Gravitation, Sundial, Astrology, Kepler

School Science Lessons
(UNPh36.1)
2025-09-07

Gravity
Contents
36.1.0 Astrology
36.2.0 Gravitational attraction of the Earth
36.3.0 Observing the Moon
36.4.0 Sundials
36.5.0 Transit of Venus

36.2.0 Gravitational attraction of the Earth
36.2.1 Gravity
36.2.2 Gravitational field of the Earth
36.2.3 Gravitational potential energy
36.2.4 Mass
36.2.5 Newton's universal law of gravitation
36.2.6 Satellite in stable orbitt
36.2.7 Weight
36.2.8 Wind farms and Earth rotation

36.3.0 Observing the Moon
36.3.1 Altitude of the Moon and the Sun
36.3.2 Cold moon
36.3.3 Eclipse of the moon
36.3.4 Lunar month
36.3.5 Measure the distance to the Moon, parallax
36.3.6 Man in the Moon illusion
36.3.7 Moon slows the Earth
36.3.8 Moon watch
36.3.9 Moonbows
36.3.5 Parallax
36.3.10 Phases of the Moon
36.3.11 Phases of the Moon and lunar eclipses
36.3.12 Positions of the Moon
36.3.13 Rising and setting Moon
36.3.14 Rising moon illusion
36.3.15 Sun and Moon diameter illusion
36.3.16 Supermoon

36.4.0 Sundials
36.4.1 Build a sundial
36.4.2 Demonstration sundials
36.4.3 Find due north with a sundial, shadow stick
36.4.4 Flowerpot sundial
36.4.5 Lengths of sundial shadows during the year
36.4.6 Make a sundial for your home
36.4.7 Measurements using the Sun
36.4.8 Pocket sundial
36.4.9 Sundial for the Northern Hemisphere
36.4.10 Sundial for the Southern Hemisphere
36.4.11 Universal globe sundial

36.2.1Gravity
Gravity, weight of an object and g
Gravity, gravitational force, is the attractive force by which bodies are attracted towards the centre of the Earth.
The intensity of gravity is measured by the acceleration produced by the force of gravity on that object and is measured in newton, N.
The value of g decreases with altitude.
If at Earth's surface, g = 9.8 m.s^-2, at 1000 km above surface, g = 7.33 m.s^-2.
The value of g is less at the equator than at the north and south poles, because of the inertia produced by Earth's rotation that works against g and the equatorial bulge of the Earth at the equatorial region.
At the equator, g = 9.780 m.s^-2.
At Hong Kong, g = 9.819 m.s^-2.
At the north and south poles, g = 9.832 m.s^-2.
Gravitation is the attractive force exerted by each particle of matter on every other particle.
The law of gravity states that the force of gravity between two particles is directly proportional to the product of their masses and inversely proportional to the distance between them.
F = m₁m₂ / r², where m₁ and m₂ = the masses of the particles, and r = distance between the centres of the masses.

36.2.2 Gravitational field of the Earth
Gravitational field of the Earth, g, mass of the Earth
The law of gravitation states that the force between two bodies is directly proportional to the product of their masses and inversely proportional to the distance between them.
The gravitational constant is = 6.67 × 10^-11 N m²kg ^-2.
The force of gravity exerted by the earth pulls everything down with the same force no matter what is the mass with a constant acceleration of 9.8 ms^-1.
Near the surface of the Earth, g = 9.8 N / kg, acting towards the centre of the Earth
The weight of an object, W = mg newton, where g = G × Me / Re² newton / kg.
The weight of an object on the Earth, W = Fg = G × mMe / Re², where G = universal gravitational constant, G = 6.67 × 10^-11 N m² kg^-2, m = mass of the object in kilograms,
Me = mass of the Earth in kilograms, Re = radius of the Earth in metres.
The gravitational field of the Earth at that place, g = G × Me / Re².
Near the surface of the Earth, the gravitational field of the Earth, g = 9.8 N / kg acting towards the centre of the Earth.
The magnitude of g diminishes as you get further from the Earth, because r increases in the equation, F = G (m₁ × m₂) / r².
Gravitational field strength
The strength g of a gravitational field is the force acting on an unit mass at any point in the gravitational field.
From Newton's law of gravitation, the gravitational field around mass m₁ can be found by treating m₂ as the unit mass g = F / m = - GM / r²
In this equation the negative sign, -, shows that this force is an attracting force.
If r is measured outwards from m₁, the field is in the opposite direction.
At the Earth's surface, g = 9.81 N kg^-1.
Mass of the Earth
To calculate the mass of the Earth, where radius of the Earth = 6.38 ×10⁶ m (6, 371 km), g = - GM₁ / r²
So M₁ = g × r² /G
M₁ = 9.81 Nkg^-1 × (6.38 × 10⁶ m)²/ 6.67 × 10^-11 N m² kg^-2 = 5.99 × 10²⁴ kg.

