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Astro Stats and MeasurementMeasurement Part B : The atmosphere and how to avoid it

Andy Lawrence, University of Edinburgh

Sept-Dec 2013

We can’t measure any light until it has passed through the atmosphere, or we have risen above theatmosphere to meet it. The atmosphere blocks light, distorts it, and creates a bright backgroundthat makes it hard to see faint objects. All these problems depend strongly on wavelength (or typeof particle). In this chapter we look at how these effects arise, and how to get around them. Wewill examine in turn transmission, sky backgrounds, refraction and seeing, and the pros and cons ofworking in space.

1 Transmission through the atmosphere

1.1 Physical effects in atmospheric transmission

A rather small fraction of the electromagnetic spectrum makes it through the Earth’s atmosphere.Some of the incoming radiation is absorbed, and some is scattered or reflected. The net effect of re-moval of light by the combination of true absorption and scattering is known as extinction. Across thewhole EM spectrum, a number of different physical processes contribute. We need to consider threemajor categories of physical process - atomic and molecular absorption, scattering, and refraction.

Atomic and molecular absorption. The atmosphere is dominated by Nitrogen and Oxygen, andespecially the molecular forms N2 and O2, so these species feature strongly in atmospheric absorp-tion. However, some molecules, such as H2O, CO2, and O3, while less common, have such largecross sections that they have a significant effect on atmospheric absorption. Several different kindsof transition can absorb light. The simplest kind is discrete transitions between atomic energy levels,causing narrow absorption lines.These can be annoying if they occur just at the wavelength you wantto observe, but they do not remove much light in total. Molecules however have additional energylevels corresponding to many different modes of quantised vibration and rotation, leading to broadabsorption bands removing large chunks of the spectrum in the IR and submm (see next section).Molecules can also suffer photo-dissociation, i.e. photons of sufficient energy can split them apart.For example, the process N2 → 2N requires photons of energy E > 9.76eV, i.e. λ < 1270A, andO2 → 2O requires λ < 2408A. The ozone molecule O3 has many different dissociation pathways,with the biggest effect cutting in at λ < 3100A. Finally we need to consider photo-ionisation, wherethe absorption of a photon completely ejects an electron. For example, atomic Nitrogen can be ionisedby photons with E > 0.4keV, i.e. λ < 31A. For both photo-dissociation and photo-ionisation, thecross-section decreases at energies larger than the minimum necessary, so like molecular bands, eachspecies tends to remove a middling-sized chunk of wavelength.

Scattering. First we need to consider Thomson scattering, which involves EM waves and free elec-trons. You can think of the electron as a driven oscillator; the E-field of the incoming wave drives anoscillation in the electron. However an accelerating charge radiates; the effect is a dipole pattern ofradiation at the same frequency as the incoming wave - so a wave in a single direction is re-radiatedinto a wide range of directions, i.e it is scattered. The cross-section σT is the same for all frequen-cies and the frequency of the scattered light is unchanged. This process requires free electrons, andthe atmosphere largely speaking is not ionised, so for most of the EM spectrum it’s not an impor-tant effect. However, for high energy (X-ray) photons, where the photon energy is larger than the

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Astro Measurement B : Atmosphere 2

binding energy of the electron in the atom, even bound electrons behave as if they were free, andelectron scattering becomes important in the atmosphere. However, the classical picture of Thomsonscattering becomes complicated in two ways, essentially because we need to take into account themomentum as well as the energy of the photon. First, the scattered photon does not stay at the samefrequency, but loses energy to the electron. This is the process of Compton scattering. Second, thecross-section becomes energy dependent, gradually decreasing below σT at higher energies (this isthe Klein-Nishina cross-section). The effect is that Compton scattering becomes much less importantat E > 100keV.

Next, we need to consider Rayleigh scattering. Above we talked about atomic and molecular transi-tions absorbing light, but they can also cause scattering. An absorption leaves an atom or molecule inan excited state; this could potentially de-excite in many different ways, but one possibility is simplyto drop back down and emit a photon of the same frequency. This is known as resonant scattering. Aphoton at a frequency not exactly equal to the resonant frequency still has a chance of absorption andre-emission, but with decreasing probability as we get further from the resonant frequency. The atombehaves like a driven but damped oscillator, which gives a generic solution for all transitions - at largedistances from the resonant wavelength, the intensity of scattered light goes as λ−4. Many differentatomic and molecular species contribute to this effect, but they all have the same λ−4 dependence.

Finally, under the general heading of scattering we need to consider large particle scattering. Theatmosphere contains many different species of quite macroscopic particles including water droplets,large molecules, dust, pollen, and bacteria. Most of these are small enough that they are suspendedin the atmosphere as aerosols, but occasionally larger particles are temporarily driven into the at-mosphere and then gradually fall out - for example the observatory on La Palma in the Canaries isoccasionally affected by sand blown by the wind from the Sahara. Large enough particles essentiallypresent a problem of geometrical scattering; smaller ones need an explicit treatment of the interactionof radiation with the material through Maxwell’s equations. For particles which are approximatelyspherical, this interaction is well described by an approximation known as Mie theory.

Refraction. This is the bending of light waves due to a change of velocity. At UV-optical-IR wave-lengths the effect is small, and produces a change in the apparent directions of astronomical objects,and a systematic blurring of light passing through the atmosphere - effects which we will considerlater in this chapter. For long wavelength radio waves passing through ionised regions however, theeffect can be so large that it effectively produces reflection, preventing waves from getting through theatmosphere. The effective refractive index is less than 1, so that waves bend away from the normal.As the wave passes through layers of changing density, the waves can be completely turned round,i.e. reflected. This works from both sides of an ionised layer. Incoming UV and X-ray light from theSun ionises material in the upper atmosphere, producing several distinct layers known collectivelyas the ionosphere. The lower D and E layers, which reflect terrestrial signals back towards Earth,are ionised only during the day time, and their height changes. The higher F layer is permanentlyionised. During the night it reflects terrestrial signals (which is why we can receive more distant AMstations at night), but it reflects astronomical signals at all times.

