Chapter 2 · The Human Eye and the Colourful World

2.3 Atmospheric Refraction and Scattering of Light

The air above you is a giant, restless lens and a sorter of colours. Find out why stars twinkle, why the Sun is seen before it rises, why the sky is blue but clouds are white, and why danger lamps are red.

On a clear night, lie on your back and watch the stars: they flicker, brighten and dim, sometimes even flash a little colour. Now find a bright planet such as Jupiter or Venus — it shines with a steady light. In the morning, the Sun peeps over the horizon a couple of minutes before it has geometrically risen, and in the evening it lingers for a couple of minutes after it has really set. All three effects have the same cause: the air bends light.

The air also does something quite different to sunlight: it scatters it. Tiny molecules send blue light off in all directions, so the whole sky glows blue, while big cloud droplets scatter every colour alike and look white. In this section you will meet atmospheric refraction and the scattering of light, and use them to explain twinkling stars, early sunrises, the Tyndall effect, the blue sky, white clouds and the colour of danger signals.

Atmospheric Refraction and Twinkling Stars

Air is a transparent medium with a refractive index very slightly greater than 1 — about $1.0003$ at sea level. Colder, denser air has a slightly higher refractive index than warmer, thinner air. Because the atmosphere becomes thinner as you go up, and because pockets of warm and cool air are always moving about, light crossing the atmosphere passes through layers of slightly different refractive index and is bent a little at each one.

📘 Definition 2.9 — Atmospheric refraction
Atmospheric refraction is the refraction of light by the Earth's atmosphere, caused by layers or pockets of air with different optical densities (refractive indices). Light from a star entering the atmosphere passes from rarer air high up into denser air near the ground, so it bends gradually towards the normal along a gently curved path.

You can see atmospheric refraction on a small scale above a fire, a hot road or the exhaust of a jet engine: objects seen through the rising hot air seem to waver. The hot air above the fire has a lower refractive index than the cooler air around it, and the boundaries keep moving, so the light from the objects behind is bent by a changing amount.

On a large scale, starlight bends downwards as it enters the denser air near the ground. Your eye traces the light back along the direction in which it arrives, so you see the star slightly higher than its true position. The effect is largest near the horizon (about half a degree) and zero for a star directly overhead, whose light enters along the normal.

📐 Result 2.6 — Why stars twinkle and planets do not

A star is so far away that it is effectively a point source of light. As its light passes through moving air of changing refractive index, the amount of bending keeps changing. So

  • the apparent position of the star shifts very slightly from moment to moment, and
  • the amount of starlight entering the eye keeps changing — the star looks brighter, then fainter.

This flickering is twinkling. A planet is much closer and looks like a tiny disc, an extended source made of many point sources. The flickers from the different points are independent and average out, so the total light reaching the eye stays nearly constant: planets do not twinkle.

✏️ Example 1 — The steady planet
On a winter evening a student sees two bright "stars" low in the west. One flickers strongly; the other shines with a steady light. Which is more likely to be a planet? Explain, and say why the flickering object twinkles more when it is close to the horizon than when it is high in the sky.
Identify The steady one is the planet. It is close enough to appear as a small disc; the changing bending of light from different points of the disc averages out.
Twinkling star The star is a point source: when the moving air bends its single beam slightly more or less, the light entering the eye changes, so it flickers.
Near the horizon Light from a low star travels a much longer path through the atmosphere, crossing many more pockets of moving air, so the fluctuations are larger: it twinkles more.
✏️ Example 2 — No twinkling in space
Astronauts on the International Space Station, about $400\ \text{km}$ above the ground and outside almost all of the atmosphere, report that the stars do not twinkle. Explain.
Idea Twinkling is caused by the changing refraction of starlight in moving air.
Answer Above the atmosphere there is (almost) no air, so the starlight reaches the astronaut without any changing refraction. Its brightness and direction stay steady: no twinkling. This is also why telescopes are sent into space — their images are not blurred by the atmosphere.

Advance Sunrise and Delayed Sunset

The same bending works on sunlight. When the Sun is just below the horizon, its light enters the atmosphere almost horizontally, is bent downwards along a curved path, and reaches your eye. You see the Sun in the direction from which the light arrives — just above the horizon — although geometrically it is still below it.