36.2.3 Gravitational potential energy
1. The energy an object possesses, because of its position in a gravitational field is called its gravitational potential energy.
On the Earth the gravitational acceleration is about 9.8 m / s².
The potential energy of an object at a height h above the ground = the work required to lift the object to that height.
The force required to lift the object = its weight, so gravitational potential energy = the weight of an object × times the height it is lifted.
In space, the force approaches zero for large distances.
So the gravitational potential energy near a planet is negative, because gravity does positive work as a mass approaches.
The small mass approaching the large mass of a planet it bound to it unless it can get access to enough energy to escape.
The general form of the gravitational potential energy of mass m is: PE = -GM1m² / r, where G = the gravitation constant, M = mass of the planet, m = mass of the approaching object, r = distance between the centres of the planet and the approaching object.
2. An object of weight W = mg newton can be raised to a height by either, (a) lifting it vertically or (b) pushing it up a frictionless ramp.
* By applying a force equal and opposite to the weight, the object could be lifted directly through the height.
Work done = force × height = mg h joule.
Increased gravitational potential energy of the object at height h = mg h joule.
* By applying a force equal and opposite to the component of the weight acting down the slope, the object could be pushed up the slope.
Work done = force × distance up the slope = (weight × sin angle of slope) × (height / sin angle of slope) = mgh joule.
The method of raising an object vertically, or via any ramp, does not change the amount of work required to be done, and does not change the increase in gravitational potential energy, Ep = mgh joule.
3. On the surface of the Earth, the weight of an object is constant, and any change in gravitational potential energy depends on mass, g, (constant at the place), and height, Ep = mgh joule.
However, a satellite launched from the Earth has a changing gravitational force on it, falling to zero at infinity.
Gravitational binding energy is the extra energy an object needs to escape from the Earth.
A mass on the Earth's surface must be launched with sufficient kinetic energy, EK, to overcome the binding energy and escape from the Earth.
Escape velocity is the velocity needed to escape and is the same for all masses of objects.
If a satellite is given just enough energy to escape from the Earth, it will remain in the Earth's orbit, but on the opposite side of the Sun from the Earth.
The farthest the satellite can escape from the Earth without escaping from the Sun is in the Earth's orbit on the other side of the Sun.
The orbiting satellite needs extra energy to escape from the Sun.

36.2.4 Mass
Mass, inertial mass and gravitational mass
The mass of a body is the quantity of matter it contains, as measured by its acceleration produced by a force, or by the force exerted on it by a gravitational field.
Every body or particle within a body has mass and is attracted towards the centre of the Earth by a force = mg, where m = mass, and g = the attractive force by which objects are attracted towards the centre of the Earth.
The intensity of gravity is measured by the acceleration caused by the gravitational force.
Inertial mass
The standard of mass is the kilogram, based the existence of a particular cylinder of platinum-iridium alloy.
This standard can be referred to as the inertial mass, m_i.. So mass is defined by its inertia.
Gravitational mass
However, mass is conveniently measured by using the weight, W, of the body, i.e. the force of gravity attracting it to the Earth.
W = mg, where g is the acceleration of fall that varies slightly in different places on the surface of the Earth.
To define mass in terms of the gravitational force it can produce, i.e. gravitational mass, m_g, use the formula: m_g = Fd² / MG where M is a standard body distance d from another body of mass m_g, F is the gravitational force between the bodies and G is the universal gravitational constant.
However, m_i = m_g.

36.2.5 Newton's universal law of gravitation
Newton's universal law of gravitation, universal gravitational constant, G
This law explains the acceleration caused by the gravitational attraction of all massive bodies.
1. Every object in the universe attracts every other object in the universe with a force, F_g, that is directly proportional to the product of their masses, and inversely proportional to the square of the distance between their centres of mass.
So, if two masses m₁ and m₂ attract each other with a force F, in the inverse square law, F = G × m₁m₂ / r₂, where r = distance between the centres of the masses, and where the universal gravitational constant, G = 6.67 × 10^-11 N m² kg^-2.
In the equation, m is in kilograms, r is in metres and F is in newtons.
2. The universal law of gravitation states that every object in the universe attracts every other object in the universe with a force, Fg, that varies directly with the product of the masses, and varies inversel with the square of the distance between the centres of the two masses.
So Fg = G × (m1 × m2) / r², force, where Fg is in newtons, m is in kilograms, r is in metres, and the universal gravitational constant, G = 6.67 × 10^-11 N m² kg^-2.

36.2.6 Satellite in stable orbit
Satellite in stable orbit, geostationary orbit
For a satellite to remain in a stable circular orbit around the Earth at a fixed radius, r_s, the required centripetal force, Fc, must be supplied by the gravitational force, Fg.
So Fg = Fc = G × m_s Me / r² = m_s 4 π²r_s / T_s², where m_s = mass of satellite and T_s = the period of revolution of the satellite.

36.2.7 Weight
Weight is the quantity of a substance as measured by the downward force exerted on its mass by a gravitational field.
The weight of an object is the force of gravity on that object, measured in newtons, N.
The weight of an object, W = F_g = G × m × Me / Re², where m = mass of the object in kilograms, Me = mass of the Earth in kilograms, Re = radius of the Earth in metres.
Weights are pieces of metals or other substances, each having a definite amount of weight.
The amount of weight is determined by weighing using a balance, e.g. a beam balance, which balances only when the weight of the substance to be weighed in a pan at the end of one arm is equal to the total weights in a pan at the end of the other arm,
Use whichever method gives the largest uncertainty.
In the history of calculation of the fundamental constants, note the current values are compared to the range of uncertainty of the early measurements.
Measure a 1g M&M three times on an electronic balance and it may show 1.0 g three times.
So one can be fairly certain that the weight is between 0.9 & 1.1 g.

36.2.8 Wind farms and Earth rotation It is unlikely that the construction of wind farms affects the rotation of the Earth.
The relative forces are not comparable.
Some people have suggested that half the wind farms could face east and the other half face west to counteract any effect on the rotation of the Earth!

36.3.1 Altitude of the Moon and the Sun
See diagram 36.9: Simple astrolabe (sextant).
1. Cut out a rectangular piece of cardboard slightly larger than a protractor.
Trace the shape of the protractor on the cardboard and mark the main points of a scale at 10 degree intervals.
Start with zero degrees at the bottom of the scale.
Punch a small hole through the cardboard at the point corresponding to the position of the cross hairs of the protractor.
Attach a drinking straw to the edge of the cardboard closest to the hole.
Attach a washer as a plumb bob to one end of a piece of string.
Thread the other end of the string through the hole in the cardboard and tie a knot at the end.
The plumb bob should swing freely from the cross hairs.
Sight through the drinking straw at any object, e.g. top of a tree, and measure the angle showing the altitude of the object above the ground.
At night, use the simple astrolabe to measure the altitude of the Moon.
2. Measure the altitude of the Sun during the day.
Cut out a 4 cm × 4 cm piece of cardboard.
Punch a hole in the middle to form a tight fit over the drinking straw.
Attach the cardboard to one end of the drinking straw.
With the back to the Sun, adjust the alignment of the astrolabe so that the shade forms a shadow on a screen.
When you can observe a point of light in the middle of the shade patch, you can read the altitude of the Sun.