1.2 Frequency dependence : atmospheric windows

Figure 1 shows how the absorption of electromagnetic radiation by the atmosphere varies with wave-length, with Fig. 2 and Fig 3 showing more detail in the optical and submm regions. The atmosphereis opaque at most wavelengths. Through the UV and soft-X-ray region, broad photo-dissociation andphoto-ionisation features overlap completely, so that astronomy at these wavelengths is only possibleby getting completely above the atmosphere, in spacecraft. In harder X-rays, Compton scatteringand various nuclear processes likewise completely absorb incoming photons; only by the gamma-rayregion does the atmosphere begin to become partially transparent.

Returning to longer wavelengths, most of the absorption is in broad bands caused by O3, H2O, and


Figure 1: Cartoon indicating absorption by the atmosphere versus wavelength, and the height needed to carryout astronomy at different wavelengths. Figure kindly provided by Dr Adam Woodcraft.


Figure 2: Components of transmission in the optical region. See text for discussion. The upper panel showsthe location of the standard Johnson filters - see Chapter 5. Based on a figure from Patat et al 2011, kindlyprovided by Dr Fernando Patat, ESO.

CO2, with a few relatively clear gaps. The broadest and most consistently transparent of these gapsis in the optical region, where the Sun emits most of its radiation and human eyes work. Furtherrelatively clear “windows” occur spread through the IR and submm. These clear windows are usedto define standardised passbands used in ground-based astronomy. (See Chapter 5 for more detail.)Note that a large swathe of the far-infrared (FIR) region is, like UV and X-ray wavelengths, accessibleonly from space. Moving to longer wavelengths, the molecular absorption bands cease, and theatmosphere is almost completely transparent to radio waves between wavelengths of ∼ 1cm and∼ 10m (the long wavelength end changes with ionospheric conditions). At longer wavelengths, theionosphere completely reflects radio waves.

Fig 2 shows how the transmission of light in the visible light window involves several componentswhich can vary independently. As well as the absorption features due to water and oxygen, visiblewavelength light suffers significant scattering: by large molecules, which produce Rayleigh scatteringdominating in the blue, and by a variety of aerosols, which produce somewhat greyer scattering. Thevarious components can vary from night to night, and even within the night. Note the sharpness ofthe cut-off around 3500A, due to the combination of ozone and scattering; this means that measure-ments in the blue are particularly sensitive to atmospheric conditions. The complexity and variabilityof atmospheric transmission means that we need to take an empirical calibration approach, as weshall discuss in section 1.5. First however we will look a little more carefully at factors involved invariations of transmission.

1.3 Dependence on height and weather

Fig 1 indicates crudely what sort of height one needs to reach to observe at various wavelengths. Somewavelengths require observations from completely above the atmosphere, but at some wavelengthsthe atmosphere is partially transparent. This is most strikingly true for the absorption caused by water


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Figure 3: Transmission in the submm region, through the atmosphere above Mauna Kea in Hawaii, at a heightof 4200m, on two different occasions. Submm observations are only possible at all at high dry sites like MaunaKea, and only within specific atmospheric windows. Even at such a site, a value of PWV=2mm is typical; avalue of 0.5mm is exceptionally clear. Observations at 350µm are only possible on occasional nights. (Basedon data derived from the CSO atmospheric calculator - see references)

vapour in the atmosphere. Nearly all the water vapour in the atmosphere is in the troposphere, i.e.at relatively low heights, so that even observing from a high mountain top can bring a significantadvantage. The amount of water vapour also varies strongly from place to place; some locations aremuch drier than others. The air above Antarctica is particularly dry, making a very good if expensiveplace to undertake IR and submm astronomy. The amount of water vapour also varies strongly withtime, and of course can form dense localised patches of water droplets - clouds. When cloud layersform, the normally transparent optical window becomes opaque. Even in cloudless conditions, thedryness of a “clear” sky can vary considerably. The integrated amount of water above a given locationis specified by the column of Precipitable Water Vapour (PWV), which can be expressed either in kgm−2, or simply in mm. This refers to the depth of water which would result if all the water in theatmosphere descended to Earth. The average amount of water in the atmosphere is enough to coverthe Earth to a depth of ∼ 25mm; observations in the most sensitive submm windows only becomepossible when the local overhead PWV is less than a few mm. Fig 3 shows how sensitive the submmregion is to dryness.

1.4 Dependence on zenith angle

For wavelength regions which are partially transparent, the atmospheric transmission will also de-pend on the zenith angle of the observation, as when we look at larger zenith angles we are lookingthrough a greater thickness of atmosphere. (See Fig. 4.) Although the nature of the absorbing/ scat-tering effects will be different at each wavelength, for different locations, and on different nights,the character of the behaviour with zenith angle is always the same, and is most simply expressed inmagnitude terms.

