Physics · Optics · Grade 7

Shadow and eclipses

Why can only people in a narrow strip see a solar eclipse as total?

The situation

In the middle of the day it suddenly grows dim, the Sun turns into a thin sickle, and in the shade of a tree hundreds of little “crescents” fall on the ground. Two weeks later the full Moon appears copper-coloured for almost an hour. A total solar eclipse is seen only by people in a strip a few hundred kilometres wide; everyone else sees a partial one. How does the straight-line travel of light explain this, and why is there no eclipse every month?

Topic

Rectilinear propagation of light, shadow. Solar and lunar eclipses: umbra and penumbra

Key ideas

Rectilinear propagation
Umbra and penumbra
Solar and lunar eclipses

Formulas

D_{s} = d \frac{a + b}{a} - D \frac{b}{a}

The diameter of the umbra on the screen (similar triangles): $D$ is the source, $d$ the ball, $a$ the source–ball distance, $b$ the ball–screen distance.

D_{y s} = d \frac{a + b}{a} + D \frac{b}{a}

The outer diameter of the penumbra. For a point source ( $D = 0$ ) the two are equal — the shadow is sharp.

L = \frac{d a}{D - d}

The length of the umbral cone ( $D > d$ ). If the screen is farther away than this, only penumbra remains in the centre — as in an annular eclipse.

h^{'} = h \frac{b}{a}, δ \approx d_{t} \frac{a + b}{a} + 2.44 \frac{λ b}{d_{t}}

Camera obscura: the height of the image and the blur spot of each point ( $d_{t}$ is the pinhole diameter, $λ \approx 550$ nm). The sharpest image is at about $d_{t} \approx \sqrt{2.44 λ b}$ .

t = \frac{1}{c} \int n d l \to \min

Fermat's principle: light travels along the path that takes the least time. In a uniform medium this is a straight line.

Theory

Light travels in straight lines

In a uniform medium light travels in a straight line: this is shown by the sharp edges of shadows, a laser beam and the camera obscura. This is one of the basic laws of geometrical optics; the others are the independence of beams (crossing rays do not interfere with one another) and the laws of reflection and refraction (the “Car mirrors and the periscope” and “Laser, water and optical fibre” experiments). The law holds when obstacles and apertures are much larger than the wavelength ( $\approx 0.5$ µm); with a very small aperture diffraction becomes noticeable.

Umbra and penumbra

A point source gives a sharp shadow. If the source is extended, then from some points of the screen only part of the source is visible: where the source cannot be seen at all is the umbra, where it is partly visible is the penumbra. The boundaries are given by the rays that graze the edge of the ball from the edges of the source (the two-coloured lines in the scene) — so the sizes are found from similar triangles. On the graph the illuminance in the penumbra rises smoothly from 0 to 100 %. The model: the source is a disc of uniform brightness, and the ball's shadow is the shadow of a disc.

Camera obscura

A small hole in one wall of a dark box forms an inverted image on the opposite wall: from each point of the object a thin beam passes through the hole. If the hole is made larger the image gets brighter, but each point turns into a large spot and the image blurs; if the hole is made too small, diffraction enlarges the spot. So there is an optimal hole that gives the sharpest image: about 0.5 mm for $b = 20$ cm. The camera obscura was described in detail by Ibn al-Haytham in the 11th century; the photographic camera began with this idea.

Solar eclipse

At new moon, if the Moon passes between the Sun and the Earth, its shadow falls on the Earth. The Moon's umbral cone is about 370–380 thousand km long — very close to the distance to the Moon (363–406 thousand km). If the Moon is nearer, the cone reaches the Earth: a total eclipse is seen in a narrow strip. If it is farther, the cone falls short — the Moon cannot cover the Sun completely and an annular eclipse occurs. The penumbra is about 7000 km wide: people there see a partial eclipse. The shadow moves over the Earth's surface at about one kilometre per second, so totality lasts only a few minutes in any one place.

Lunar eclipse and why not every month

At full moon, if the Moon enters the Earth's shadow, a lunar eclipse occurs — it is seen from everywhere the Moon is above the horizon. Even in the umbra the Moon does not vanish: the Earth's atmosphere refracts sunlight into the shadow, while blue light is scattered on the way — the Moon takes on a copper-red tint (in the scene the colour is boosted so that it is visible). The Moon's orbit is tilted ≈ 5.1° to the ecliptic, so most of the time the Moon passes above or below the shadow. An eclipse occurs when a new or full moon falls near a node — in eclipse seasons about 173 days apart. Set the tilt to 0°: an eclipse happens every month. Simplifications: the Earth's orbit is a circle; the atmosphere's enlargement of the shadow by ~2 % is ignored; the distance to the Sun is not to scale in the scene, but the shadows are calculated with real distances.

Fermat's principle

Light goes by the path for which the time taken from one point to another is the least (more precisely, stationary): $t = \frac{1}{c} \int n d l$ . In a uniform medium ( $n$ = const) the least time means the shortest path, that is, a straight line: shadows and eclipses follow from this. From the same principle come the law of reflection (the angle of incidence equals the angle of reflection) and the law of refraction $n_{1} \sin α = n_{2} \sin γ$ — see them in the “Car mirrors and the periscope” and “Laser, water and optical fibre” experiments.

Did you know?

The Sun's diameter is about 400 times that of the Moon, but it is about 400 times farther away. So in the sky both look almost the same size — about 0.5° — and a total eclipse is possible.
During a partial solar eclipse the gaps between tree leaves become tiny camera obscuras: the patches of light on the ground are not round but crescent-shaped.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Car mirrors and the periscope — Grade 7
Laser, water and optical fibre — Grade 7
The eye and spectacles — Grade 7
Measuring the speed of light — Grade 9
Optical instruments — Grade 9
Interference and diffraction — Grade 11
Polarisation of light — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11