An improved method for correcting Atmospheric Extinction Murray Forbes

An improved method for correcting Atmospheric Extinction Murray Forbes
VSS4, March 2016

Outline of talk The standard method The improved Vilnius method

The standard method Most of the plots I present will use results from the Vilnius filter system, with observations made some time ago when I was doing my PhD at VUW.

The standard method

The standard method This is the extinction for this star! 𝑚 0 =𝑚+𝑘∗𝑋

The standard method Choosing an extinction star:
Must not vary in brightness Are not in Variable star catalogues Standard star catalogues (e.g. the E-regions) are a good source Use Simbad to check the stars are not variable (e.g. not in any variable star catalogues).

Must not vary in brightness Are isolated in the sky from other stars Are not visual doubles (check double star catalogues) Check star position catalogues, photographs, your own CCD images Nearby stars (that may or may not be included in your measurement aperture or psf fit depending on seeing conditions) will cause problems so exclude stars with nearby stars (e.g. within sky background psf or aperture annulus) and are brighter than your instrument’s sky or dark current background level.

Must not vary in brightness Are isolated in the sky from other stars Is not a spectroscopic double star As would give the wrong secondary (colour) extinction coefficient and may turn out to be an eclipsing binary

Must not vary in brightness Are isolated in the sky from other stars Is not a spectroscopic double star Is a main sequence star (luminosity class V) Avoiding the instability strip (spectral classes A-F) Most main sequence stars in the instability strip are actually stable.

Must not vary in brightness Are isolated in the sky from other stars Is not a spectroscopic double star Is a main sequence star (luminosity class V) Often advised to pick extinction stars with similar brightness to the target star (to reduce effect of non-linearity in your instrument system), BUT If this means picking a faint star, have signal-noise problems And may be observing several target stars in one night So better to pick a star bright enough to give Minimises effect of sky or dark current correction Good signal to noise Easy to find Reduces measurement time Still in the linear region of your instrument

Must not vary in brightness Are isolated in the sky from other stars Is not a spectroscopic double star Is a main sequence star (luminosity class V) Often advised to pick extinction stars with similar brightness to the target star (to reduce effect of non-linearity in your instrument system), BUT Also often advised to pick extinction stars with similar colour to the target star (to reduce the effect of inaccuracies in your secondary extinction), BUT Your target star may vary significantly in colour over time And may be observing several target stars in one night So better to pick a star with zero colour (e.g. B-V = 0 in the Johnson filter system)

The standard method Rayleigh + Aerosol <click>
Rayleigh = scattering off molecules – changes with air temperature and pressure Aerosol = scattering off dust or droplets Absorption by molecules (H2O = water – highly variable, O2 = Oxygen, O3 = Ozone)

The standard method Notice that the extinction is usually much stronger on the bluer (shorter wavelength) side of the filter’s passband that on the red side. This means a star that is brighter at shorter wavelengths (e.g. an O or B spectral type) will suffer greater extinction (at the same airmass) than a star that is brighter at longer wavelengths (e.g. an K or M spectral type).

The standard method To properly allow for how the extinction is affected by the star’s spectrum, we need to know the shape of the extinction curve within the filter’s passband. <click>Say we had a star that gives this extinction coefficient - this gives us one point along the extinction curve. <click>Ideally we would also have additional filters closely spaced about our filter, so we get measurements of the extinction at different wavelengths and can trace out the actual extinction curve. As an aside, we should also have additional filters like this to use to transform from our filter system to the original (standard) filter system.

The standard method 𝑚 0 =𝑚+ (𝑘 ′ + 𝑘 ′′ ∗𝑐𝑜𝑙𝑜𝑢𝑟)∗𝑋
However I don’t know of anyone actually doing this. Instead a simple correction is made using a single colour index based on other filters in the filter system, for instance B-V in the Johnson filter system. <click> And the extinction correction equation becomes … where k-dash is the primary extinction coefficient <click> and k-dash-dash is the secondary (or colour) extinction coefficient (which is related to the slope of the extinction curve inside the filter’s passband)

The standard method Primary extinction - observing techniques:
Measure the primary extinction every night! In the best observing sites in the world (e.g. Chile or Hawaii), the atmospheric extinction may not change much from one night to another. However for the rest of us, it will be different every night.