📐 Result 2.7 — Advance sunrise and delayed sunset

Atmospheric refraction near the horizon raises the apparent position of the Sun by about half a degree — roughly the Sun's own width. Because of this

  • the Sun is seen about 2 minutes before it actually crosses the horizon at sunrise, and
  • it remains visible about 2 minutes after it has actually crossed the horizon at sunset,

so the day is lengthened by about 4 minutes. (These are typical values for places near the equator; they are larger at high latitudes, where the Sun rises at a shallow slant.)

Atmospheric refraction also explains why the Sun (or the full Moon) looks flattened into an oval when it is on the horizon: light from its lower edge passes through denser air and is lifted more than light from its upper edge, so the disc appears squashed vertically.

✏️ Example 3 — How many minutes early?
The Earth turns through $360^\circ$ in $24$ hours. Near the equator the Sun rises straight up from the horizon at the rate the Earth turns. If atmospheric refraction lifts the image of the Sun by $0.50^\circ$ at the horizon, how early is the Sun seen to rise? By how much is the day lengthened?
Rate $24\ \text{h}=24\times60=1440\ \text{min}$, so the Earth turns $\dfrac{360^\circ}{1440\ \text{min}}=0.25^\circ$ per minute.
Time To be lifted $0.50^\circ$ the Sun needs $\dfrac{0.50^\circ}{0.25^\circ/\text{min}}=2.0\ \text{min}$: it is seen $2.0$ minutes before it actually rises.
Day length The same happens at sunset, so the day is lengthened by about $2\times2.0=4.0\ \text{minutes}$.
Check The answer matches the "about 2 minutes" of Result 2.7. ✓
Interactive 1 — Starlight Through a Restless Atmosphere

Light from a star crosses layers of air that get denser towards the ground; the curved path is traced layer by layer with Snell's law. Your eye looks back along the arriving ray, so the star appears higher than it really is. Tick moving air and press Play: the layers wobble, the point-like star flickers, but the disc of a planet stays steady. Choose the rising Sun and Play to watch the Sun appear before it has truly risen. (The bending, the atmosphere's thickness and the Earth's curvature are all greatly exaggerated in the drawing; the readouts give real values.)

—
true elevation—
seen at—
lifted by—
brightness now—

Scattering of Light and the Tyndall Effect

Sunbeams slanting through a gap in the clouds, the beam of a cinema projector, a torch beam in a misty forest at dawn — you can see the path of the light only because something in the air sends a little of it sideways into your eyes. In perfectly clean air or water the beam itself would be invisible from the side.

📘 Definition 2.10 — Scattering of light
Scattering of light is the process in which very small particles in its path — molecules, dust, smoke, tiny droplets — absorb some of the light and send it out again in all directions. The colour of the scattered light depends on the size of the scattering particles.
📘 Definition 2.11 — Tyndall effect
The Tyndall effect is the scattering of a beam of light by the particles of a colloid (a mixture with particles far larger than molecules but too small to settle or to be seen, such as smoke, mist or milk diluted in water), which makes the path of the beam visible. It is named after John Tyndall, who studied it in the 1860s.

A beam passing through a true solution such as salt water or sugar water is not seen from the side: the dissolved particles are single molecules or ions, far too few and too small to scatter much. Through diluted milk, soapy water, smoke or a starch solution the beam shows up clearly. Looked at from the side, very fine particles scatter mainly blue; larger particles scatter all colours roughly equally, so the scattered light looks white.

📐 Result 2.8 — The colour of scattered light depends on particle size
  • Very fine particles — air molecules and particles much smaller than the wavelength of light — scatter light of short wavelength (blue, violet) much more strongly than light of long wavelength (red). The scattered light looks blue, and the light that passes straight on loses its blue and becomes reddish.
  • Large particles — dust, water droplets in clouds, particles bigger than the wavelength of light — scatter all wavelengths about equally, so the scattered light looks white (or grey).

Red light, with the longest wavelength, is scattered least by fog, smoke and air.

A classroom demonstration shows all of this in a single beaker. Shine a narrow beam of white light through a dilute solution of sodium thiosulphate, add a few drops of dilute hydrochloric acid and stir. Fine particles of sulphur slowly form. After a few minutes the beam seen from the side turns blue, while the spot of light that passes straight through onto a screen turns orange and then red, because the light going straight through has lost much of its blue. As the particles grow, the side-scattered light turns whitish.

