{ Practical astronomy | Astronomy | The Earth's atmosphere }

# The Earth's atmosphere

Subsections:

A planet's atmosphere is a layer of gas bound to the planet by the weight of the gas in the gravitational field of the planet. For the Earth and other rocky planets this layer is thin compared to the radius of the planet – for the Earth perhaps 10 or 100 km compared to 6400 km. The gravitational field is nearly constant at all layers of the atmosphere.

## The height profiles of pressure and density

Consider the picture of a hillside and a small parcel of air above it. The air in the parcel, as an (almost) ideal gas is characterised by its temperature T, particle density n and the equation of state for an ideal gas:

p = n k T
k = 1.380648 · 10−23 J/K

p: pressure
n: number density (particles per unit volume)
k: Boltzmann constant
T: temperature (above absolute zero)

The gas particles have an average mass μ, and so the gas has a mass density ρ = μ·n. In those terms the equation of state reads

p = ρ kT / μ
μ = 28.966 amu = 4.810 · 10−26 kg

ρ: mass density (mass per unit volume)
μ: average mass of a particle (molecule or atom) of air

The sky above a hillside and an imaginary parcel of air.

Our parcel of air has a small horizontal cross section dA and a small vertical extent dz. Combined they make a small volume dV = dA·dz. The parcel then has the small mass dm and "feels" a small gravitational force:

dm = ρ dA dz
dFg = g dm = g μ n dA dz
g = 9.80665 m/s2

g: acceleration of any mass in
the Earth's gravitational field

Our parcel of air cannot fall to the ground because of the air below it. Instead, our parcel of air puts external pressure on the air below. This is similar to a block of cheese lying on a table; rather than fall through the table to the ground, it puts pressure on the table.

In this view pressure is simply the weight g·m of the cheese or parcel of air divided by the contact area dA. Although we initially pictured the air parcel having a single pressure value p, we have to acknowledge, that due to its own weight, the pressure at its bottom is a little higher than at its top.

dp = −g dm / dA = −g ρ dz

We put a minus sign, because we measure z upward from the Earth's surface; while the pressure rises downward. This equation captures only the gravity. We have to fold into this the gas equation of state, i.e. replace ρ with p and T:

dp = −p (g μ/kT) dz

This is a differential equation in the variable p (pressure): As we move up from the surface to higher z values the pressure decreases (dp/dz is negative). The rate of decrease is not constant, however, but proportional to the pressure p itself. As differential equations go, this is a very simple one, and one that occurs in physics all the time. The solution is an exponential decline:

p = p0 · exp(−z/z0)
z0 = kT / (g μ)

z0: scale height
p0: pressure at the surface

We inadvertently made an approximation, however: We assumed a constant temperature T at all altitudes z. So long as we do this, we might note that the density profile is virtually the same, due to the equation of state:

ρ = ρ0 · exp(−z/z0)

ρ0: density at the surface

The scale height and surface pressure are difficult to calculate from first principles. Better to measure them.

Pressure versus time while walking up a hill.

Return to the picture of the hillside above. Using an app on my Android smart phone, I can monitor the air pressure while walking from the bottom of the hill up to the observatory. The figure shows the time series during the five-minute hike. From a good topographical map I can read the elevation contours to learn that I started and ended at z = 82.5 m and 122.5 m resp. above sea level. The pressure is measured as p = 101795 Pa and 101311 Pa resp. The scatter in the measurements is about 5 Pa.

The pressure change is only 0.5%, which justifies us to consider dz and z itself as small and to simply extrapolate to sea level.

dp/dz = −484/40 Pa/m = −12.1 Pa/m

p0 = (101795 + 12.1 · 82.5 Pa) = 102800 Pa

There is a weather station at the observatory; checking the Met Office website confirms the air pressure (corrected to sea level) on that day as 102900 Pa.

Since we know the Earth's gravitational acceleration g, we can calculate the gas density from the pressure gradient:

–12.1 N/m3 = dp/dz = −g ρ

ρ0 = 12.1 N/m3 / g = (12.1 / 9.80665) kg/m3 = 1.234 kg/m3

And we can calculate the scale height (using the average pressure from our hike):

dp/dz = −p (g μ/kT) = −p / z0

z0 = −p / (dp/dz) = 101553 Pa / (12.1 Pa/m) = 8390 m

If I had measured the temperature, we could even calculate the particle mass μ. The Met Office can help us out, they measured a temperature at the time of 15 °C, which is about 288 K.

z0 = kT / (g μ)

μ = kT / (g z0)
= 1.380648 · 10−23 kg · 288 / (9.80665 · 8390) = 4.83 · 10−26 kg

less than a percent off the correct value.

Gravity reduces only by a few percent towards high altitudes, which has little effect on the pressure and density profiles. The temperature changes significantly with altitude, but the significance is more for radiation, turbulence and chemistry than for the pressure and density profile. If we use T ≈ 261 K (about −12 °C), then the scale height comes out correctly for the actual atmosphere

Pressure profile after Kraus (1980). Pressure is drawn to the right, on a logarithmic scale.