Measure the primary extinction every night! Do not observe the extinction star when too low in the sky (avoid airmasses > 2). Measurement error increases with airmass. Airmass calculation not accurate due ozone & aerosol extinction. <click 1>Although observations made at larger airmasses should improve the accuracy in determining the extinction coefficient (as the line is fitted over a larger range of values), this is offset by the error in the observations increasing with airmass. Young (1974) found X ~ 2 is the optimum maximum airmass. <click 2>The formula used to calculate the airmass ‘X’ is not very accurate at wavelengths where ozone and aerosol extinction is strong. Fortunately the corrections to the standard calculation due to ozone and aerosol extinction are in opposite directions and partially cancel, and are not significant for Airmass < 2.3 (Young, 1974).

Measure the primary extinction every night! Do not observe the extinction star when too low in the sky (avoid airmasses > 2). Include observations when the extinction star crosses it’s meridian (airmass ~ 1). Fitting the extinction coefficient using only measurements at large airmasses gives an incorrect value due to the Forbes effect (no relation). Also make measurements when the star is at it’s highest in the sky (i.e. at the lowest airmass, about 1). <click 4>Remember extinction is usually stronger on the blue (shorter wavelength) side of a filter. The effect increases with increasing airmass. For example, if the extinction on the blue side of the filter is twice the extinction on the red side at some airmass, then doubling the airmass increases the blue extinction by a factor of four over the red extinction. This reduces the effect of the star’s colour on the extinction coefficient, which changes the slope of the airmass plot and is known as the Forbes Effect – Forbes (1842).

Measure the primary extinction every night! Do not observe the extinction star when too low in the sky (avoid airmasses > 2). Include observations when the extinction star crosses it’s meridian (airmass ~ 1). Get measurements when the star is rising and setting. Make measurements when the star is rising from the East towards the meridian and when the star is setting towards the West.

Measure the primary extinction every night! Do not observe the extinction star when too low in the sky (avoid airmasses > 2). Include observations when the extinction star crosses it’s meridian (airmass ~ 1). Get measurements when the star is rising and setting. Get at least five measurements. Get at least 5 measurements – to enable a good least-squares fit to the line (of the star’s magnitude vs airmass). So measure the extinction star at least once per hour (and maybe even once every 30 minutes).

Measure the primary extinction every night! Do not observe the extinction star when too low in the sky (avoid airmasses > 2). Include observations when the extinction star crosses it’s meridian (airmass ~ 1). Get measurements when the star is rising and setting. Get at least five measurements. Often given advice to choose an extinction star close in the sky to the target star, BUT; This may contradict the advice above to measure the extinction star over an airmass of 1 to 2. You will often see advice to choose an extinction star that is close in the sky to the target star. <click>However this may contradict the advice above to make measurements over an airmass range of 1 – 2. And again, you may be observing several target stars in a night. So I believe it is better to pick one star to use as an extinction star following the advice given earlier and have different stars for comparison & check stars to use for differential photometry with your target star(s).

The standard method 𝑚 0 =𝑚+ (𝑘 ′ + 𝑘 ′′ ∗𝑐𝑜𝑙𝑜𝑢𝑟)∗𝑋
(𝑌−𝑍) 0 = 𝑌−𝑍 + 𝑘 ′ + 𝑘 ′′ ∗ 𝑋−𝑌 ∗𝐴𝑖𝑟𝑚𝑎𝑠𝑠 So how do we measure the secondary extinction coefficient? Need to measure a series of extinction stars that have different colours. Each data point here is the primary extinction coefficient found for a particular extinction star, measured over the same night. In this case, they are the primary extinction coefficients for the Y-Z colour of the Vilnius filter system. The slope of this line gives the secondary extinction coefficient (k-dash-dash, equals here) that can be used when correcting for atmospheric extinction for any star with the formula here. The top equation is the general formula, that can be used for any colour in any filter system. The bottom equation shows how the Vilnius Y-Z colour is corrected for extinction. As I said before, the B-V colour is normally used for the Johnson filter system.