36.3.2 Cold moon
The so-called ‘Cold Moon’ will rise at 15:20 GMT on 15 December, 2024, and set at 09:44 on Monday morning.
The name is from its timing as the first full moon of winter, and has also been called the Long Night Moon and the Oak Moon.
It occurs once every nineteen years, because there is an 18.6-year cycle during which the exact places the moon rises and sets on the horizon waxes and wanes.
The final full moon of the year will move across the skies with a ‘major lunar standstill’.
The major lunar standstill will see the moon rises and sets at its northernmost and southernmost positions on the horizon.
Stonehenge was carefully designed to align with the movements of the moon, including the lunar standstills.
The major lunar standstill, occurs because the earth and the sun are at their maximum tilts, causing the moon to rise and set at the extremes of its range.

36.3.3 Eclipse of the moon
See diagram 36.97: Eclipse of the Sun and the Moon.
See diagram 36.85: Eclipse of the Moon.
See diagram 36.95: The Moon in the sky.
See diagram 36.34: Blood moon.
1. An eclipse occurs when a hypothetical straight-line can be drawn through the centres of the Sun, the Earth and the Moon, i.e. they all line up.
Lunar Eclipses occur when the Earth lies between the Sun and the Moon, (Sun-Earth-Moon), and the shadow of the Earth falls on the Moon, full Moon.
Direct observation of a lunar eclipse is safe.
Observe the shape of the Earth's shadow as its edge crosses the Moon as evidence that the Earth is spherical.
However, a disc-shaped Earth could cause the effect.
During an eclipse of the moon the moon is usually blood red, because the Earth's atmosphere refracts light from the Sun.
2. The appearance of a lunar eclipse in real life varies dramatically and depends to a large degree on the level of dust in the Earth's atmosphere.
The greater the quantity of dust, the more red and darker the totally eclipsed Moon will be.
Dust scatters more of the shorter wavelengths of light leaving the longer (redder) light to be bend into the darkest part of the Earth's shadow (which is where the Moon will temporarily pass in our evening twilight).
You cannot see the true colour of the Earth's inner shadow and therefore the eclipsed Moon if the Moon is rising in eclipse in the bright evening twilight.
You will get a much better view if darkness has fallen prior to the start of the total eclipse phase.
3. A tetrad is when four successive total lunar eclipses occur, with no partial lunar eclipses in between and each is separated by six lunar months, i.e. six full moons.
The last tetrad ended with a total lunar eclipse on 28 October 2004.
The current tetrad starts 14 April 2014 with the last of the four eclipses occurring on 28 September 2015.
The next tetrad will start on 25 April 2032.
4. Eclipse of the moon does not occur at every new and full Moon.
The Moon's orbit is inclined enough to cause the Moon usually to pass above or below the Earth's shadow or the region between the Earth and the Sun.
5. Measuring the distance to the moon during a total solar eclipse
Dr Graeme White, University of Southern Queensland
As there is a total lunar eclipse this Wednesday night, 13: 29: 51 UTC, 31/01/2018, we are proposing to measure the distance to the Moon by comparing its position in the sky relative to the stars as seen from different places on the Earth.
We need simultaneous digital images showing both the Moon and surrounding stars.
We can do this during a total eclipse of the Moon when the Moon's disk is clearly seen, but not too bright that we cannot see the stars.
The best way to get these images is as follows:
Use a DSLR camera on a tripod.
Use a modestly fast (high) ISO.
Use maximum zoom (up to 500 mm).
Telescope shots may be too narrow and miss the background stars.
Automatic focus may not work, so perhaps a few trial focus images will be needed.
Do not over expose the Moon image, but use the histogram function to make sure that the image is well exposed and not saturated.

36.3.4 Lunar month
Lunar month, synodic month
Earth is moving in its orbit about the Sun so the Moon has to travel more than 360^o to get to the next new Moon so the lunar month or synodic month is 29.531 days.
Artemis, the goddess Diana, called Cynthia, from Mount Cynthus in Delos, Greece, represented the Moon.

36.3.5 Measure the distance to the Moon, parallax
See diagram 36.37: Distance to the Moon.
By taking two simultaneous photographs of the Moon from two locations, Position 1 and Position 2, a big, but accurately measured distance between positions apart, and comparing the positions of the Moon relative to the fixed stars, the distance to the Moon can be calculated.
Distance to Moon = distance between positions of observers taking simultaneous photographs / tan ratio of parallax angle, p.
Parallax is the apparent displacement of an observed object due to a change in the position of the observer, e.g. the difference between the view of an object as seen through the picture-taking lens of a camera and the view as seen through a separate viewfinder.
A speedometer of a car may show 60 km per hour if you are looking straight ahead at it, but will indicate a greater speed if you head is to the left of it, or indicate a slower speed if your head is to the right of it.
Parallax
In astronomy, parallax is the apparent angular displacement of a celestial body, because it is observed from the surface of the Earth instead of from the centre of the Earth, i.e. geocentric parallax.
The apparent movement of a star caused by the motion of the Earth is called parallax.
One second of parallax of arc = 3.258 light years or 3.086 X 10¹³ km.