Density profile after Kraus (1980). Density is drawn to the right, on a logarithmic scale.

p = p0 · exp(−z/z0)
p0 = 101325 Pa
z0 = 7640 m

The same scale height applies to the density profile, but for the value at ground level we better use a temperature of 288.15 K (+15 °C):

ρ = ρ0 · exp(−z/z0)
ρ0 = p0 μ / (kT) = 1.225 kg / m3

The two graphs show the profiles of pressure and density versus altitude. The convention for altitude profiles is to draw the free variable altitude upwards and the measured quantity to the right. Further, where a quantity changes by orders of magnitude, it is common to draw its decadic logarithm rather than its "linear" value. This shows relative changes equally well at all values.

Often the logarithmic scaling also makes the plot linear: Up to altitude 100 or 120 km the blue lines are approximately straight lines. The slope of this line represents the scale height z0. Both profiles – pressure and density – show very similar slopes, because they are both governed approximately by the same scale height.

### The column density

The pressure at the bottom of the atmosphere, at sea level, is simply the result of the weight per unit area of the column of air above. The base pressure is well known to be 101325 N/m2. The value of gravitational acceleration g tells us that each weight of 9.80665 N corresponds to a mass of 1 kg, hence the column density of the atmosphere is 10330 kg/m2. While we stand upright, our body has a horizontal cross section of perhaps 0.5 m2 so that there is a weight of 5 tons of air pushing down on us!

We think of the atmosphere as a thin and fragile shield against the evils of space, namely high-energy photons and particles that the universe might throw at us. Indeed, in the ocean, where water has a density of about 1000 kg/m3, the same column density is added by going just 10.33 m below the surface.

But we can also argue that the atmosphere is a very thick layer – in comparison to the space beyond. The interstellar medium in a galaxy has a density of only 1,000,000 hydrogen atoms per cubic metre, or 1.660539 · 10−21 kg/m3. With such a low density, a column of 10330 kg/m2 would have to be very long indeed, much longer than the diameter of a galaxy.

## Chemical composition

The Earth's atmosphere is mostly molecular nitrogen (N2, 78% "by volume", meaning by number of particles) and molecular oxygen (O2, 21%). The gas equation of state applies not only to the gas as a whole, but separately to each constituent species of molecules or atoms.

pi = ni k T
i = N2, O2, CO2, CH4, H2O, etc.

All species have a common temperature, so that for each species pressure and density are proportional and so that the sum of all partial pressures pi and the sum of all particle densities ni add up to the overall pressure and density. Apart from water vapour and the overall small concentration of ozone, the composition is the same from the ground to 100 km altitude; we speak of "dry air" for excluding H2O from the list (Kraus 1980, p.12):

• 78.1% N2, molecular nitrogen, 28.016 amu.
• 20.9% O2, molecular oxygen, 32.000 amu.
• 0.93% Ar, atomic Argon, atomic mass 39.944.
• 0.041% CO2 (Dlugokencky and Tans 2017), carbon dioxide, 44.010 amu.
• 0.0018% Ne, atomic Neon, 20.182 amu.
• 0.00052% He, atomic Helium, 4.003 amu.
• 0.00018% CH4, methane, 16.0 amu.
• ...

The average particle mass of dry air is 28.966 amu.

## The temperature profile

Temperature profile after Kraus (1980). Temperature is drawn to the right.

The temperature of the atmosphere at various altitudes depends on absorption of sunlight at each altitude. At the top, the atmosphere is very thin and easily heated by sunlight, leading to high temperatures. Further down, there are two layers where a great deal of absorption and heating occurs. First is the ozone layer, where the density of the atmosphere and of its molecular oxygen is sufficient that it absorbs much of the UV light from the Sun and forms ozone. Second is the ground, where most of the solar radiation is absorbed.

As a result, the temperature profile has two distinct local maxima or warmer layers. One is at the ground, from where the bottom layer of the atmosphere is heated; the other is the ozone layer. In addition, the temperature rises at the highest altitudes with altitude. Between these three warm or hot layers are two cold layers. One is at 10 to 20 km altitude, the other at 80 to 90 km altitude. This profile is the basis of dividing the atmosphere into layers with names ending in "sphere". The upper boundaries have names ending with "pause".

Temperature, pressure, density and the precise chemical composition of the atmosphere differ depending on altitude, but also depending on geographic latitude, time of year, time of day, etc. Meteorologists have invented the standard atmosphere, in particular the 1976 US Standard Atmosphere (NOAA et al. 1976), to get a simple handle on the overall average properties of the atmosphere. The standard atmosphere gives us altitude profiles of various quantities, in particular temperature, but also pressure, density and number density. The following table after Kraus (1980, p.99) shows for the three main layers the points where the temperature profile changes slope. In the thermosphere the temperature gradient is not constant; rather the temperature rises to an asymptotic value of 1000 K.

US Standard Atmosphere 1976.
z
km
T
K
p
Pa
ρ
kg/m3
n
m−3
layer
1000.00 1000.00 7.51 · 10−9 3.56 · 10−15 5.44 · 1011 thermosphere
500.00 999.24 3.02 · 10−7 5.22 · 10−13 2.19 · 1013
200.00 854.56 8.47 · 10−5 2.54 · 10−10 7.18 · 1015
100.00 195.08 3.20 · 10−2 5.60 · 10−7 1.19 · 1019
91.00 186.87 0.154 2.86 · 10−6 5.96 · 1019
86.00 186.87 0.373 6.96 · 10−6 1.45 · 1020 mesosphere
71.80 214.65 3.96   6.42 · 10−5 1.34 · 1021
51.41 270.65 66.9     8.62 · 10−4 1.79 · 1022
47.35 270.65 111        1.43 · 10−3 2.97 · 1022 stratosphere
32.16 228.65 868        1.32 · 10−2 2.75 · 1023
20.06 216.65 5475        8.80 · 10−2 1.83 · 1024
11.02 216.65 22600        0.364   7.57 · 1024 troposphere
0.00 288.15 101300        1.225   2.55 · 1025

Looking briefly back at the column density and comparison with the ocean and intergalactic space, observe that even at the altitudes of artificial satellites (a few hundred km) the particle density is still 1015 or 1014 molecules per m3, compared to 106 in interstellar space.