The standard method Secondary extinction - observing techniques:
Pick stars as you would to be a primary extinction star, except Include stars with different spectral types (colours) <click 1> Obviously you need to have extinction stars that cover a wide range of colours

Pick stars as you would to be a primary extinction star, except Include stars with different spectral types (colours) They should be close together in the sky They need to be close to each other in the sky, to minimise any errors in calculating the star’s primary extinction coefficient that night due to variations in extinction across the sky

Pick stars as you would to be a primary extinction star, except Include stars with different spectral types (colours) They should be close together in the sky And each (hourly?) extinction measurement made on all stars ‘simultaneously’ Each of the regular extinction measurements should be made on all of the stars at the ‘same’ time, again to minimise errors in calculating the star’s primary extinction coefficient that night due to variations in extinction during the night. You may have noticed in the earlier slide that I’d picked stars from the same E region of standard stars, in this case E6. Another possibility, especially if you’re using a CCD, is to find a suitable open cluster of stars.

Pick stars as you would to be a primary extinction star, except Include stars with different spectral types (colours) They should be close together in the sky And each (hourly?) extinction measurement made on all stars ‘simultaneously’ Measure the secondary coefficients at monthly – yearly intervals So how often do you need to measure the secondary extinction coefficients? These coefficients only depend on the slope of the extinction curve vs wavelength and shape of the filter’s passband. Neither of these are expected to change very quickly, so you don’t have to make the nightly measurements required for the primary extinction coefficient. The advice I seen ranges from measuring the secondary extinction coefficient once per month, to once per year.

The standard method 𝑠𝑙𝑜𝑝𝑒=𝑘= 𝑘 ′ + 𝑘 ′′ ∗𝑐𝑜𝑙𝑜𝑢𝑟
To summarise; The slope of the extinction star’s magnitude vs airmass gives the extinction coefficient k for that star. Subtract the secondary extinction coefficient k-dash-dash times the colour of the extinction star to get the primary extinction coefficient k-dash. To correct each observation of the target star for extinction, use the primary and secondary extinction coefficients, the colour of the target star and the target star’s airmass X. Given measurements like those here, why would we ever want to have an improved method for correcting for extinction ? ….. 𝑚 0 =𝑚+(𝑘′+ 𝑘 ′′ ∗𝑐𝑜𝑙𝑜𝑢𝑟)∗𝑋

The improved Vilnius method
Because we frequently get results like this instead, where the measurements when the star is setting do not lie on the same line as when the star was rising earlier. The standard method is based on two assumptions; The extinction is the same throughout the entire night, and The extinction is the same no matter where you look in the sky On this particular night, one of these assumptions was probably wrong.

Time-varying extinction Horizontal gradient in extinction Here I’ve shown three possible explanations for these measurements <click 1>The extinction slowly changes during the night. Here the curve shows the extinction assuming it changes by a small, constant amount per hour. <click 2>Another possibility is that the extinction is different when you look at the zenith compared to looking lower in the sky (rather than changing with time). Here the curve shows the extinction assuming it changes with the zenith angle of the star. <click 3>The third possibility is that the instrument system is changing during the night – perhaps it is sensitive to temperature and things got cooler as the night worn on. Here I’ve modelled the zero-point of the instrument changing by a small, constant amount per hour. Another possibility, which produces identical results, is that the extinction star’s brightness was changing. As you can see, these simple models produce equally good fits to the measurements. In fact, if you’ve only used one star to measure extinction during the night then you can not tell what was actually causing these odd results. Now the improved method I’m going to describe was developed by the Vilnius Observatory and is intended to deal with the first possibility, i.e. where the extinction changes during the night but is uniform across the sky at any moment in time. Unstable instrumental system