36.3.6 Man in the Moon illusion
1. Observe the craters and flatter areas, "seas" (Mare), and oceans (Oceanus).
The space craft Apollo 12 was launched 14 November 1969 and landed on the Oceanus Procellarum on 19 November 1969.
Then the astronauts walked to the remains of previous lunar probe Surveyor 3 and retrieved some pieces of it.
The arrangement of craters, sea oceans and other features allow different people and cultures to see figures in the Moon.
Although "the man in the Moon" in the Northern Hemisphere looks like an old man walking away carrying sticks or leaning on a fork, some people can see different faces and figures, even a frog.
In China and Japan they see a large rabbit stretched across the Moon with the ears pointing down from the upper right and the legs crossed at the lower left.
The rabbit is making something in a box.
Most figures can be seen only at or near a full Moon.
Some of these figures appear differently in the Northern and Southern Hemisphere.
Stare at the Moon at different phases until you can see figures.
Record the figures on a Moon diagram and note the time and date of the observations.
2. Measure the angular diameter (visual angle), of the Moon, about 29.3 ' to 34.1'.
Its actual diameter is about 3476 km (equatorial), and about 3472 km, (polar).

36.3.7 Moon slows the Earth
The Earth is slowing down, because the Earth spins 'beneath' the Moon faster than the Moon revolves around it.
The Moon's gravity creates a tidal bulge on the Earth.
This bulge attempts to rotate at the same speed as the rest of the planet.
As it moves 'ahead' of the Moon, the Moon attempts to pull it back.
This slows down the Earth's rotation.
In the Universe, if individual pieces speed up, slow down, or change direction, the sum total of angular momentum cannot change.
The Earth loses angular momentum when the Moon slows it down, so the Moon has to gain angular momentum by moving further away in its orbit.
The Moon is currently receding from the Earth by about one and a half cm per year.

36.3.8 Moon watch
At the same time each evening, e.g. 8.00 p.m., record the date, time, apparent shape (full, gibbous, half, crescent, new, crescent, half, gibbous, full), azimuth and altitude.
Draw a Moon each night so that the lunge remains white and the rest of the Moon is shaded black.
When the Moon is a gibbous Moon, use circles to represent the Moon and show the orientation of the terminator of the gibbous Moon through the night, i.e. when the Moon is in the east, north and west.
Record the dates of the phases.
Make these observations during four weeks.
Always observe from the same place.
Consult an almanac so you can begin the observation on the date when the crescent Moon is just visible in the evening, two or three days after a new phase.
The horns of the crescent Moon are turned away from the sun.
A lunar month is from new Moon to new Moon, about 29.5 days, i.e. the time taken for the Moon to revolve around the Earth.
However, most people think of the lunar month as a period of 28 days, (Waxing, Old English weaxan to increase), (Waning, Old English wanian to lessen).
Table 36.3.2
Phase First......Waxing......Full............Waning..........Last...............Waning.......New...........Waxing ........ First.....
..........quarter...gibbous.....Moon..........gibbous........quarter...........crescent......moon.......... cresent.........quarter.
Date...July ..... -................August ..........-................. August........... -................August.........-...................August.
...........29...........................6.....................................14......................................20....................................27........

36.3.9 Moonbows
A moonbow, lunar raibow, is caused by moonlight being refracted by water doplets in the air.
Moonbows are fainted than raibows and are not seen often,
The moonbow seen at Brisbane at 8 pm on 6/03/2023 was close to its fullest phase, in a dark black sky, at about 50 degrees elevation, with 3 moon diameters of black sky around the moon, then the circular rainbow about 3 moon diameters wide, with the red of the spectrum on the outside of the circular rainbow.

36.3.10 Phases of the Moon
See diagram 36.28: Phases of the Moon.
See diagram 36.28.1: Phases of the Moon in a classroom.
1. Phases of the moon
The phases of the Moon are visible, because different portions of the illuminated and non-illuminated parts of the Moon are facing towards Earth at different times.
The Moon shines, because it reflects light from the Sun.
At any particular time, half the Moon is illuminated by the Sun.
The Moon takes 27 days, 7 hours and 43 minutes to travel around the Earth.
The rotation of the moon is synchronized in that it rotates around its axis in the same time it takes to orbit the Earth.
So the same side of the Moon is always facing the Earth and we cannot see the other side of the Moon from the Earth.
On 2006-09-22 the Moon was farthest from Earth (apogee), at 406 498 km.
On 2006-09-08 the Moon was closest to Earth (perigee), at 357 174 km.
The "phase" refers to the illuminated part of a celestial body.
The different relative positions of the Moon and Sun cause the phases of the Moon (new, crescent, half, gibbous, full Moon).
When the Moon and Sun are on opposite sides of the Earth, you see sunlight reflected from all of the face of the Moon, a full Moon.
When the Sun is on the same side of the Earth as the Sun, little light is reflected back towards the Earth, a new Moon.
When the angle made by the Sun and the Moon at the Earth is between 0^o and 180^o, you see the light from only a part of the Moon, a crescent Moon.
From just after the new Moon, the crescent shape changes into a quarter Moon, then a gibbous Moon, and finally into a full Moon.
Then the changes reverse.
2. Blue moon
A "blue Moon" means a second full Moon in the same calendar month about 7 times in each 19 years, i.e. "once in a blue Moon".
The Moon has no atmosphere so you see a clear separation between the lit and unlit portions of its surface, the terminator.
It is an arc of an ellipse.
A lune or crescent is the area enclosed by the terminator and the nearer edge of the Moon.
3. Harvest moon
A "harvest Moon" is the full Moon nearest to the autumn equinox, during 22 September, 2008, in the Northern Hemisphere and 20 March, 2008, in the Southern Hemisphere.
4. Direction of the moon
From the Southern Hemisphere, the Moon appears to move around the Earth in a clockwise direction, while from the Northern Hemisphere, the Moon appears to move around the Earth in an anticlockwise direction.
The Moon rises about 50 minutes later each day.
For a few days after the new Moon to a few days before the full Moon, the Moon appears to move clockwise from west to east and can be seen in the morning during school time.
The best time to observe the Moon is 7.00 p.m.
The waxing crescent Moon is visible low in the western sky, the first quarter is visible high in the Northern sky and the full Moon is visible low in the eastern sky.
5. Lunation
Lunation is the mean time between successive new Moons, i.e. for one lunar cycle, 29.530589 days.
6. Torch moon in the classroom
Simulate the phases of the Moon in the classroom.
the Southern Hemisphere, assume that one end of the classroom is approximately north.
Use the ball to simulate the Moon and a big electric torch (flashlight), to simulate the Sun.
One student will carry the "Moon" around the class with the torch always pointing at the "Moon" in a north to south direction.
The rest of the class remains in the centre of the classroom on the "Earth". By starting from the north end of the classroom, with the torch pointing south behind the Moon, the students can only see a weak rim of light illuminating the periphery of the "Moon", a new Moon.
By moving to the right-hand side of the classroom, east, with the torch still pointing south at the "Moon", the students see half the "Moon", first quarter.
By moving to the south end of the classroom with the torch still pointing south at the Moon, students see the whole Moon illuminated by the torch, full Moon.
By moving to the west side of the classroom with the torch still pointing south, see half the "Moon", third quarter.
7. Atmosphere
Early researchers deduced that the Moon has no atmosphere, based on the sharpness with which it occults starlight at its edge.
8. Whole moon
The whole hemisphere of the Moon can be seen even when it is in its first or last quarter, because the part of the Moon that does not receive any light directly from the Sun does receive sunlight reflected from the Earth and some of this light is again reflected from the Moon to the Earth to make the dark part faintly visible.