## The troposphere

### Water

Phase diagram of water, after Grimm (2016) and Kraus (1980). The green curve represents the Standard Atmosphere (total pressure, not partial pressure of water).

Water exists in significant amounts (up to 4%) only in the troposphere. This can be in any phase: gas (also called vapour or steam), fluid (such as drops and droplets, also lakes and oceans below the atmosphere), or ice (snow flakes or hail stones). Note that you cannot see steam, what emerges visibly from a steam locomotive is (i) smoke from burnt coal and (ii) droplets condensed from the steam that drives the engine.

The molecular mass of water is 18.015 amu. Since this is less than the average particle mass of dry air (28.966 amu), the more water vapour is contained in the air, the less dense and lighter it.

At this point we should study the phase diagram of water. This plots temperature and pressure on the two axes and divides this parameter space into areas where water can exist in one particular phase and is not inclined to change phase. If p and T lie on a boundary curve between two of these three areas, then those two phases usually co-exist.

There is an important difference between the gas phase on one hand and the liquid and solid phases. The gas phase functions independent of the other species in the air, so that the partial pressure pH2O of the water is the important quantity. But the liquid and solid phases feel the whole air pressure as an external pressure on the surface facing the air. You can see this every day, when a kettle boils at 100 °C, which is the point on the liquid/gas boundary curve where the pressure reaches 101300 Pa.

A few examples should help:

• At the base of the Standard Atmosphere we have p = 101300 Pa and T = 288 K. This is firmly in the liquid area of the phase diagram. We should not be surprised that lakes and oceans exist and do not disappear into thin air.
• At 288 K the steam curve – the boundary curve between steam below and liquid or ice above – reads a pressure of 1700 Pa.
• Recall that vapour is not subject to the air pressure only to the partial pressure of the vapour. This means that at 288 K any lake is expected to "build up steam" in the air above it to the point where its partial pressure reaches 1700 Pa. This is 1.7% of the total air pressure at sea level. Hence the air immediately above the lake – cross winds permitting – should contain 1.7% of water vapour.
• More water vapour is not possible: This would put the water vapour above the steam curve and demand condensation into liquid water. The steam curve defines the saturation pressure as a function of temperature.
• If we think that temperatures of 303 K (about 30 °C) are reasonably common and that its saturation pressure is 4240 Pa (4% of air pressure at sea level), then we understand why at best only 4% water vapour content can exist in the atmosphere. The colder the air, the less water content is possible without water condensing or freezing out.
• Consider liquid water in a kettle. Starting out at 20 °C, the air in the kettle may hold 2340 Pa of water gas. As the water is heated, this increases. However, the water feels the total air pressure and evaporation is slow. When the temperature reaches 100 °C, the water in the phase diagram moves across into steam territory even for the air pressure of 101300 Pa. The liquid water will now evaporate almost instantaneously. Tea and coffee drinkers call this "boiling the kettle".
• The phase diagram shown includes the US Standard Atmosphere, i.e. a curve representing the combination of its T and p data. The troposphere is at the top, firmly in the territories of liquid water and ice. Only in the lower troposphere can water be liquid. Above 2 km it is too cold and ice would form instead.
• The most common measure of humidity is to state the water vapour partial pressure as fraction of the saturation pressure. E.g. if the relative humidity is stated as 60% and the temperature is 10 °C (283.15 K, with a saturation pressure of 1230 Pa), then the actual water vapour partial pressure is 738 Pa. The percentages given are far higher than the fraction by volume of water in the atmosphere; in this case that would be as small as 0.73%. The relative humidity is more informative: when it approaches 100% vapour begins to fall out as liquid droplets or snow flakes.
• Another interesting measure of humidity is the dew point. This answers the question, at what (lower) temperature the current vapour partial pressure would be the saturation pressure. Continuing the previous example, 738 Pa would be saturation if the temperature dropped to 2.5 °C (275.65 K). As it gets colder in the evening, fog might form, because when the dew point is reached, condensation begins and dew forms.

### Stable and instable temperature gradients

Example of adiabatic lapse rate and temperature gradient of the surroundings (after Kraus 1980, p.123).

Stability here is the question, whether the upward motion of a rising parcel of air is counteracted by the vertical temperature profile of the surrounding air, or whether the upward motion is accelerated by the surroundings.

In the Standard Atmosphere, the general temperature gradient is −6.5 K/km. In reality, it can be steeper, shallower, or even inverted. An inversion is when at some altitude the temperature gradient is positive, i.e. warmer air is on top of colder air. These are ultra-stable conditions and make vertical movement of air parcels quite difficult.

