.

25. Zenith Zenith Parallactic Angle Horizon Horizon