36.3.11 Phases of the Moon and lunar eclipses
See diagram 36.95: The Moon in the sky.
1. Fix an electric torch to shine full on a white ball as a Moon.
Hold an Earth ball in position to view the white ball Moon from different directions and see crescent quarter phases, gibbous, and full Moons.
Rotate the Earth globe to show how the times of rising and setting of the Moon are closely related to the phase.
For example, the first quarter Moon rises about noon, is highest in the sky at sunset, and sets about midnight.
By sighting across the position on the globe corresponding to your own geographic locality, simulate the relationship of the Moon to the horizon for Moon rise and Moon set positions.
2. Place the white ball Moon in the shadow cast by the Earth globe to simulate a partial or total lunar eclipse.
Place the Moon between the electric torch and the globe so that its shadow is cast on the Earth.
Show that an eclipse of the Sun is not visible over as great an area of the Earth as an eclipse of the Moon, which is seen from the entire half of the Earth that is towards the Moon.

36.3.12 Positions of the Moon
1. On the first night, draw the position of the Moon relative to prominent landmarks, e.g. above a tower or church steeple.
Measure its height above the horizon in degrees, using your fist or your fingers extended e.g. a fist at arm's length = 100, a span of a thumb and little finger = 200.
Record these measurements and the time on a sketch.
Also, record the direction of the horns of the Moon, and the shape of the crescent.
Two hours later, repeat the observations and note the time.
2. Make repeated observations in the same way every night for two weeks.
Record the following observations:
* How the shape of the Moon's illumination changes from night to night,
* How its apparent location changes,
* How its horns, or cut-off edge, are oriented relative to the position of the Sun below the western horizon
* How the Moon changes position during one night.
A drawing of an "impossible Moon" shows its horns pointing down!

36.3.13 Rising and setting Moon
36.3.5a Rising and setting Moon times (Table)
During the last quarter phase of the Moon, make the above observations during the morning and compare them with the same observations during the evening.

36.3.14 Rising moon illusion
See diagram: Moon illusion.
See diagram 36.32.1: Rising Moon illusion.
See diagram 36.32.2: Ebbinghaus illusion.
The rising Moon appears to be much bigger on the horizon than the Moon high in the sky, but if you photograph both Moons they are the same size in the photograph.
The rising Moon optical illusion is caused by the adjacent buildings and trees that appear to be close to it.
This illusion may be called the angular size illusion and optical scientists are still discussing the best explanation for this common illusion.
It may be partly explained by the Ebbinghaus illusion where, for most people, black circle "A" looks bigger than "B".
The explanation that the moon illusion is caused by also seeing adjacent buildings and trees may be refuted by the observation of the moon illusion at sea.

36.3.15 Sun and Moon diameter illusion
The Sun and the Moon appear to be greater in diameter at sunrise and sunset than when vertically overhead.
The appearance of the size of an object depends on the angle subtended by the object, α, and the distance from the object, d.
The distance may be known or may be estimated from the size of other familiar objects.
However, if there are no other objects in view the distance is usually underestimated.
We estimate that the sky is a saucer-shaped covering, so when the Sun or Moon is at its zenith, we estimate that it is closer than when at the horizon.
So its size is estimated as being greater that at the horizon.

36.3.16 Supermoon
1. The distance between the Moon and the Earth varies by about 50, 000 kilometres, because of its elliptical orbit.
A supermoon is when a full Moon occurs when the Moon is at its closest approach to the Earth.
The most impressive view is when the Moon was close to the horizon, but that is mostly due to the Moon illusion.
On 24-06-2013 the Moon appeared to be up to 15% larger than normal, a "supermoon", when the Moon is 353, 000 km from the Earth, much closer than the average 380, 000 km orbit.
In 2014 the biggest supermoon was on 10 August when the Moon is closest to the Earth.
When it first appeared in China it was quite elliptical.
On 24 June, 2013, the supermoon caused a larger than usual king tide along the Australian eastern coast resulting in beach erosion in holiday areas, e.g. the "Gold Coast" near Brisbane.
On 14 November, 2016, the "superest" supermoon was the biggest and brightest for almost 70 years, when it was 8 per cent larger than normal as it orbited 356, 512 km from Earth.
The term "supermoon" is popular, but is not used by astronomers who call the phenomenon a "perigee full moon".
The perigee is the point nearest to the Earth in the path of a body orbiting the Earth.
There is no evidence that the full moon causes insanity, although this is a popular belief among police and health professionals who claim to be busier on full moon nights.
The reason may be that more crimes occur in the moonlight and some people my lose sleep in the moonlight and then have psychiatric problems.
2. As the rare super blue blood moon put on a spectacular show on 01-02-2018, Australians and people around the world flocked outside to capture the unusual lunar spectacle.
It is the first time in 35 years a blue moon has synced up with a super moon and a total lunar eclipse, or blood moon, because of its red hue.
A blue moon is a full moon that occurs twice in a month, while a blood moon takes on a reddish hue during an eclipse.
The "super moon" applies when the moon appears bigger, because it is close to the Earth.
The red appearance is caused by the light filtering and bending properties of our atmosphere.
NASA is calling it a "lunar trifecta", the first super blue blood moon since 1982.
That combination won't happen again until 2037.