Suppose the true flux density of a star in some passband is F0, i.e. this is what would be measuredabove the atmosphere. If the column of air we see it through has N extinguishing particles of some


Figure 4: Geometry of extinction. If we approximate the atmosphere as a simple uniform parallel slab, thepath length through the atmosphere is proportional to sec z where z is the zenith angle. The path lengthvertically through the atmosphere is referred to as one airmass.

kind, the reduced flux density will be F = F0e−aN where the constant a depends on the precise

physics. Then lnF = lnF0 − aN . The change in the log of flux can be expressed as a change inmagnitude, remembering to switch to base 10 logs and multiply by 2.5 : ∆m = (2.5× log10 e× a)×N . We can just re-express this as

∆m = bN,

i.e. the extinction in magnitudes is proportional to the amount of matter we are looking through.However, the column N varies with zenith angle. For a simple plane atmosphere, we would havecos z = Nvert/Nz whereNvert is the column looking straight up, andNz is the column when lookingat zenith angle z. (See Fig. 4). This remains correct for a stratified but still plane-parallel atmosphere,which you can see by imagining adding up lots of segments. It breaks down at large zenith anglesbecause you have to take the Earth’s curvature into account, but astronomers will anyway avoidobserving at large zenith angles. Merging the unknown values b and Nvert we simply end up with

∆m = k sec z = kA,

where the constant k is the zenith extinction in magnitudes, andA = sec z is the number of airmassesbeing looked through. Straight upwards is one airmass; at z = 30◦ we are looking through 1.15airmasses; at z = 60◦ through 2.0 airmasses; and at z = 75◦ through 3.86 airmasses.

1.5 Calibrating extinction

The dependence with zenith angle gives us a way to calibrate the extinction at a specific wavelengthon any given occasion. The value of the zenith extinction k will be different at different observingsites, different from night to night, and different at different wavelengths. However, on any oneoccasion it can be easily measured. If we observe a standard star at two different airmasses A1 andA2, i.e. at two different times of night, and measure magnitudes m1 and m2 then we can solvefor the constant k and thus for the true magnitude of any observed object m0 = mobs + ∆m. In


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practice what is usually done is to measure at a range of airmasses, fit a straight line to the data, andinterpolate/extrapolate as appropriate.

The zenith extinction measured in this way is a sensitive function of wavelength, as shown in Fig5. Because of Rayleigh scattering and ozone absorption it is much worse in the far blue part of thevisible light window. This means that observations in the far-blue have to be performed relativelyclose to the zenith. In the far blue, a zenith extinction of 0.5 magnitudes corresponds to 1 magnitudeat a zenith angle of 60◦, i.e. a factor of 2.5.

2 Backgrounds

The detection of faint astronomical sources can be made much harder by the presence of backgroundlight, as explained in Chapter 1. We will look briefly at the relevant backgrounds in each majorwavelength region, which come from a variety of environmental effects, including the atmosphereitself.

2.1 Background light from the atmosphere

In the optical and NIR, the light from the sky is a mixture of scattered light and airglow. Thescattering is by the same agents that cause extinction of starlight on the way down – large molecules,dust, and aerosols – which means that the scattered light problem is worst in the blue. The worst lightpollutant is the Moon; the faintest objects have to observed in “dark time”. The second worst pollutantis streetlights; this why telescopes need to be in remote locations. Airglow however is unavoidable. Itis due to atoms and molecules being excited by particles from the solar wind hitting the atmosphere.(See section 2.3 for a discussion of these particles).

Figs 6 and 7 show night sky spectra from two dark sites : La Palma in the Canary islands, and Mauna


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Figure 6: The night sky spectrum in the optical region, from La Palma in the Canaries. The underlyingcontinuum is scattered light; the spectral features are airglow due to atoms and molecules excited by fastparticles. Based on data provided by Dr C.Benn, ING, La Palma Observatories.

Kea on Hawaii. In the middle of the visible range, the two strongest airglow features are emissionlines from atomic oxygen at 556 nm and 630nm. These are the features which make the beautifulgreen and pink curtains of light in aurorae, but they are there at a much lower level in every nightsky. In the red-visible and near-IR, the airglow is much brighter and composed of broad bands madeup of many transitions due to OH and H20. These airglow bands also have a spatial structure andvary rapidly during the night. IR observing therefore typically requires short exposures so that thechanging background can be subtracted.

As we move into the mid-IR (>2µm) we see another problem : thermal emission. Note that at anormal Earth temperature of 20C = 293K, the peak of blackbody emission given by λT = 2900µm isλ ∼ 10µm. This means that the telescope structure, the instrument, and the human observers are allglowing brightly in the mid-IR. The solution is to make everything as cold as possible. The detectoritself can be inside a vacuum structure cooled by liquid Nitrogen or liquid Helium. The telescopecan be placed on a cold mountain top, and the observers can be kept in a separate thermally isolatedroom.

2.2 Terrestrial interference

At long radio wavelengths, terrestrial radio signals bounce back from the ionosphere, especially inthe day when the ionosphere is lower, making a severe background. Even at higher frequencies, radiofreqency interference comes from all around – television signals, electric motors, mobile phones,power transmission lines, computers, and all sorts of other things.

These problems have led the International Telecommunications Union (ITU) to allocate certain re-served frequency ranges to be for the purpose of radio astronomy, thus defining some standard radioastronomy bands – for example L-band (1.4GHz), C-band (5 GHz), and X-band (8 GHz). This isonly partly successful, so radio telescopes, like optical and IR telescopes, tend to be built in remoteplaces, such as the New Mexico desert, where the Very Large Array (VLA) is located.


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Figure 7: The night sky spectrum in the IR, from Mauna Kea in Hawaii. The solid line is the emission fromthe sky itself, based partly on data from the Gemini Observatory website (see references), and partly on theatmospheric model of Lord (1992). It assumes a precipitable water vapour column of 1.6mm, an effective skytemperature of 273K, and applies to what would be seen looking at the zenith, i.e. 1.0 airmasses. The emissionis averaged over bins of size 0.05µ m, which smooths over many narrow lines at a time. The dashed lineindicates the additional contribution due to thermal emission from the telescope and other local structures,modelled as a blackbody with temperature 273K. The normalisation of the telescope thermal component de-pends very strongly on how well the IR camera is shielded, the emissivity of the telescpe mirror, and so on. Thelevel shown indicates typical performance from a well designed IR telescope.