𝑚 0 𝑠𝑙𝑜𝑝𝑒=𝑘 The Vilnius method is all about working out the actual magnitude of the star, m-zero. So why is that important? Remember the extinction correction equation. <click>Well, that can be easily re-arranged so that we can calculate the extinction coefficients <click>k at every time we’ve made a magnitude measurement <click>m by calculating the airmass of the star <click>X at those times, provided we already know the star’s actual magnitude <click>m-zero. 𝑚 0 =𝑚+𝑘∗𝑋 ↔𝑘= 𝑚 0 −𝑚 𝑋

Extinction star 1 2 2 And that is what the Vilnius method aims to do. It uses two extinction stars, labelled here ‘extinction’ and ‘control’. The stars are chosen so at the beginning of the night the extinction star is near the meridian and the control star is rising from low in the sky to the east. <click>At this moment the two stars are ‘simultaneously’ measured, as indicated by the crosses each labelled with a ‘1’ . When the control star is crossing its meridian and the extinction star is setting, the two stars are again ‘simultaneously’ measured – again indicated here by crosses labelled with a ‘2’. 1 Control star

Extinction star 1 2 2 As I’ll show you shortly, it only takes these four measurements to calculate the extinction star’s actual magnitude. <click>However once you know that, you need regular measurements of the extinction star throughout the night in order to calculate the extinction coefficients throughout the night. 1 Control star

Extinction star 1 2 2 I have also proposed a slight modification of this method, where the control star is also measured at regular intervals throughout the night – with each control star measurement again being made at the same time as an extinction star measurement. 1 Control star

Choosing an extinction and control star: Pick stars as you would to be a primary extinction star, except the two stars should have the same spectral types (colours) The extinction and control stars should be chosen in the same way you normally would for a primary extinction star, except <click>If the stars have different colours, then correcting for the difference in the secondary (colour) extinction between the stars complicates the calculation.

Choosing an extinction and control star: Pick stars as you would to be a primary extinction star, except the two stars should have the same spectral types (colours) the two stars should have similar magnitudes To reduce errors due to any non-linearity in your system, the two stars should be a similar brightness

Choosing an extinction and control star: Pick stars as you would to be a primary extinction star, except the two stars should have the same spectral types (colours) the two stars should have similar magnitudes Assumes: The extinction is the same for both the extinction & control stars when each pair of measurements are made, i.e. it is uniform across the sky at any one moment BUT does NOT need the extinction to be the same all night The standard method requires the extinction to be both uniform across the sky and constant for the entire night, while the Vilnius method only has the uniformity requirement

The method’s algorithm: Get the extinction star’s approximate magnitude 𝑚 0,𝑒𝑥𝑡𝑛 using the standard method

The method’s algorithm: Get the extinction star’s approximate magnitude 𝑚 0,𝑒𝑥𝑡𝑛 using the standard method Average the extinction star’s approximate magnitude from each night in the observing run, to give 𝑚 0,𝑒𝑥𝑡𝑛 Improve the accuracy of your approximate value by taking the average of these values from several nights. Remember what you’re measuring is the star’s magnitude in your instrumental system, which may slowly change as, for example, your filters age. You can get an idea of how stable your system is by looking at if, and how fast, the exact magnitude of the extinction star and control star found each night appears to change over the months – which we’ll get to in a few more steps.

The method’s algorithm: Get the extinction star’s approximate magnitude 𝑚 0,𝑒𝑥𝑡𝑛 using the standard method Average the extinction star’s approximate magnitude from each night in the observing run, to give 𝑚 0,𝑒𝑥𝑡𝑛 For every simultaneous measurement of the extinction star ( 𝑚 𝑒𝑥𝑡𝑛 𝑖 ) and control star ( 𝑚 𝑒𝑥𝑡𝑛 𝑖 ), calculate the control star’s approximate magnitude 𝑚 0,𝑐𝑛𝑡𝑙 𝑖 = 𝑚 𝑐𝑛𝑡𝑙 𝑖 − 𝑚 𝑒𝑥𝑡𝑛 𝑖 − 𝑚 0,𝑒𝑥𝑡𝑛 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑐𝑛𝑡𝑙 𝑖 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑒𝑥𝑡𝑛 𝑖