36.4.1 Build a sundial

36.4.1 Build a sundial
See diagram 36.70.5: Horizontal sundial.
The gnomon is the part of the sundial that produces the shadow.
The top edge of the gnomon must slant upward away from the base, or horizontal, at an angle equal to the latitude of the observer and towards the South for an observer in the Southern Hemisphere.
The gnomon must be aligned along the N-S meridian.
The hour lines are marked on the other part of the sundial, called the time plane.
The configuration of the gnomon and the time plane identifies the type of sundial constructed.
In the diagram, the shaded area represents a sundial.
The top edge of the gnomon is parallel to the Earth's axis and the angle, γ (gamma), between the top edge of the gnomon and the horizontal is equal to the latitude of the observation site.
A horizontal sundial has the hour lines marked on a time plane horizontal to the Earth's surface.
You can use the data to construct your horizontal sundial.
The table contains hour angles for some cities and towns by using spherical trigonometry.
Thhe hour angles vary with latitude.
Table 36.70.0: Hour angles for the horizontal sundial contains: Time, Places, Hours
Ffor example: B = Brisbane, R = Rockhampton, M = Mackay, C = Cairns, T = Toowoomba, L = Longreach, M = Mt. Isa
On a square sheet of cardboard draw a line perpendicular to one edge to represent the 12 h 00 m hour line.
Use a protractor to draw lines spreading out from the 12 h 00 m hour line at the angles in the table if you are in one of the places in Table 3.
Outside these places, find out the latitude of your place and estimate the hour angles.
Label the hour lines as in the diagram.
Use another piece of cardboard to cut out the gnomon with one angle equal to the latitude of your location.
Attach the gnomon to your sundial base along the 12 h 00 m hour line, with the angle equal to your latitude pointing North.
The angle shown in Figure 3 is the latitude of Brisbane.
Align the gnomon along the N - S meridian.

36.4.2 Demonstration sundials
See diagram 36.68: Shadow stick sundial, Circular plate sundial.
1. Make a shadow stick sundial.
Demonstrate a simple sundial by placing an upright stick in the ground so that it is not in the shade.
At hourly intervals, mark the position of the shadow from the top of the stick on the ground.
2. Make a simple dial from a circular metal or plastic plate divided into 24 equal arcs.
Push a steel knitting needle through the centre of the plate so that the plane of the plate is at right angles to the needle.
Fix the plate so that the gnomon (i.e. the needle), points towards the celestial pole.
If the noon position of the shadow of the gnomon falls on the XII marking, the shadow will then fall on the other markings, close to correct time.
Mark the plate on both sides, because the shadow of the gnomon will move from one side to the other as the Sun's declination changes.

36.4.3 Find due north with a sundial
See diagram 36.13: North-south meridian.
1. Find due north to align the gnomon of the sundial along the north-south meridian.
Draw a circle on a cardboard base.
Attach a shadow stick to the base at the centre of the circle and put the apparatus in a sunny location.
Use a plumb bob to check that the shadow stick is vertical.
Mark Point M where the shadow just touches the circle in the morning.
Mark Point A where the shadow just touches the circle in the afternoon.
The line drawn from the shadow stick to the midpoint of MA represents due north-south.
Shadow stick
2. Use a shadow stick to find the shortest shadow of the day.
The direction of the shortest shadow is due north-south.
3. Set up the sundial so that the shadow is aligned with local apparent time of 10 h 15 m at exactly 10 h 30 m zone time, so that the gnomon is pointed due north-south.
Use a shadow stick to find the direction of the Earth's daily rotation.

36.4.4 Flowerpot sundial
Use a stick fixed through the hole in a flowerpot.
Mark the position of the shadow on the flowerpot rim each hour.

36.4.5 Lengths of a sundial shadow during the year,
See diagram 36.6.1: Length of a sundial shadow during the year.
See diagram 36.6.2: Analemma curve.
The apparent path of the Sun in the sky during the year is called the analemma.
The path can be shown by direct photographs of the Sun each day or by recording the position of the shadow of a vertical rod or gnomon at the same time each day, e.g. at noon.
Analemma
The analemma curve is the figure of eight formed by plotting the position of the Sun at the same place and at the same time during the year.
The variation along the long axis of the 8 is caused by the Earth's axial inclination.
It is highest at the summer solstice and lowest at the winter solstice.
Variation across the short axis is caused by the eccentricity of the Earth's orbit.