✏️ Example 4 — Which beams can you see?
A torch beam is passed in turn through four glass tanks: (a) pure water, (b) salt solution, (c) water with a few drops of milk, (d) water with a little smoke bubbled through it. In which tanks is the path of the beam seen from the side? Name the effect.
(a), (b) Not visible (or only very faintly): pure water and a true solution have no particles large enough to scatter much light.
(c), (d) Visible: milk droplets and smoke particles form colloids that scatter light sideways into the eye.
Effect The Tyndall effect. Seen from the side, the dilute milk may look slightly bluish while the beam coming out of the far end looks yellowish-orange.

Blue Sky, White Clouds and Red Danger Signals

Now apply Result 2.8 to the whole atmosphere.

✏️ Example 5 — A much longer path near the horizon
Model the air as a uniform layer $8.0\ \text{km}$ thick over an Earth of radius $6400\ \text{km}$. Find the length of the path of light through this layer (a) from a star or the Sun overhead and (b) from a star or the Sun on the horizon. Use the result to explain why a star low in the sky twinkles more than one overhead.
(a) Overhead, the light crosses the layer straight down: path $=8.0\ \text{km}$.
(b) Geometry On the horizon, the ray grazes the Earth at your position, so it is a tangent. The line from the Earth's centre to you (radius $6400\ \text{km}$) is perpendicular to the ray, and the line from the centre to the top of the layer is $6408\ \text{km}$. By Pythagoras, the path from you to the top of the layer is $$L=\sqrt{6408^2-6400^2}\ \text{km}=\sqrt{102\,464}\ \text{km}\approx320\ \text{km}.$$
Compare $\dfrac{320\ \text{km}}{8.0\ \text{km}}=40$: light from the horizon passes through about $40$ times as much air as light from overhead.
Meaning Light from a star on the horizon crosses about $40$ times as many pockets of moving air, so the changes in its bending — and in its brightness — are much larger: it twinkles more. The same long path makes atmospheric refraction, and scattering, strongest near the horizon. (What this long path does to the colour of the rising or setting Sun is in the 🚀 box below.)
✏️ Example 6 — A world without air
Describe how sunrise, the colour of the sky and the stars at night would look to an observer on the Moon, which has no atmosphere. Give a reason for each.
Sunrise No advance sunrise or delayed sunset: with no atmosphere there is no refraction to lift the Sun's image.
Sky The daytime sky is black: there is nothing to scatter sunlight into the observer's eye from other directions.
Stars They shine steadily and do not twinkle, because their light crosses no moving air. They are visible even in the daytime, against the black sky.
Interactive 2 — The Scattering Lab

Tyndall tank: a beam of white light crosses a tank of water containing tiny particles. Change the particle size with the slider: compare the colour you would see from the side (the scattered light) with the colour of the spot on the screen (the light that gets through). Sky and Sun 🚀: the slider sets the Sun's height; the colour of the sky (and of the Sun) is computed from how much blue and red light is scattered along the path through the air. The colour of the low Sun is not in the board exam — see the 🚀 box below. Press Play to sweep. (A simple model: very small particles scatter blue far more than red; large ones scatter all colours alike.)

—
particle size—
seen from the side—
spot on the screen—
blue : red scattered—
⚠️ Common mistakes

1. "The sky is blue because it reflects the sea." No — the sky is blue over deserts too. It is blue because air molecules scatter blue sunlight most strongly.

2. Mixing up refraction and scattering. Twinkling, advance sunrise and the flattened Sun are refraction effects (bending). Blue sky, white clouds, the visibility of red danger lamps and the Tyndall effect are scattering effects.

3. "Red is used for danger because it is scattered most." The opposite: red is scattered least, so it travels farthest through fog and smoke.

🚀 Beyond the board syllabus (enrichment only) — optional: the red Sun at sunrise and sunset, and how much more blue is scattered

Not listed in JEE Main 2026, NEET 2026 or JEE Advanced 2026, and the CBSE 2026-27 syllabus explicitly leaves out the colour of the Sun at sunrise and sunset — read this for understanding only.

The red rising and setting Sun. Example 5 showed that light from the horizon crosses about $40$ times as much air as light from overhead. Along such a long path most of the blue and other short wavelengths are scattered out of the beam (Result 2.8), and mainly red and orange reach you, so the Sun looks orange-red near the horizon. At noon the path is short, little is lost, and the Sun looks white. The Sun itself does not change colour; the scattered blue goes to brighten the sky for people elsewhere. The orange-red spot in the sodium thiosulphate experiment is the same effect in a beaker. After a large volcanic eruption, fine dust high in the atmosphere scatters out even more blue, and sunsets around the world look unusually deep red for months.