The method’s algorithm: Plot the airmass ratio 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑐𝑛𝑡𝑙 𝑖 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑒𝑥𝑡𝑛 𝑖 on the horizontal axis and the approximate magnitude of the control star 𝑚 0,𝑐𝑛𝑡𝑙 𝑖 on the vertical axis Exact magnitude of control star 𝑚 0,𝑐𝑛𝑡𝑙 = intercept <click>The intercept is the exact magnitude of the control star, while <click>the slope is the difference between the exact magnitude of the extinction star and its approximate magnitude <click>The original Vilnius method would have only had these two (circled) points – by including further control star measurements for the least squares fit, the accuracy is improved and <click>the standard deviations give an indication of how accurate the ‘exact’ magnitudes are. Exact magnitude of extinction star 𝑚 0,𝑒𝑥𝑡𝑛 = 𝑚 0,𝑒𝑥𝑡𝑛 + slope

The method’s algorithm: Get the extinction star’s approximate magnitude 𝑚 0,𝑒𝑥𝑡𝑛 using the standard method Average the extinction star’s approximate magnitude from each night in the observing run, to give 𝑚 0,𝑒𝑥𝑡𝑛 For every simultaneous measurement of the extinction star ( 𝑚 𝑒𝑥𝑡𝑛 𝑖 ) and control star ( 𝑚 𝑒𝑥𝑡𝑛 𝑖 ), calculate the control star’s approximate magnitude 𝑚 0,𝑐𝑛𝑡𝑙 𝑖 = 𝑚 𝑐𝑛𝑡𝑙 𝑖 − 𝑚 𝑒𝑥𝑡𝑛 𝑖 − 𝑚 0,𝑒𝑥𝑡𝑛 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑐𝑛𝑡𝑙 𝑖 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑒𝑥𝑡𝑛 𝑖 Plot the airmass ratio 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑐𝑛𝑡𝑙 𝑖 𝐴𝑖𝑟𝑚𝑎𝑠𝑠 𝑒𝑥𝑡𝑛 𝑖 on the horizontal axis and the approximate magnitude of the control star 𝑚 0,𝑐𝑛𝑡𝑙 𝑖 on the vertical axis 𝑚 0,𝑐𝑛𝑡𝑙 = intercept 𝑚 0,𝑒𝑥𝑡𝑛 = 𝑚 0,𝑒𝑥𝑡𝑛 + slope Average the ‘exact’ magnitudes of the extinction star 𝑚 0,𝑒𝑥𝑡𝑛 and control star 𝑚 0,𝑐𝑛𝑡𝑙 from each night to further improve their accuracy 5. Finally, improve the accuracy of the ‘exact’ magnitudes of the extinction and control stars by averaging these values that have been found for each night over, say, the last month.

Now that we know the exact magnitude of both the extinction star and control star, we can use these to calculate the extinction coefficient for every time we measured those stars. This is from the night that I talked about earlier, where the star had a ‘U’ shaped curve in the magnitude verses airmass plot. We can see now that the extinction slowly decreased as the night went on. Apparently this is due to a slow fall-out of aerosols in the atmosphere during the night as the convention driven by solar heating, which was stirring up the aerosols, ceases. Those of you who are still awake may have noticed I have three stars plotted here. These observations were made in April, where the nights are about 12 hours long. However it only takes about 6 hours for a star to set after cumulating so the extinction star, here HD086629, sets about midnight. By this time the control star was crossing the meridian, so it became the extinction star of an extinction-control pair of stars. In the middle of winter, it can be possible to use yet another pair of extinction-control stars for the last part of the night.

An improved method for correcting Atmospheric Extinction Murray Forbes

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An improved method for correcting Atmospheric Extinction Murray Forbes

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