36.4.6 Make a sundial for your home
See diagram 36.69: Sundial for the Northern Hemisphere.
Make the base with a flat rectangular piece of wood, metal or polystyrene.
The gnomon ABC consists of a thin triangular piece of metal or plastic and such that angle ABC = latitude of the place at which the dial is being set up and angle ACB = 90^o.
Use a spirit level to test that the base is horizontal.
The central line must lie along the north south line, i.e. the meridian.
Erect the gnomon vertically so that the hypotenuse points towards the Pole Star in the Northern Hemisphere and the celestial south pole in the Southern Hemisphere.
For approximate results, make the hour markings by noting the position of the shadow of the gnomon at hourly intervals, using a watch set to local mean time.
You can obtain more accurate results if the markings are made on 15 April, 15 June, 1 September or 24 December, when there is no difference between watch time and dial time.
Errors of up to 16 minutes are possible if you make markings on other dates.
For accurate hour markings, find the angles the markings make with BC using the following the formulae:
tan IOC = tan 15^osin lat., tan IIBC = tan 30^osin lat, tan IIIBC = tan 45^osin lat., tan IVBC = tan 60^osin lat, tan VBC = tan 75^osin lat., tan VIBC = tan 90^osin lat.
Since the markings are symmetrical about the central line XY, you do not need to calculate other angles.
If the base of the dial is erected vertically, then the angle between the gnomon and the base must equal 90^ominus latitude of that place.

36.4.8 Pocket sundial
Cut a wire coat hanger in half and set the angle to the latitude of your location.
Attach the coat hanger to a cardboard base marked with the hour lines and align the gnomon north south.
Use the sundial to investigate the altitude of the Sun and the passage of time during the day.
Maintain daily records of the progress of sunrise and sunset to the North and South.

36.4.9 Sundial for the Northern Hemisphere
See diagram 36.69: Sundial for the Northern Hemisphere.
Make the base with a flat rectangular piece of wood.
The gnomon ABC is a thin triangular piece of metal.
Angle ABC = latitude and angle ACB = 90^o.
Use a spirit level to test that the base is horizontal.
The central line must lie along the north-south line, i.e. the meridian.
Fix the gnomon vertically so that the hypotenuse points towards the pole star in the Northern Hemisphere and the celestial south pole in the Southern Hemisphere.
For approximate results, make the hour markings by noting the position of the shadow of the gnomon at hourly intervals, using a watch set to local mean time.
Get more accurate results by making the markings 15 April, 15 June, 1 September or 24 December, which is when there is no difference between watch time and dial time.
The markings are symmetrical about the central line XY so do not calculate other angles.
If the base of the dial is made vertical, then the angle between the gnomon and the base must equal 90^ominus the latitude.

36.4.10 Sundial for the Southern Hemisphere
See diagram 36.68.
Place an upright metre stick in the ground so that it is not likely to be shaded from the sun.
Mark the position of the top of the metre stick on the ground at hourly intervals.

36.4.11 Universal globe sundial
See diagram 36.70A: Universal globe sundial.
Use a globe of the Earth to make a sundial that shows the season of the year, the regions of dawn and dusk, and the hour of the day wherever the Sun is shining.
The globe is rigidly oriented as an exact model of the Earth in space, with its polar axis parallel to the Earth's axis, and with your own town "on top of the world".
First turn the globe until its axis lies in your local meridian, in the true north and south plane.
Find this by observing the shadow of a vertical object at local noon, or by observing the Pole Star on a clear night, or by consulting a magnetic compass.
You should know the local variation of the compass, the magnetic deviation.
Turn the globe on its axis until the circle of longitude through your home lies in the meridian.
Tilt the axis around an east west horizontal line until your home town stands at the very top of the world.
Now your meridian circle connecting the poles of your globe lies vertically in the north south plane.
A line drawn from the centre of the globe to your local zenith will pass directly through your home spot on the map.
Lock the globe in this position and let the rotation of the Earth do the rest.
Be patient and do not turn the globe at a rate greater than that of the turning of the Earth.
However, it will take a year for the Sun to tell you all it can before it begins to repeat its story.
When you look at the globe fixed in this proper orientation, you can see half of it lighted by the Sun and half of it in shadow.
These are the actual halves of the Earth in light or darkness at that moment.
An hour later, the circle separating light from shadow has turned westward and its intersection with the equator having moved 15^oto the west.
On the side of the circle west of you, the Sun is rising, on the side east of you, the Sun is setting.
You can count the hours along the equator between your home meridian and the sunset line and estimate how many hours of sunlight remain that day.
Look to the west of you and see how soon the Sun will rise there.
As you watch the globe day after day, you will become aware of the slow turning of the circle northward or southward, depending upon the time of year.

36.1.0 Astrology
Astrology and the zodiac
The ecliptic is divided into 12 equal sections of 30^o, each containing a 'zodiacal' constellation, a sign of the zodiac.
On or near 21 March each year, the Sun moves into 0^o of Aries, first point of Aries, which defines the start of the tropical year of 365.242194 mean solar days.
The timetable for the Sun passing through the 12 signs of the zodiac as follows, may vary plus or minus 1 day depending on leap years:
Aries (Ram) 21 March to 20 April, Taurus (Bull) 21 April to 20 May, Gemini (Twins) 21 May to 21 June, Cancer (Crab) 22 June to 23 July, Leo (Lion) 24 July to 23 August, Virgo (Virgin) 24 August to 23 September, Libra (Scales) 24 September to 23 October, Scorpius (Scorpion) 24 October to 22 November, Sagittarius (Archer) 23 November to 22 December, Capricornus or Capricorn (Goat) 23 December to 20 January, Aquarius (Water carrier) 21 January to 19 February, Pisces (Fish) 20 February to 20 March.
The zodiac is the circular band of stars seen along the same path as the Earth's orbit around the Sun.
It is a belt on the celestial sphere 8^o on either side of the ecliptic, forming a background to the motion of the Sun, moon and planets.
In twelve groups, these stars make up the twelve signs of the zodiac, each 30^o long.
They are named after the constellations identified during the time of the ancient Greek astronomers.
Astrologers believe that the positions of heavenly bodies when you were born influence what you are so they match zodiac signs with human characteristics.
Some traits associated with signs of the zodiac:
Aries: aggressive, courageous, self-motivating, impulsive.
Aries was the first constellation of the zodiac, but the vernal equinox, the point at which the Sun crosses the celestial equator from south to north, also called the spring equinox and the first point of Aries, is now moved into Pisces, because of precession causing the movement westwards by one seventh of a second of arc daily.
Taurus: determined, practical, unemotional, calm, Taurus contains Alderbaran and the Pleiades, the sign of the zodiac which the Sun enters about 22 April, Gemini: versatile, restless, talkative, superficial,
Cancer: persistent, possessive, changeable, moody,
Leo: leadership ability, self-concerned, generous, egotistical,
Virgo: modest, diligent, picky, intellectual snob,
Libra: fair minded, diplomatic, hesitant, lover of peace,
Scorpio: subtle, determined, possessive, intense,
Sagittarius: friendly, optimistic, enthusiastic, restless,
Capricorn: resent all interference,
Aquarius: remarkable spiritual healers.
(The "age of Aquarius" is a time of freedom, including sexual freedom, and general brotherhood.)
Pisces: creative, changeable, emotional, devious.
Experiment
List which of the traits in the list describe (a) yourself (b) a friend.
Then ask the friend to make a similar list.
How many traits in the list were according to the astrological prediction?