How much more is blue scattered? For particles much smaller than the wavelength $\lambda$ (such as air molecules), the intensity of scattered light varies as $\dfrac{1}{\lambda^4}$ — Rayleigh's law, worked out by Lord Rayleigh in 1871. Comparing blue light ($450\ \text{nm}$) with red ($700\ \text{nm}$):

$$\frac{I_{\text{blue}}}{I_{\text{red}}}=\left(\frac{700}{450}\right)^4\approx5.9 .$$

Blue is scattered about six times as strongly as red, and violet ($400\ \text{nm}$) about $9.4$ times as strongly. The sky is not violet because sunlight contains less violet than blue, some violet is absorbed high in the atmosphere, and the eye is far less sensitive to violet. You will meet Rayleigh scattering again in Class 12 Physics.

Exercises

I
Atmospheric refraction.
  • 1 What is atmospheric refraction? What causes it?
  • 2 Why do objects seen through the hot air above a fire or a bonfire appear to waver?
  • 3 Is the apparent position of a star higher or lower than its true position? Explain with the direction of bending.
  • 4 For which star is the apparent position the same as its true position? Why?
  • 5 Why do stars twinkle?
  • 6 Why do planets not twinkle?
  • 7 Why does a star near the horizon twinkle more than one high in the sky?
  • 8 Why does the Sun look slightly oval, flattened from top to bottom, when it is setting?
1 The bending of light by the Earth's atmosphere, caused by layers or pockets of air of different optical density (refractive index), which in turn come from differences in temperature and pressure.
2 The hot air above the fire is optically rarer than the cooler air around it, and it keeps moving, so light from objects behind it is refracted by a constantly changing amount; the apparent position of the objects keeps shifting.
3 Higher. Starlight passes from rarer to denser air and bends towards the normal (downwards); the eye traces the ray back along its final direction, which points above the true position.
4 A star directly overhead (at the zenith): its light enters along the normal to the air layers and is not bent.
5 A star is a point source. Moving air of changing refractive index keeps changing the bending of its light, so its apparent position and the amount of light entering the eye fluctuate.
6 A planet appears as a small disc — many point sources. Their fluctuations are independent and average out, so the total light reaching the eye is steady.
7 Its light travels a much longer path through the atmosphere and crosses many more moving air pockets, so the fluctuations are larger.
8 Light from the lower edge travels through denser air near the horizon and is lifted more than light from the upper edge, so the disc appears squashed vertically.
II
Sunrise, sunset and path lengths. Assume a place near the equator where the Sun rises vertically, and take the Earth's radius as $6400\ \text{km}$.
  • 1 Through how many degrees does the Earth turn in one minute? In one hour?
  • 2 Atmospheric refraction lifts the Sun's image by $0.60^\circ$ at the horizon. How many minutes early is sunrise seen?
  • 3 If the lift is $0.55^\circ$, find the advance of sunrise and the total lengthening of the day.
  • 4 The Sun is seen $2.5$ minutes before it truly rises. By what angle does refraction lift its image?
  • 5 Model the atmosphere as a layer $10\ \text{km}$ thick. Find the path length through it of light from a star on the horizon, and how many times longer this is than for a star overhead.
  • 6 Repeat part 5 for a layer $6.0\ \text{km}$ thick.
  • 7 If the Earth had no atmosphere, would the day be longer or shorter, and by about how much?
Check your answers: light's path through a layer of air
path length—
times the overhead path—
1 $\tfrac{360^\circ}{1440\ \text{min}}=0.25^\circ$ per minute; $15^\circ$ per hour.
2 $\tfrac{0.60^\circ}{0.25^\circ/\text{min}}=2.4\ \text{min}$.
3 $\tfrac{0.55}{0.25}=2.2\ \text{min}$ early; the day is lengthened by $2\times2.2=4.4\ \text{min}$.