Figure 8: Radio interference spectrum as observed at Jodrell Bank. Taken from the website of the Jodrell BankCentre for Astrophysics - see references.


Figure 9: Illustration of the structure of the Van Allen radiation belts, drawn roughly to scale compared to thesize of the Earth. (The magnetopause is at roughly 10 Earth radii). This figure is taken from the web site ofthe NASA Van Allen Probes mission (see references), and shows the orbits of the two spacecraft. (Which are ofcourse not to scale!) Credit : NASA

Terrestrial interference also affects gravitational wave experiments. Astronomers are attempting todetect passing gravitational waves via very subtle movements of test masses in interferometer arms.Such tiny motions are very hard to distinguish from the vibration noise caused by people, passing cars,air motion, etc. Gravitational wave experiments are therefore inside tunnels in remote locations, andif a real signal is one day seen, can test for coincidence between experiments in different countries;the signal will be the same in the two locations whereas the noise will not be.

2.3 Particle backgrounds in space astronomy

The same particles that produce the airglow also produce problems for UV, X-ray, and gamma-rayastronomy. Some particles arriving at Earth are cosmic rays, very high energy particles that comefrom throughout the Galaxy and possibly beyond. However, most particles in the near-Earth envi-ronment come from the solar wind, which originates in the solar corona. The majority of these arethermal with a temperature of around a million degrees giving keV particle energies, but a minorityare accelerated to very high energies, up to several hundred MeV. Most of these solar wind particlesare deflected by the Earth’s magnetic field, but a minority make it through. The density of particles isstrongly enhanced in regions where particles become trapped on specific field lines, forming the VanAllen radiation belts. There are two main belts, as illustrated in Fig. 9. The inner proton belt is at aheight of 1000 – 5000 km (compared to the Earth’s mean radius of 6378km). The outer electron beltis at 15000 – 25000 km. Many spacecraft will try to stay below the inner proton belt; unfortunatelypart of this belt, known as the South Atlantic Anomaly, dips lower down towards the Earth, so itis hard to avoid protons for the whole orbit. The distorted magnetic field and radiation belts forma highly dynamic system. The density of particles varies by several orders of magnitude from oneplace to another, and from one time to another.

The particles in the near-Earth environment can have several bad consequences for astronomicalobservations made from spacecraft, and even from the ground.

(i) They can damage instruments, for example degrading sensitivity, or reducing the charge transferefficiency of CCDs (see Chapter 4). Delicate instruments therefore need shielding from energeticparticles.

(ii) They can excite upper atmosphere atoms and cause airglow. This is a significant problem for UVastronomy in space. The most common upper atmosphere species are Hydrogen and Helium. These


Figure 10: Countrate from the EPIC X-ray detector on board the XMM spacecraft, over half a day. This isthe count rate over the whole detector. Most of the events are caused by particles. The rate is quiescent forthe latter part of the observation, but suffers flaring in the first 15,000 seconds, thought to be due to protons of∼100 keV energy, just low enough energy that they can reflect from the mirrors and be focused on the detector.(Figure kindly provided by Prof. M.Watson, University of Leicester).

can be ionised by incoming particles and then glow brightly in the Lyα transitions of neutral Hydro-gen and Helium (HI and HeI) and singly ionised Helium (HeII), making a bright UV background, butone that is concentrated in narrow spectral features. This is known as geo-coronal emission, and it isstrongest towards the poles of the Earth’s magnetic field.

(iii) High energy particles can also mimic X-ray and gamma-ray photons, by creating events in detec-tors. These can be distinguished from real photons to some extent, for example with anti-coincidencetechniques (see Chapter 4), but the separation is never perfect. The level of particle activity in thenear-Earth environment goes up and down by a large amount on all timescales - see Fig 10. X-raydetectors monitor this background, and standard X-ray analysis normally starts by cutting out timewindows of high particle background, where the data are close to useless.

(iv) Some particles – mostly not solar-wind related particles, but the very high energy “cosmic rays”originating outside the solar system – make it through the atmosphere. These can make occasionalfake events in CCD cameras at all wavelengths, so that a long integration visible light image can bepeppered with bright spots caused by these cosmic rays. They do not suffer the same atmosphericblurring as light, so can usually be quite well separated in analysis, as they look sharper than stars.

3 Systematic Distortion effects

Two effects significantly alter the apparent position of astronomical objects on the sky. The first isaberration, caused by the motion of the Earth, which we will deal with briefly before moving on tothe second effect : atmospheric refraction. In both cases the size of the effect depends on the local co-ordinates of the object, so that a uniform grid of stars would be distorted. Telescope pointing systemsneed to take these distortions into account, otherwise they can be out by 1-2 arcminutes. Even fromone side of a CCD image to the other, the distortion is significant. The amount of refraction alsovaries with wavelength, making it hard to make sharp images across a wide wavelength range.


Figure 11: Atmospheric refraction by a single uniform layer of air.