Consider the "dry-adiabatic" rise of a parcel of air. Here, dry does not mean free of water vapour; it means that no condensation occurs and that any water vapour present simply remains in its gaseous form. Adiabatic means that the air parcel does not lose energy to the surroundings nor gains energy from it. As the parcel rises, it remains in pressure equilibrium with the surroundings, meaning that its pressure drops. By the gas law, the density must then also drop, the parcel must expand. This expansion requires energy, and the temperature of the parcel must drop to deliver that energy. It turns out that between the gravitational constant and the specific heat (energy required per unit mass and unit of temperature increase) of air, the parcel will cool by −10 K/km. This is the dry adiabatic lapse rate.

This is steeper than the surrounding's temperature average gradient. During the rise, the parcel therefore cools below the surroundings. Since its pressure is determined by the surroundings, its density must be higher than in the surroundings, by the gas law. As a result, the parcel has negative buoyancy and gravity will pull it back down, the temperature layering is stable, on average.

Also consider the moist-adiabatic rise of a parcel of air. By moist is meant that the temperature is at dew point and condensation is occurring. It is again the case that the parcel has to expand, but here some of the energy comes from the condensation of water. Less cooling is required and the moist adiabatic lapse rate is shallower than the dry rate.

The moist adiabatic lapse rate depends on pressure and temperature. It is steepest at high pressure and low temperature, reaching −8.55 K/km at 100000 Pa and −20 °C (253.15 K). At sea level and +20 °C (293.15 K) it is as shallow as −4.26 K/km.

It is then likely that the surrounding temperature gradient is between the dry and moist adiabatic lapse rates. Water-saturated air will then spontaneously move vertically while air warmer than the dew point (not saturated) will not do so. But it can also happen that the surrounding gradient is steeper even than the dry lapse rate and that all air parcels take up vertical speed spontaneously.

When unsaturated air rises it will first cool down by the dry lapse rate. But it will eventually reach its dew point, start to condensate, and rise even faster.

On the other hand, an inversion will always suppress vertical motion of air parcels.

These considerations are interesting, of course, because condensation at altitude means the formation of a cloud and perhaps the precipitation of water, snow or hail to the ground. Interesting and important as this is, the evaluation of circumstances is also quite complicated. This is not even taking into account local pressure differences (areas of low and high pressure), wind, sunshine, heat capacity of the ground (land or ocean).

Forecasting the weather is a very complex business.

The figure shows an example of dry and moist adiabatic rise of an air parcel. Drawn in blue is the temperature gradient of the Standard Atmosphere, +15 °C at sea level, −15 °C at about 4.6 km. The red line is the actual temperature profile on a summer day, with an inversion at height. Below the inversion layer, the T gradient is steeper than standard, particularly steep below 100 m altitude due to heating from the warm ground. A parcel of air can ascend from the surface along the green curve. Initially it helps that the surrounding air gradient is extra steep; the air parcel cools much slower than the surrounding temperature falls, propelling the parcel's ascent. Above 100 m, the parcel warms at the same rate as before, which is now the same as the surroundings get warmer, but this still means that the parcel has buoyancy. At 1 km, the water vapour partial pressure in the air parcel reaches saturation, in other words its temperature has dropped to the dew point. As the parcel rises and cools further, vapour must condense, a cloud forms. The condensation releases heat, so that the parcel cools more slowly than before and again more slowly than the surroundings; the parcel's buoyancy increases. In this example, at 2 km the T profile reverses in an inversion layer that accommodates the transition to generally warmer air above 2.5 km. In this example, the air parcel cools below the surroundings below the top of the inversion. The air parcel cannot rise further and the cloud extends only to this height.

### The tropopause

At the top of the troposphere, at the tropopause, the temperature begins to turn around and the temperature gradient becomes very shallow. As we have just seen, a shallow gradient or even inversion is a strong inhibitor of vertical exchange of air parcels. This makes it difficult in particular for water vapour to move up into the stratosphere. At the tropopause, the pressure is 22600 Pa and the temperature 216.65 K (−56.5 °C). From the phase diagram, the water saturation pressure then is 2 or 3 Pa and the highest possible vapour content is on the order of 0.01% by volume (by number of particles).

There is very little water content above the troposphere, and little that there is, is very unlikely to saturate and condense or sublime into snow or ice.

## The troposphere in photographs

### Crepuscular rays

Crepuscular rays are created by the irregular boundary between cloud and clear sky. The result is a difference in illumination further away from the Sun. With particles present to scatter light towards us, we can see this play of light and shadow even though we are not looking toward the sun. The "rays" converge on the Sun. Sometimes the rays can be seen converging on the anti-solar point (with the Sun behind the observer. They are then called anti-crepuscular rays.

The upper image shows the impossible. One day I flicked into a TV programme about clouds, just when the presenter showed a similar painting where a large cloud surrounded by clear sky produced the rays. The cloud expert said this was not possible, the rays were made by clear sky surrounded by cloud and not by a cloud surrounded by clear sky.

For further reading see Cowley (2019).

Crepuscular rays from a cloud edge, 2009-05-29.

Crepuscular rays near sunset, 2010-07-20.

Physical parameters:

• Altitude: ~1 km

As with most imaging of the sky during the day, the landscape will probably come out too dark; and there may be trouble getting the auto-focus and automatic exposure to co-operate. If the Sun is not obscured by cloud, try to place it outside the frame or to hide it behind an object that doesn't look too bad in silhouette – a house, tree, or road sign.