4 $2.5\ \text{min}\times0.25^\circ/\text{min}\approx0.63^\circ$.
5 $L=\sqrt{6410^2-6400^2}=\sqrt{128\,100}\approx358\ \text{km}$; $\tfrac{358}{10}\approx36$ times longer.
6 $L=\sqrt{6406^2-6400^2}=\sqrt{76\,836}\approx277\ \text{km}$; $\tfrac{277}{6.0}\approx46$ times longer.
7 Shorter, by about $4$ minutes: there would be no advance sunrise and no delayed sunset.
III
Scattering and the Tyndall effect.
  • 1 What is meant by scattering of light?
  • 2 What is the Tyndall effect? Give two examples from daily life.
  • 3 Why can the path of a beam be seen through dilute milk but not through salt solution?
  • 4 How does the colour of scattered light depend on the size of the scattering particles?
  • 5 Why is the clear sky blue?
  • 6 Why are clouds white?
  • 7 In the sodium thiosulphate experiment, what colour is (a) the light seen from the side, (b) the spot on the screen, after a few minutes? Explain.
  • 8 Why are the beams of light visible in a dusty cinema hall or a smoke-filled room?
1 The sending of light in all directions by very small particles (molecules, dust, droplets) in its path.
2 Scattering of light by the particles of a colloid, which makes the beam's path visible. Examples: sunbeams through the canopy of a misty forest; a torch beam in fog; sunlight through a window into a dusty room.
3 Milk droplets in water form a colloid whose particles are big enough to scatter light; salt solution is a true solution of ions far too small to scatter appreciably.
4 Very fine particles scatter mainly short wavelengths (blue); larger particles scatter all wavelengths about equally (white).
5 Air molecules are much smaller than the wavelength of visible light and scatter blue light much more strongly than red; the scattered light reaching us from all over the sky is mainly blue.
6 Cloud droplets are larger than the wavelength of light and scatter all colours nearly equally, giving white.
7 (a) Blue: fine sulphur particles scatter blue most; (b) orange-red: the light going straight through has lost much of its blue.
8 Dust or smoke particles scatter part of the beam sideways into our eyes (Tyndall effect).
IV
Applications and challenges.
  • 1 Why are danger signals and stop lights red?
  • 2 Why can you see the path of sunbeams streaming through a gap in the clouds or through the canopy of a misty forest?
  • 3 Why does the sky appear dark to an astronaut in space?
  • 4 A pilot flying at a very great height notices that the sky above looks dark blue to almost black. Explain.
  • 5 On a hazy, polluted day the sky looks milky white instead of deep blue. Suggest why.
  • 6 State whether each is caused mainly by refraction or by scattering: (a) twinkling, (b) blue sky, (c) advance sunrise, (d) white clouds, (e) wavering view above a hot road.
  • 7 Challenge. At the very moment of sunset you see the whole Sun just sitting on the horizon. Is the real Sun above or below the horizon at that moment? Explain.
1 Red has the longest wavelength and is scattered least by fog, smoke and air, so it can be seen from far away.
2 Tiny water droplets, dust and smoke in the air scatter part of the sunlight sideways into your eyes, so the path of the beam becomes visible (Tyndall effect).
3 There is no atmosphere to scatter sunlight into the eye from other directions.
4 At great height most of the air is below the aeroplane, so there is very little air above to scatter sunlight; the sky looks dark.
5 Haze contains dust, smoke and droplets much larger than air molecules. These larger particles scatter all colours about equally, adding white light to the blue scattered by the air, so the sky looks whitish.
6 (a) refraction; (b) scattering; (c) refraction; (d) scattering; (e) refraction.
7 Below the horizon: refraction lifts the Sun's image by about half a degree, about the Sun's own width, so the real Sun has already set geometrically.
🎯 Exam corner — board practice Board