3.1 Aberration

We are all familiar with the phenomenon of running into a rain shower; it makes the rain appear to becoming towards us even though it may be falling vertically. In a similar fashion the apparent directionof incoming light is altered by the Earth’s motion v in space. When v/c is small, the change of angleis ∆θ = (v/c) sin(θ) where θ is the angle between v and the true direction of the incoming light.Several different components of the Earth’s motion may be relevant depending on the astronomicalobject emitting the light - for example the Earth’s orbital motion; the orbit of the Sun around theGalaxy; and the motion of the Galaxy with respect to the Cosmic Microwave Background. Thismeans that we never see the true positions of objects, only their apparent positions. Nearly all ofthose motions change very slowly, so this is just taken as a fact of life. However, the orbital motionof the Earth produces an effect with varies around the year, and which depends on RA and Dec.Given that vEarth = 29.8 km s−1, the maximum size of the effect is 20.5′′. The rotation of the Earthalso produces a diurnal aberration effect of maximum size 0.32′′. The same motions - rotation of theEarth, orbital motion of the Earth, Galactic orbit of the Sun, and the motion of the Milky Way - alsoproduce Doppler shifts, so that stellar velocities and galaxy redshifts have to be corrected for theseeffects.

3.2 Refraction

The refractive index of air (n∼1.0003) is very small compared to water (n∼1.33) or glass (n∼ 1.5) butthe effects are nonetheless significant. For simplicity, let us first take the atmosphere to be a simplesingle layer. Consider a light ray that arrives at the top of this layer at zenith angle z, and is refractedtowards the vertical so that the apparent zenith angle is z′ (See Fig. 12). Then Snell’s law tells us thatthese angles are related by

sin z

sin z′=n2n1,

where n1 = nvacuum = 1 and n2 = nair = n so that sin(z) = n sin(z′). For a star at the zenith thereis no effect, whereas for z = 45◦ we find z′ = 44.9828. It is useful to work in terms of the angle ofrefractionR = z−z′. Then we can write z = z′+R and so sin(z) = sin(R) cos(z′)+cos(R) sin(z′).If R is small then sin(R) ∼ R and cos(R) ∼ 1 and so

sin(z) = R cos(z′) + sin(z′).


Figure 12: Atmospheric refraction by a series of plane-parallel layers.

Putting this into the refraction formula sin(z) = n sin(z′)we get

R = (n− 1) tan(z′).

The value of n depends on temperature and pressure. An empirical formula that works reasonablywell is

(n− 1) = 7.89× 10−5Pmbar

T,

where Pmbar is the pressure in millibars and T is the air temperature in degrees Kelvin. Convertingfrom radians to arcseconds, we get

R′′ = k tan(z′) where k = 16.27′′ × Pmbar/T.

For P=1 bar and T = 20◦C = 293◦K, k=55.53, and so z = 30◦ causes a refraction angle of R=32′′and z = 60◦ gives R=96′′. This refraction is towards the local vertical.

3.3 Multi-layer refraction

The atmosphere is not of course a single layer, and the pressure and temperature of the air changesby a large amount with altitude. However, it is easy to show that only the local conditions matter. Wecan think of the atmosphere as a series of layers, each with a different refractive index n. Consideringthree layers (see Fig. 12) we have

sin i1sin r1

=n2n1

andsin i2sin r2

=n3n2.

However i2 = r1 and so we find that

sin i1sin r2

=n3n1,


i.e. the effect of the intermediate layer cancels out and the net effect depends only on the first andlast layers. A little algebra will show that if you have N layers, this cancellation continues to be worklayer by layer. If the first layer is vacuum with n = 1 and i1 = z the true zenith angle, then iN = z′,the apparent zenith angle, and we simply recover the result of the previous section, that the angle ofrefaction is R′′ = k tan(z′) where k depends only on the local atmospheric conditions.

3.4 Chromatic differential refraction

Refractive index is a function of wavelength. This means that the red image of a star and the blueimage of a star are refracted by different amounts. This effect is known as chromatic differentialrefraction, or atmospheric dispersion. Near the zenith this effect is small, but at large zenith angles itcan be quite significant, as shown in Fig 13. This shows the position of star images at four differentwavelengths for various zenith distances and typical conditions.

There are two ways to handle this problem. The first is to use relatively narrow-band filters, so thespread in wavelength is not too big. The second is use a device called an Atmospheric DispersionCorrector (ADC). These typically use two overlapping prisms, whose separation can be adjusted toproduce an effect which precisely counterbalances the differential refraction.

Differential refraction is also a problem for spectrographs, as one cannot get the light from all dif-ferent wavelengths through a narrow slit at the same time. One solution is to use a wide slit, but thislowers the spectral resolution (see Chapter 5). Another solution is to orient the slit vertically, so thatall the light goes through the slit, but produces a curved rather than straight image of the spectrum onthe detector.

4 Random blurring by the atmosphere : seeing

For a telescope of diameter D diffraction should lead to images of size θ ∼ λ/D. For a 4m telescopeand visible wavelength light with λ=0.5µm , this gives θ=0.03′′. However real observations from theground give images that are ∼1′′ across. This is because the air that causes atmospheric refraction isnot stable, but turbulent, especially in particular layers of the atmosphere. Fig 14 shows that for veryshort exposures the image of a star appears as a series of spots known as “speckles”, but that overlonger integrations, the speckles move around, produce the net blurring that we refer to as “seeing”.How does this come about ?

4.1 Air Turbulence

Turbulent motions can arise three main ways. (i) By convection, as slightly warmer air rises fromthe telescope mirror and observatory building - collectively known as dome seeing. (ii) By windcolliding with structures, such as boulders and buildings - known as ground layer turbulence. (iii)Through shearing motions, because the wind is travelling different speeds in different parts of theatmosphere - atmospheric turbulence. Modern observatory design aims at minimising dome seeingand ground layer turbulence, for example by keeping the dome air at a constant temperature, and byflushing air through the dome in a laminar manner. However atmospheric turbulence is unavoidable.