Image parameters (top):

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 55 mm
• Field of view: 23 × 16°
• Aperture: f/16
• ISO: 400
• Exposure: 2 ms
• Location: Edinburgh, Scotland

Image parameters (bottom):

• Camera: Panasonic Lumix DMC-TZ8
• Detector: 5.8 × 3.9 mm
• Focal length: 6.5 mm
• Field of view: 45 × 30°
• Aperture: f/3.6
• ISO: 100
• Exposure: 50 ms
• Location: Edinburgh, Scotland

### Contrail

Contrail is a made-up word meaning "condensation trail". It forms behind an aircraft, where water vapour may condense to liquid droplets and then likely freeze if the air is cold enough (above 2 or 3 km altitude it should be). The aircraft engine itself adds vapour to the atmosphere, but the turbulence caused by the aircraft may also trigger condensation of the vapour already in the air. Some distance behind the aircraft, the droplets or ice particles will evaporate again. The length of the contrail depends on how dry the atmosphere is and how quickly this therefore happens.

Contrails forming behind the engines of an aircraft, 2002-10-06.

Long contrails from the left and right wings of an aircraft, suffering wind shear rather than dispersal or evaporation. 2010-07-11.

Physical parameters:

• Altitude: about 5 to 10 km

Contrails tend to be high in the sky, and you will probably zoom in by a certain amount rather than go for the widest field of view. The image will then only contain clear sky and the contrail, so that automatic exposure should be fine. Auto-focus should not cause too much of a problem, as the contrail gives significant contrast on small scales. Imaging the aircraft in any detail will require a very high zoom factor of 10 to 20 over the widest field of view. Even then, you will probably crop away much of the resulting frame in later processing.

Image parameters (top):

• Camera: Sony Mavica MVC-FD88
• Detector: 5 × 4 mm
• Focal length: 43 mm
• Field of view: 7 × 5°
• ISO: 100
• Location: Glen Lyon, Scotland

Image parameters (bottom):

• Camera: Panasonic Lumix DMC-TZ8
• Detector: 5.8 × 3.9 mm
• Focal length: 32 mm
• Field of view: 9 × 6°
• Aperture: f/6.3
• ISO: 80
• Exposure: 1.2 ms
• Location: Muiravonside, Scotland

### Belt of Venus

The "belt of Venus" is the fancy name for a quite common phenomenon. Shortly after sunset or before sunrise, look in the opposite direction. The horizon should be noticeably darker than the sky some distance above. This is simply because the atmosphere low on the horizon is in the Earth's shadow, while higher above the horizon it is in sunlight. The boundary layer can have a pink colour. This is sunlight that has passed over us and is scattered back. It therefore has had a very long path through the atmosphere and is deeply reddened.

For further reading see Cowley (2019).

The Earth's shadow to the left, day side to the right. 2010-12-08.

Physical parameters:

• Altitude: ~5 km

The twilight sky is in general very easy to photograph, except that there is little contrast for auto-focus to make use of. It may be a good idea to take the image not directly away from the Sun, but at somewhat over 90° away from the Sun. The boundary between light and shadow should then be inclined, making it more recognisable.

Image parameters:

• Camera: Panasonic Lumix DMC-TZ8
• Detector: 5.8 × 3.9 mm
• Focal length: 4 mm
• Field of view: 70 × 50°
• Aperture: f/3.3
• ISO: 100
• Exposure: 0.1 s
• Location: Edinburgh, Scotland

### Rainbow

To observe a rainbow, turn away from the Sun. The rainbow is a circle of 42.5° radius around the anti-solar point. It is vital to have drops of water in the air to see the rainbow, for it is these water drops that reflect the sunlight towards you. It is also vital that the Sun shines on those drops. The proverbial "colours of the rainbow" come about, because the different colours that make up white sunlight are reflected slightly differently. You therefore see each colour at a slightly different angular distance from the anti-solar point. Rainbows are best near sunset or sunrise, because then the Sun is low on the horizon, the anti-solar point is not far below the horizon, and the top of the rainbow is high in the sky.

If what the raindrops do were a simple reflection, there would be no colours. The reflection occurs at the back of the raindrop; sunlight enters the sphere of water (the drop), some of it does not leave at the back but is reflected back into the interior of the drop, and emerges heading in a direction similar to where the original light came from. The two passages from air into water and from water into air involve refraction, and that is what is colour-dependent and separates the colours.

The rainbow is not just a ring of light. What we see is the bright rim of a disc of light. In strong rainbows, you can see that the interior is brighter than the exterior.

There can be a second rainbow outside the primary one. It has a radius of 51°, 9° more than the primary. Again, this is a bright rim of an area of light. But is is the inner rim of that area, meaning the area between the two bows is darker than the area outside the secondary. You could say, the radius is really −51° or 129° from the Sun. It is then perhaps not surprising that the colours of the secondary are ordered opposite to the primary. The primary has red on the outside, the secondary has red on the inside.

The secondary rainbow is due to light that is reflected twice in the drop. Most light leaves the drop after the first reflection, but some goes through a second reflection within the drop and exist in a different direction. Because most light does not take this path, the secondary rainbow is much fainter than the primary.

For further reading see Cowley (2019).

Primary and secondary rainbow, 2006-09-12.

Primary rainbow, 2011-05-12.

Primary and secondary rainbow, 2011-06-18.