How this is examined: atmospheric refraction and scattering are board-exam topics (2–3 mark explanations, assertion–reason and case studies); they are not listed in the JEE Main 2026 or NEET 2026 syllabus.

E1. Board · 3 marks Explain why stars twinkle but planets do not. Why is the apparent position of a star different from its actual position?

Position Starlight entering the atmosphere passes into denser air and bends towards the normal; the star appears slightly higher than its actual position.
Twinkling The atmosphere is not steady, so the bending keeps changing; for a point-sized star the apparent position and the light entering the eye fluctuate.
Planets A planet is an extended source (a collection of point sources); the fluctuations average out and it shines steadily. Exam tip: the words "point source" and "extended source" are what the marking scheme looks for.

E2. Board · 2 marks What is meant by advance sunrise and delayed sunset? By about how much do they lengthen the day?

Idea Because of atmospheric refraction, the Sun is seen about 2 minutes before it actually crosses the horizon at sunrise and about 2 minutes after it actually crosses it at sunset.
Answer The day is lengthened by about 4 minutes. Exam tip: draw the Earth, the atmosphere, the curved ray, and the "apparent Sun" above the horizon with the "actual Sun" below it.

E3. Board · Case study During a science fair, students pass a beam of white light through a glass tank of sodium thiosulphate solution, then add a little dilute hydrochloric acid. After a few minutes visitors looking at the side of the tank see a bluish glow, and the spot on a screen beyond the tank turns orange-red. (i) Name the phenomenon and explain the bluish glow. [2 marks] (ii) Why does the spot of light on the screen turn orange-red? [1 mark] (iii) Why does the side glow turn whitish later, as the particles grow? [1 mark]

(i) Scattering of light (Tyndall effect). Very fine sulphur particles form; they scatter short wavelengths (blue) much more than long ones, so the scattered light seen from the side is blue.
(ii) The light going straight through the tank has lost much of its blue (and other short wavelengths) by scattering; mainly the longer orange and red wavelengths are left to reach the screen.
(iii) Larger particles scatter all colours about equally, so the scattered light becomes white. Exam tip: always link the colour to the particle size.

E4. Board · 1 mark To an astronaut in a spaceship far above the Earth's atmosphere, the sky in daytime appears (A) blue (B) red (C) white (D) black

Idea The sky's colour is light scattered by the air; above the atmosphere there is no air to scatter.
Answer (D). Exam tip: "no scattering medium ⟹ dark sky" also explains the black sky on the Moon.

E5. Board · 1 mark Which one of the following is not caused by atmospheric refraction? (A) twinkling of stars (B) advance sunrise (C) flattening of the Sun's disc near the horizon (D) blue colour of the clear sky

Idea (A), (B) and (C) are all bending effects. The blue sky comes from the scattering of blue sunlight by air molecules — no bending is involved.
Answer (D). Exam tip: sort each effect: bending → refraction; colour of the sky, clouds or a beam seen from the side → scattering.

E6. Board · 1 mark Assertion (A): Danger signal lights are red in colour. Reason (R): Red light is scattered the most by fog and smoke, so it is easily noticed. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

Idea A is true. R is false: red light is scattered the least, which is why it penetrates fog and smoke and can be seen from far away.
Answer (c). Exam tip: watch for "most" and "least" swapped in assertion–reason items.
🔬 Research–Teaching NexusResearch-tutored · optional, about 45 minutes

Research question. How does the colour of the sky change with air quality — and can a phone camera measure it?

Why it matters. The fine particles in polluted air (called PM2.5, smaller than $2.5$ micrometres) are a major health risk in large cities from Delhi and Lahore to Cairo and Mexico City. Because these particles scatter light, satellites such as NASA's MODIS instruments and ground stations of the worldwide AERONET network estimate the amount of haze from how strongly the sky scatters and dims sunlight. On a clean day the sky is deep blue right down to the horizon; on a hazy day it fades to milky white, because the larger haze particles scatter all colours.

What to do.

  1. On five or more different days, at the same time (for example one hour before sunset), photograph the same view of the sky about $30^\circ$ above the horizon and away from the Sun, with your phone's camera on fixed (not automatic) exposure if possible. Never point the camera or your eyes at the Sun.
  2. Use any free colour-picker app or website to read the red, green and blue values (0–255) at the same spot in each photo. Compute the "blueness" $\dfrac{B}{R}$.
  3. Look up the PM2.5 reading for your town for each day from an official air-quality website.
  4. Plot $\dfrac{B}{R}$ against PM2.5 and describe the trend.

Tools: a phone camera, a colour-picker app, an air-quality website, a spreadsheet.

Hand in: your photographs (small), a table of $R$, $G$, $B$, $\tfrac BR$ and PM2.5, the graph, and a paragraph that uses Result 2.8 to explain what you found.

Going further. On the same days, also photograph the sky straight overhead and compare its $\tfrac BR$ with the sky near the horizon. Which is bluer, and does the difference grow on hazy days? Use the path lengths of Example 5 in your explanation.

Instructor note: expect $\tfrac BR$ of roughly $1.5$–$2.5$ for a clean blue sky and close to $1$ for a hazy, whitish sky; the trend with PM2.5 is usually clear but noisy (humidity, clouds and camera processing matter). Automatic white balance can distort colours — encourage a fixed setting or the same camera mode every day. Stress the safety rule about not looking at the Sun.

🧪 Self-check quiz

0/5 correct

Q1. Stars twinkle because

Q2. Because of atmospheric refraction, the day is longer than it would otherwise be by about

Q3. The path of a beam of light is clearly visible through

Q4. The clear sky looks blue because

Q5. Clouds look white mainly because

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