Turbulent processes result in a power-law spectrum of perturbations of temperature and pressure,from some outer scale L ∼ 10–1000m down to an inner scale of a mm or less where viscosity finallydissipates the energy. This in turn produces a spectrum of values of refractive index n. Kolmogorovtheory says that for two regions of air a distance r apart, the typical value of the square of the differ-ence in n is


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c)

Zenith angle, degrees

Figure 13: Chromatic differential refraction. The upper plot shows the angle of refraction vs zenith distancefor two wavelengths - 360nm (solid line) and 790nm (dotted line). The lowerplot shows the relative positionsof a stellar image at wavelengths of 360nm (violet), 440nm (blue), 550nm (green), 640nm (pink), and 790nm(red). The circles indicate a seeing size of 1′′.


Figure 14: Montage of a series of images of a bright star showing seeing speckles. Each image has anexposure of 1/100th second and they were taken about once every second, using a webcam. Figure kindlyprovided by Dr Bill Keel, University of Alabama.

Dn = 〈(n1 − n2)2〉 = C2nr

2/3,

where C2n can be anywhere in the range 10−14 to 10−18 in different parts of the atmosphere. So the

further apart two pieces of atmosphere are, the more different they are in refractive index, until youget to the outer scale L, after which the variations are just white noise. The overall variance in n is∼ 10−6, which you can compare to the typical value n ∼ 1.0003. So these variations are very small,but they make a big difference, as we will see. They also vary with time, as turbulent structures areblowing sideways across the field of a view of a telescope with a typical wind speed of 10 m s−1.The largest structures, of the order ∼ 10m will then change on a timescale of 1 second, and thesmallest structures that we care about, which as we will see below are of the order 10cm, change ona timescale of a few milliseconds.

4.2 Wavefront bending

The changes in refractive index bend the light to and fro. Consider a wave which hits an interfacebetween a region with index n1 and a region with index n2. Following the terminology of section3.2, the wave will be bent by an angle

R = tan i

(n1n2− 1

).

Because we are talking about random turbulent cells, the average angle of incidence is 45◦ and sotan i = 1. If we write n1 = n2 + ∆n we find that the typical bend angle as a wavefront crossesa turbulent cell is just 〈R〉 = ∆n. Above we learned that the dispersion in n is σn ∼ 10−6, so thetypical bend angle is ∼ ±10−6 radians ∼ ± 0.2′′ . Now imagine a series of wavefronts arriving atthe telescope aperture. Each one makes a close to diffraction-limited spot, but in slightly the wrongplace. As the turbulent structures blow across, the bend angle keeps changing and the spots swingaround, making a net smearing with a FWHM of around 0.5′′ . This overall bending of the wavefrontis known as the tip-tilt effect.


Figure 15: Schematic illustration of wavefront tip-tilt. In the left hand figure, parallel wavefronts are proceed-ing through a uniform atmosphere, including any systematic refraction. The telescope focuses these wavefrontsto a point at the focal plane. In the middle figure, a slightly denser patch of air causes extra refraction, and thefocused spot moves slightly. The right hand figure shows the effect of multiple patches bending the wavefront byslightly different amounts at spatially different places. Each section of the wavefront produces a spot (speckle)in a slightly different location on the focal plane. Note that the degree of bending is hugely exaggerated in thisillustration, or it would not be visible!

However, at any one time, different parts of the wavefront are passing through different bits of at-mosphere. Imagine dividing the telescope aperture up into a series of sub-apertures and consideringthe relevant sections of wavefront above them. Each wavefront section has a slightly different tipand tilt. The result is that the wavefront is no longer plane-parallel, but corrugated. The effect inthe image plane is that rather than a single spot, at each instant in time there are a large number ofmini-spots - the speckles. Over time, the individual speckles come and go, and because of the maintip-tilt effect, the whole pattern of speckles swings around. The net effect is smearing of the order of1′′, but varying considerably from one site to another, and from night to night.

4.3 Phase delays and coherence length

The bending and crinkling of the wavefronts, while very important, is tiny. However, the effect onthe phase of the incoming waves is very large. Consider a wavefront tilted at an angle of 0.5′′. At adistance x along the wavefront, the vertical difference from an untilted wavefront is h = θx (Notethat θ is small). For a telescope of aperture 2m that gives h=5µm , which is ten times larger than thetypical wavelength of incoming light. In other words, at any one time, different parts of the telescopeaperture are not even measuring the same wavefront. Another way of looking at the phase effect isthat as light passes through two neighbouring sections of thickness d with indices n1 and n2 theyhave a difference in optical path length ∆l = d∆n, which introduces a phase difference. When theeffects of turbulence are modelled, this is usually done by introducing a “phase screen”, which canthen be moved laterally across the line of sight.

If we consider a small enough section of the incoming wavefront, it will be reasonably coherent. Forseeing size θs the typical tilt is θs/2. If we ask for the phase difference to be less than π, i.e. halfa wave, then we need to consider sections of the wavefront of size less than r = λ/θs. Consider asub-aperture of this size; it would produce a diffraction-limited image of size θdiff ∼ λ/r ' θs. Fora sub-aperture any smaller than r ∼ λ/θs, the atmospheric effects don’t matter, as the diffraction sizeis bigger than the seeing. This scale size can therefore be seen as a characteristic scale for the effectsof turbulence, and is known as the Fried parameter. The full theory gives

r0 = 0.1

[λ

0.5µm

]6/5m,

where r0=10cm is a typical value, but varies from one site to another and from night to night. The


Figure 16: Phase change caused by the tilt of a wavefront. If a section of a wavefront is tilted at a smallangle θ, the path difference caused at a distance x is h = xθ. At location A, the path difference is less than awavelength, whereas at the more distant pointB the path difference is several times as large as the wavelength.Waves arriving at points A and B are thereforely severely out of phase.

seeing is then

θ = λ/r0 = 1′′ × λ−1/50.5 ,

where λ0.5 is the wavelength in units of 0.5µm . The expected seeing in the infrared is thereforemoderately better than in the optical, and the scale length over which infra-red light remains coherentis substantially better.