Physical parameters:

• Distance: ~1 km
• Apparent radius of primary: 42.5°
• Apparent radius of secondary: 51°
• Colour order in primary: red outside
• Colour order in secondary: blue outside

The biggest problem in imaging rainbows is that they last only a few minutes. You have to have a camera in easy reach and ready to shoot within a minute or so. Usually, you will use the widest field available, but the whole rainbow will not fit. As with many daytime objects in the sky, the landscape below tends to be recorded very dark and with little contrast. In later processing, one tends to make this worse when trying to increase the contrast between the rainbow and its sky background. When taking the image, automatic focus may be difficult on the sky; focusing (and adjusting exposure) on the horizon before re-framing higher up may cause overexposure.

Image parameters (top):

• Camera: Handspring Treo 650
• Field of view: 40 × 30°
• Location: Edinburgh, Scotland

Image parameters (middle):

• Camera: Panasonic Lumix DMC-TZ8
• Detector: 5.8 × 3.9 mm
• Focal length: 4.4 mm
• Field of view: 65 × 45°
• Aperture: f/4
• ISO: 100
• Exposure: 2.5 ms

Image parameters (bottom):

• Camera: Panasonic Lumix DMC-TZ8
• Detector: 5.8 × 3.9 mm
• Focal length: 4.1 mm
• Field of view: 70 × 50°
• Aperture: f/3.3
• ISO: 100
• Exposure: 13 ms
• Location: Heilshorn, Germany

As can be seen, including the whole rainbow is a challenge and would require an extreme wide angle lens.

### Glory

A glory forms similar to a rainbow and is centred on the anti-solar point. However, the water drops are much smaller, making the idea of refraction and reflection too simplistic. For light that enters the drop grazing on its periphery, entry may be delayed by one or two wavelengths. The same can happen in the internal reflection at the back and on egress. For the large water drops involved in a rainbow, the wavelength of the light is minuscule. But when the droplets are only 20 μm or less, a wavelength of 0.5 μm is no longer negligible.

Such small droplets are not all that uncommon. Clouds – unless frozen – are full of them, because all large drops have dropped out of the cloud. (I don't think this is a pun!)

A glory is quite commonly seen from an aircraft looking down on cloud. The glory diameter in the upper image is about 3°, implying a droplet size of about 20 μm. The size of the glory is inversely proportional to the size of the droplets. In the lower image, the glory is about 40% larger, hinting at droplets of only 15 μm diameter.

For further reading see Cowley (2019).

Glory, 2006-01-26 above Edinburgh.

Glory, 2006-01-26 above Amsterdam.

Physical parameters:

• Distance: ~1 km
• Apparent radius: ~1.5° or more

Photographing a glory is not difficult, provided you have access to your camera in the aircraft cabin. The cloud has to be low enough to be liquid. Above a certain altitude, the air temperature is below 0 °C and the clouds are ice.

Image parameters:

• Camera: Handspring Treo 650
• Field of view: 40 × 30°
• Location: Edinburgh, Scotland; Amsterdam, Netherlands

### Iridescent cloud

Irisation is caused by similar particles that cause glories (small droplets), but here we look close to the Sun rather than at the anti-solar point. The affected parts of the cloud can be much brighter and can show subtle colour.

For further reading see Cowley (2019).

Iridescent cloud, 2007-10-04.

Physical parameters:

• Altitude: ~1 km

The landscape will most likely come out too dark, even the blue sky and regular cloud may appear too dark so as to avoid overexposing the coloured irisation. There may be trouble getting the auto-focus and automatic exposure to co-operate. Hide the Sun behind an object that doesn't look too bad in silhouette – a house, tree, or road sign.

Image parameters:

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 55 mm
• Field of view: 23 × 16°
• Aperture: f/18
• ISO: 100
• Exposure: 1.2 ms
• Location: Edinburgh, Scotland

### Halo and sun dogs

Somewhere around 2 km altitude the temperature drops below the freezing point of water. Many clouds are higher than this and therefore do not contain drops of liquid water but particles of frozen water. The optical phenomena caused by ice crystals are very different from those of spherical drops or droplets of liquid water. There is a complex system of bright circles, arc and rays, coloured or white, that come under the term of halos.

Most common is the circular halo with a radius of 22° around the Sun. The sun dogs are also quite common. They are brighter areas on the circular halo to the left and right of the Sun. Sun dogs often show colours.

The Moon can also produce the 22° circular halo.

For further reading see Cowley (2019).

Circular halo, 2004-11-20.

Physical parameters:

• Altitude: about 5 to 10 km

As with most imaging of the sky during the day, the landscape will probably come out too dark; and there may be trouble getting the auto-focus and automatic exposure to co-operate. Hide the Sun behind an object that doesn't look too bad in silhouette – a house, tree, or road sign.

Image parameters:

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 18 mm
• Field of view: 65 × 45°
• Aperture: f/16
• ISO: 400
• Exposure: 1.6 ms
• Location: Edinburgh, Scotland

### Lightning

Lightning is a discharge between a negatively charged bottom layer of cloud and the ground. It is not entirely clear how charges are separated in clouds into positively charged tops and negatively charged bottoms. It will have to do with the presence of liquid water and of ice, with the tallness of thunderclouds, with the upward flow of moist air and the downward fall of ice particles and liquid drops. Cloud particles will collide and may transfer electric charge in the process.

The electric field between the cloud bottom and ground is never sufficient for direct discharge. Rather, the path of the lightning to be becomes ionised by collisions of cosmic rays with air molecules. The ionised channel then carries the current.