4.4 Atmospheric wavefront bending in radio astronomy

In section 1.1 we described how refraction in the ionosphere can be so strong that it causes radiowaves of low enough frequency to be completely reflected from ionised layers. At frequencies a littleabove the critical frequency, radio waves are transmitted but still suffer substantial refraction, so thatthe positions of sources can swing around, in a manner analogous to optical seeing. However theeffect can be much larger - of the order of several arcminutes.

5 Correcting seeing : adaptive optics

Is there a way we can minimise the blurring caused by turbulence ? There are several techniques.

5.1 Control the environment

Turbulence and other effects inside the telescope dome can be as bad as upper atmosphere turbulence,but can at least be minimised by keeping the dome air at a constant temperature, and by flushing airthrough the dome in a laminar manner. In the last two decades, simple measures such as this haveimproved the quality of astronomical pictures greatly.


5.2 Shift and add

If we have a bright object, we can take exposures fast enough to “freeze” the speckles. The turbulentcells high in the atmosphere are moving quite fast, typically v ∼ 10 m s−1. The timescale onwhich they move is therefore roughly r0/v ∼ 10 msec, so an object needs to be bright enough tobe detected in that short a time. This is almost never enough time to capture a detailed (high signal-to-noise) image, but can be enough to measure the centroid of an image. In software one can thenre-centre all the images and gradually stack up an exposure. Alternatively one can keep the goodimages and throw away the bad ones (”lucky imaging”). A related more sophisticated method isto Fourier Transform the image time series, apply a high pass filter, and transform back. All thesetechniques are successfully used on bright objects, but can’t be applied to very faint ones.

A variant of the shift-and-add technique is possible in low frequency radio astronomy. Because phasesensitive detection is possible (see Chapter 4), erratic phase shifts caused by the ionosphere can bedirectly measured in recorded data and corrected for.

5.3 Tip-tilt correction

Even if the target object is faint, there may be a bright star nearby which one can monitor for imagemovement. Then one can move the secondary mirror of the telescope in two co-ordinates (tip and tilt)to compensate, physically moving the image on the focal plane. As well as correcting for the first-order distortion of the incoming wavefronts, this technique corrects for other image motion problems,such as wind-shake.

5.4 Full wavefront correction

As described above, the turbulence produces crinkled wavefronts, with each short section focusingto a slightly different spot. The idea of adaptive optics is to take the light from a nearby bright guidestar, detect its light before focusing, calculate the crinkling, and somehow unbend the light from thereal target in an inverse fashion.

One method is for the wavefront sensor to consist of an array of lenslets. A plane wave arriving atthis lenslet array would then be imaged as a series of spots in a regular grid. A crinkled wavefrontwill produce a distorted grid. In the light path for the target is a wavefront corrector which consistsof a deformable mirror in an array of tiltable sections. The sections need to tilt in the manner whichwould bring the wavefront sensor spots from the guide star back to the regular grid positions.

As with the shift-and add technique, the corrections have to be applied very rapidly, on msec timescales,because the distortions are constantly changing; however, only the guide star, rather than the muchfainter target, needs to be measured on these fast timescales. For a full correction, one needs to con-sider changes in the atmosphere down to scale sizes of the order of r0. In the infrared, r0 is muchbigger; full AO corrections are therefore easier to apply in the infrared.

A problem is that the guide star has to be close on the sky to the target object, so that its light issampling approximately the same path length through the atmosphere, and so the same turbulentcells. This is referred to as the isoplanatic patch and is of the order an arcminute across. Brightenough guide stars are relatively rare, so that in practice full AO correction is limited to∼ 10% of thesky. An alternative approach is to make an artificial guide star using a laser resonating at a specificheight. This however is expensive - it is labour intensive because of the related safety issues.


Figure 17: An Adaptive Optics (AO) correction system.

6 Getting above the atmosphere : space astronomy

Some types of astronomy - X-ray astronomy, far-IR astronomy - can only be done from space, as thelight concerned is absorbed by the atmosphere. As we have seen however, the atmosphere producesother problems - refraction, extinction, seeing, and background light. Rather than work so hard tryingto get round the problems of the atmosphere, why not avoid them by flying above it ? We will firstlook briefly at the pros and cons of space astronomy, and then get a feeling for our surroundings andhow the environment varies. Appendix B provides a little background in orbital mechanics.

6.1 Advantages of space-based astronomy

There are several very important advantages of carrying out astronomy from space.

(i) No atmospheric blurring. HST can get images of resolution 0.1′′, whereas even good observingsites have natural seeing almost ten times worse.

(ii) No atmospheric absorption. Some wavelengths – X-rays, UV – cannot be done at all from theground.

(iii) Much lower sky background. This can make detecting very faint objects much faster, as we sawin Chapter 1. Because images are sharper, this advantage is even stronger – the issue is the level ofbackground under a point source. The area under a 0.1′′ image is 100 times less than under a 1′′image.

(iv) No phase scrambling. As we will discuss in Chapter 3, optical interferometry on the ground islimited by the fact that wavefronts even a short distance apart are in phase for only a very short periodof time.

6.2 Disadvantages of space-based astronomy

The list of disadvantages or difficulties for astronomy in space is unfortunately quite long.