The thunder is generated when the current stops. While the current lasts, its magnetic field confines the hot plasma. When the magnetic field suddenly disappears the hot plasma expands and generates a shock wave – essentially a strong sound wave – into the surrounding air.

Lightning, 1983-08-20.

Physical parameters:

• Distance: ~1 km
• Altitude: ~1 km
• Electric field: 40 kV/m
• Voltage: 10 MV
• Current: 20 kA
• Temperature: 30,000 K

Image parameters:

• Camera: Praktica VLC2
• Film: 36 × 24 mm, Agfaortho 25
• Focal length: 50 mm
• Field of view: 40 × 25°
• Aperture: f/1.8
• ISO: 25
• Exposure: 5 to 60 s
• Location: Bonn, Germany

To image lightning, it is best to wait for an intense thunderstorm nearby and at night. The darkness permits long exposure times and increases the contrast between the brief flash of lightning and the accumulated brightness of the quiet sky. A minimal ISO setting will further help lengthen the exposures without overexposing the images. A good tactic is to use "bulb" exposure, to open the shutter at some time and to leave it open until a lightning has occurred, then to close the shutter to avoid unnecessary exposure of the background.

## The stratosphere

The stratosphere is defined as the layer above the troposphere where the temperature rises with height. In the standard atmosphere it extends from 11 to 50 km altitude. A significant part of the sunlight coming from beyond the atmosphere is absorbed at the bottom of the atmosphere, in the ozone layer (15 to 35 km altitude). The absorbed UV radiation causes the production of ozone (O3) from regular molecular oxygen O2. The ozone content is quantitatively small at only 10 parts per million, but this is high compared the content of 0.3 ppm elsewhere. Further, the absorption of sunlight is what heats the stratosphere and makes it warmer than the troposphere below.

There is not much to see or photograph in the stratosphere. Rarely, during twilight, there may be nacreous clouds, which appear very bright, smooth and still.

For further reading see Cowley (2019).

## The mesosphere

The mesosphere is defined as the layer above the stratosphere where the temperature decreases again with altitude. This makes the mesopause at 85 km the coldest layer at any altitude. Ironically, the seasonal temperature changes at this altitude are such that the coldest period is in summer.

### Noctilucent clouds

Noctilucent clouds (NLC) can occur where and when it is coldest in the atmosphere. It is not quite clear why small amounts of water vapour come to exist at this altitude. Nor is it quite clear what seed particles are responsible for the vapour's deposition into ice crystals. What is clear, is that this can occur only when this coldest layer of air reaches a seasonal low in temperature far below the annual average of −85 °C. Contrary to intuition, this happens in summer and not in winter. Some years it may not get cold enough at all.

The clouds that form some nights of most summers are so thin that they cannot be seen in a daytime sky. They could also not be seen without sunlight falling on them. These noctilucent clouds can then only be seen from certain geographical latitudes where the ground level is in twilight while the mesosphere toward the North is still bathed in sunlight. The clouds are then seen by the forward scattering of sunlight by the ice crystals. The stripy or stringy appearance of these clouds is due to how the crystals form and dissolve. Nucleation particles at higher level drift down to an altitude where the water vapour can deposit. The forming ice crystals fall while they grow, and soon they have fallen through the mesosphere into warmer air where they sublimate again. Combine this with high-speed winds and you find small cloud-lets stretched to the extreme in a horizontal direction.

Noctilucent cloud, 2009-06-17/18.

Noctilucent cloud, 2006-07-12/13.

Physical parameters:

• Altitude: 84 km
• Air temperature: −135 °C
• Air pressure: 0.00003 Pa
• Water vapour: 5 ppm

The technique for taking pictures of NLC is similar to photography of landscape and large phenomena like rainbows or halos; a wide field of view is preferred. The time to observe is strictly limited to the interval when the Sun is between 4 and 16° below the horizon. This is dusk or dawn, when the brightness of the sky changes rapidly. Use a wide aperture, moderate ISO, and exposures between one and 30 s. The longest sensible exposure depends on how deep twilight it is. Very bright NLC can be much brighter than that and may require exposures at the shorter end of the range. A very dark site is not necessary; during much of twilight the light pollution has only a small effect.

Image parameters (top):

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 18 mm
• Field of view: 65 × 45°
• Aperture: f/3.5
• ISO: 200
• Exposure: 2 s
• Location: Edinburgh, Scotland

Image parameters:

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 18 mm
• Field of view: 65 × 45°
• Aperture: f/3.5
• ISO: 400
• Exposure: 4 s
• Location: Edinburgh, Scotland

Since 2009, I have been running an automatic camera during the NLC season from mid May to mid August, to record any occurrence of noctilucent clouds. The detailed results are available in an appendix.

NLC were first observed in the late 19th century, when the 1883 eruption of the Krakatoa volcano sparked interest in the atmosphere at twilight. Historical research indicates that the absence of reports before then is due to absence of NLC and not simply lack of interest. It is thought that the increasing occurrence of NLC may be an indicator of global warming since the industrial revolution of the 19th century.

### Meteors

A meteor is the streak of light we see at night when a meteoroid burns up in the Earth's atmosphere. A meteoroid is a small solid body in the solar system that collides with the Earth. As it descends into the atmosphere it begins to heat up the air in its path. The meteoroid itself also begins to evaporate, burn and fragment. The resulting meteoric dust can remain suspended in the atmosphere. If a sizeable body survives the process and falls to the ground, it is called a meteorite.