(i) Launching things is expensive, especially heavy things. The price quoted by various space agen-cies is usually in the range $10,000–$25,000 per kg for a launch to Low Earth Orbit (LEO). For a10 tonne spacecraft like the Hubble Space Telescope that is of the order $100M. This limits the sizeof space observatories. For example the HST has a diameter of 2.4m, whereas on the ground webuild 10m telescopes. The successor to the HST, the James Webb Space Telescope (JWST) is aimingto deploy a bigger mirror by using a design that is folded up on the launchpad and is unfolded to alarger size once in space; but this doesn’t alter the fundamental weight limitation set by the size ofthe payload bay.

(ii) Building things to space quality is expensive. Space instrumentation has to be super-reliablebecause you can’t fiddle with it once its up there; and it has to survive a rather severe radiationenvironment, as discussed in section 2.3.

(iii) Space missions have a limited lifetime. Sometimes this is because things gradually break down;sometimes its because you run out of expendable supplies, like manoeuvring gas or coolant; andsometimes its because you run out of money for the expensive operations - i.e. the money to pay forpeople to run the spacecraft.

(iv) You can’t alter things, repair instruments, or upgrade parts - or at least only if you go to verylarge extra expense, as was done with HST and the Space Shuttle.

(v) The space environment is not completely benign. As well as the energetic particles discussed insection 2.3, which can damage instruments and cause fake photon detections, thermal effects can bedifficult to handle. The side of a spacecraft facing the Sun gets much hotter than the opposite side,producing thermal stresses. Even Earth-heating is significant.

(vi) Spacecraft have to be provided with power. This is normally done using solar panels facing theSun. Achieving this while avoiding thermal stresses and keeping the spacecraft small is very tricky.

(vii) The bandwidth for communication with spacecraft is limited. For the cheapest orbits - those atlow altitudes - a given ground station can only see the spacecraft for a small fraction of the orbit - therest of the time it is below the horizon.

The various disadvantages are not the same for all orbits - some are much more expensive thanothers, some are easier for communication purposes, the particle background varies enormously, andthe thermal issues also vary considerably. In the final section we look at different types of orbit,within the context of the Earth’s surroundings.

6.3 Types of orbit and the Earth’s surroundings

Fig. 18 shows a schematic summary of the space environment surrounding the Earth, roughly to scale.The average radius of the Earth is RE = 6371km. For an orbital height below about 200km, orbitaldecay due to atmospheric drag is substantial. A typical Low Earth Orbit (LEO), such as used by atelecommunications satellite, is more typically at a height of h ∼ 500km. This barely skims over thesurface of the Earth, at a radius of r = 1.08RE . The main disadvantages of a LEO are therefore thatthe spacecraft can only be seen by a single ground station for a fraction of the orbit, and atmosphericdrag is still not negligible. Getting to a higher orbit has a higher energy cost. However, one canachieve a greater maximum height for a given energy by using an elliptical orbit. The maximumadvantage is a factor two in height (see Appendix B). Orbits are classified as “LEO” up to heights ofabout 2000km.

Above this height, we hit a new problem - the radiation belts discussed in section 2.3. The innerradiation belt, the proton belt, varies in height from h = 1000− 5000km, i.e. roughly r = 1.2RE −1.8RE . The outer, electron, belt is typically at r = 3.3RE − 4.9RE . Intermediate Circular Orbits orMedium Earth Orbits (MEO) therefore tend to aim to be between the two belts.


Figure 18: A schematic look at the space environment. L1 and L2 label the Earth-Sun Lagrangian points -the Earth-Moon Lagrangian points are not shown.

Continuing outwards, to reach a geostationary orbit, where the orbital period is the same as therotation period of the Earth, requires a radial distance of r = 6.65RE . This has the significantadvantage that a spacecraft remains in the same location relative to the Earth’s surface, in permanentcontact with a single ground station. More generally, an elliptical orbit of a similar height can begeosynchronous, oscillating around a fixed point.

A little bit further still is the distance of the magnetopause, marking the boundary within which theEarth’s magnetic field dominates the behaviour of incoming solar wind particles. This can changedramatically from day to day or even hour to hour, and is far from spherical, but is typically atr = 10RE . Outside the magnetopause, the radiation environment is much worse. In the anti-sundirection, the protected region stretches in a long “magnetotail” to a very large distance, making thisa desirable location for some spacecraft.

Continuing outwards again we reach the orbit of the Moon around the Earth, at r = 61.34RE . As aspacecraft approaches the Moon, there are key points where the gravitational pulls of the Earth andMoon partially cancel. If you map out the net gravitational potential of the two bodies, there are five“Lagrangian points” where the potential has a local minimum. The simplest points to understand arethose on the straight line joining the two bodies. The first two Lagrangian points, L1 and L2, are60,000km either side of the Moon, at r = 51.92RE and r = 70.75RE . The third point, L3, is in theopposite direction from the Moon.

As we travel further out still, there are also Sun-Earth Lagrangian points. The first two points, L1 andL2 are each at r = 236.44RE , one on the line towards the Sun, and the other on the line away fromthe Sun. (See Fig. 18.) L3 is on the far side of the Sun. Stationing a spacecraft at L2 can be verydesirable, as it is in the shadow of the Earth, shielded from most of the heating effect of the Sun. Thismakes thermal control much simpler, and makes it easier to achieve low temperatures, desirable forIR astronomy, with passive cooling. The spacecraft is also protected from much of the solar wind.On the other hand, L1 is an excellent place to study the interaction of the Sun-Earth system. TheLagrangian points track the orbit of the Earth around the Sun, remaining on the Sun-Earth line, so areeffective sun-synchronous orbits. The Lagrangian points are unstable equilibria, so the usual practiceis to design an orbit which loops around the point in a Lissajous pattern.

Space exploration involves a variety of other orbits, but the types above - LEO, MEO, elliptical,geosynchronous, and sun-synchronous - are the most important for astronomy.

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