Meteoroids can be tiny rocks similar to asteroids or minor planets, or they can be dust of cometary origin. The meteor showers usually derive from comets. They recur every year when the Earth passes close to the comet's orbit. The dust has been blown off the comet by radiation pressure from the Sun during previous orbits.

Shower meteors can be recognised by their direction across the sky. When traced back across the sky, all meteors of a particular shower intersect in one position, the radiant.

Hourly count of Leonid meteors, corrected to zenithal hourly rate. 2002-11-18.

Perseid meteor to the right of Ursa Major, 2010-08-14.

Physical parameters:

• Altitude: 50 to 120 km
• Velocity: ~30 km/s
• Magnitude: −4 to +6

The eye has the clear advantage in detecting meteors, as it does not rely on integrating the constant light of, e.g., stars for seconds or minutes. A meteor moves very quickly across the sky. For a camera to record it at all, it has to be bright, say, 0 mag. Even then it will not look as impressive as it does to the human eye. Sporadic meteors are completely unpredictable and recorded only by chance. For shower meteors, one can pick a night close to the predicted maximum activity and hope to have a reasonable rate of meteors. Go for maximum ISO, open aperture and wide field. Use "bulb" exposure and close the shutter after a bright meteor has occurred. Many frames will have no meteors or may be overexposed on the light-polluted sky.

An viable alternative is video recording with a very sensitive converted night security camera. These operate similar to the human eye, and frame by frame, a meteor will be recorded fairly compared to a star of similar magnitude.

The graph is the result of counting meteors visually in 15-minute intervals. The counts were then scaled to hourly rates and corrected for the changing altitude of the shower radiant.

The photograph was taken effectively at random near a 15-minute time mark. A bright meteor happened to appear in the field of view, just to the right of the box of the Big Dipper.

Image parameters (plot of hourly rate):

• Optics and detector: naked eye
• Location: Earlyburn, Scotland

Image parameters (photograph):

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 18 mm
• Field of view: 65 × 45°
• Aperture: f/3.5
• ISO: 400
• Exposure: 30 s
• Location: Stichill, Scotland
• Processing: linear stretch to boost the brightness

Meteoric dust is one candidate for the seed particles on which the NLC ice crystals form.

## The ionosphere

Like the ozone layer, the ionosphere does not quite fit in our layering of the atmosphere from the temperature profile. The ionosphere consists of several layers where a relatively high concentration of ions and electrons occurs.

The lowest D layer occurs only on the day side of the Earth. It lies at the top of the mesosphere between 60 and 90 km. Here, the intense Lyman-&alhpa; extreme UV line radiation from the Sun is absorbed. To us below this layer is an absorber of medium and high frequency radio waves, which explains the failure during the day to receive distant AM radio stations.

The E and F layers are at higher altitude of 100 to 300 km, in the thermosphere. These are due to absorption of even shorter wavelength radiation: extreme UV and soft X-rays. Radio waves from surface stations can be reflected by these layers, explaining the ability to receive shortwave radio (at night) from stations across the globe.

## The thermosphere

The thermosphere is defined as the layer where the temperature finally increases with increasing altitude. Below is the coldest layer at −85 °C on average. At 100 km altitude it is 0 °C, at 170 km it exceeds 500 °C.

### Aurora

Aurora (aurora borealis in the northern hemisphere, aurora australis in the southern hemisphere) is a bright glow in the atmosphere in a wide range of altitude above the mesosphere, 90 to 120 km. The light is emitted by particles of air after they have been excited or ionised by collisions that are ultimately caused by the continuous stream of ionised particles from the Sun known as the solar wind. Due to the way the magnetic fields of Sun and Earth interact, the aurora is limited to an oval region of the Earth some distance from the magnetic poles. Related to solar activity, the intensity and spread of this aurora oval can temporarily increase. Aurora can then be seen further from the magnetic pole, e.g. in Britain and the mainland United States, as opposed to Scandinavia and Canada.

The intensity of the oval changes with the time of day, and the best time to observe aurora is from midnight onwards. But this also depends very much on the timing of activity in the solar wind.

Relatively quiet aurora can be an extended green glow changing very little for an hour or more. More active aurora will exhibit vertical rays and curtains of such rays. Individual rays can form and disappear in a matter of seconds or less, and the curtains can move on times scales of between seconds and minutes.

Aurora, 2005-01-21/22.

Aurora, 2012-09-30/31.

Physical parameters:

• Altitude: 100 km
• Air temperature: −80 °C
• Air pressure: 0.0003 hPa

The technique for taking pictures of the aurora has similarities to photography of noctilucent clouds. Use a wide field of view. The best time of day to observe (for Britain and places of similar geomagnetic latitude) are the hours around and after midnight. But it is more important that the aurora be at its most active. There are forecasts and near real-time measurements of magnetic disturbance online. As aurora at its best changes very rapidly, use high ISO rating and widest aperture, so that the exposure time can be as short as possible. Depending on the brightness of the display, exposures will be between a few and a few tens of seconds.

Image parameters:

• Camera: Canon EOS 300D
• Detector: 22 × 15 mm
• Focal length: 18 mm
• Field of view: 65 × 45°
• Aperture: f/3.5
• ISO: 1600, 400, resp.
• Exposure: 5 s, 8 s, resp.
• Location: Roslin, Edinburgh, resp.