36.45 Simulated solar eclipse
See diagram 36.96: Simulated solar eclipses.
Represent the Sun with an opal electric bulb shining through a circular hole 5 cm diameter in a piece of blackened cardboard.
Draw the corona in red crayon around this hole.
The Moon is a wooden ball, 2.5 cm diameter, mounted on a knitting needle.
View the eclipse through any of several large pin holes in a screen on the front of the apparatus.
The corona becomes visible only at the position of total eclipse.
Adjust the Moon's position with a wire bicycle spoke attached to the front of the apparatus.

36.48 Lengths of day and night
See diagram 36.99: Differences in the length of day and night.
Days and nights are of equal length only at the equator.
Draw a large circle to represent the Earth's orbit.
Draw two lines perpendicular to each other through the centre.
Where they cut the circle, label the intersections in counter clockwise order: 20 March, 21 June, 23 September, 21 December.
These are positions of the Earth in relation to the Sun on these dates.
Draw a small circle for the Earth at the 21 June position.
The north pole will be off centre about the radius of the circle, towards the Sun.
For any other date or orbital position, which can be located by using a protractor, the Earth circle and pole will have the same orientation.
The Arctic circle, tropic of Cancer, and equator can be drawn in.
Then a line through the centre of the Earth circle and perpendicular to the Earth-Sun line will be the boundary between daylight and darkness.
From such a diagram, estimate the duration of sunlight at different latitudes for any date.
For example, 1 August at the Arctic circle, the Sun would be estimated as up for about 18 hours, but up only up to six hours on 1 November.

36.49 Angle of the Sun's rays on the Earth
If the rays from the Sun are assumed to be parallel, then the heating effect of the Sun on the Earth can be seen to be greatest at the equatorial region, not because the equator is closest to the Earth, but because the Earth curves less in the equatorial region.
Also, the rays of the Sun have less atmosphere to pass through in the equatorial region.
Experiment
1. Show the effect of the angle of the Sun's rays on how much heat and light received by the Earth.
Bend a piece of cardboard and make a square tube 2 cm × 2 cm × 32 cm.
Use a piece of very stiff cardboard and cut from this a strip 23 cm long and 2 cm wide.
Paste this to one side of the tube with 15 cm extending.
Rest the end of the stiff cardboard on the table and incline the tube at an angle of about 25^o.
Hold a flashlight or lighted candle at the upper end of the tube and mark off the area on the table covered by the light through the tube.
2. Repeat the experiment with the tube at an angle of about 15^o.
3. Repeat the experiment with the tube vertical.
Compare the size of the three spots and find the area of each.
Show the analogy between this investigation and the way in which the Sun's rays impinge on the Earth's surface.
Note whether the amount of heat and light received per unit area from the Sun is greater when the rays are slanting or direct.

36.49.1 Model of the solar system, orrery
In 1700, the Earl of Orrery in Ireland ordered the construction of a model of the universe made of wood and brass.
festival days "leap" a day.
The Gregorian calendar was not adopted in Great Britain until January, 1752.
The Jewish Calendar dates from the Creation fixed at 3761 B.C.
James Ussher, Archbishop of Armagh in 1658 published his calculation of the date of creation being 4004 BC.
This calculation has been ridiculed by some modern scientists, but it did involve a great deal of scholarship based on the Bible.
The Mohammedan Calendar dates from 16 July 622, the date of the Hegira.
29th February is called an intercalary day.
In England, the new style quarter days are Lady Day 25 march, Midsummer day 24 June, Michaelmas Day 11 October, "Old Christmas Day 6 January.
Midsummer is the weekly period around the summer solstice 21 June.
The term millennium (1 000 years), comes from St. John's gospel of the Bible and refers to the period of a thousand years when Christ will return to Earth and live with his saints and finally take them to heaven.
2. In AD 325 the Council of Nicaea determined that Christmas Day be celebrated always on the 25th December, an immovable feast, but Easter day remained a movable feast.
Easter Day is now determined in the United Kingdom by the Calendar (New Style), Act of 1750, as the first Sunday after the full Moon that happens upon or next after the twenty first day of March, and if the full Moon happens upon a Sunday, Easter Day is the Sunday after.
So Easter Sunday can be from 22 March to 25 April.
Most countries use the Gregorian Calendar and the date of Easter used in the United Kingdom.
However, Eastern Orthodox Churches may still use the Old Style Calendar and have a different date for Easter Sunday.
Their Xmas Day is 7 January.
Yule (Norse: jól), was a pagan festival for the winter solstice, but nowadays Yule refers to Christmas festivities.
3. The 21st century started in 2001, because the first century started in AD 1.
The year before it was 1 B.C., so there was no "year 0".
B.C. stands for "before Christ" so the years are numbered backwards.
AD stands for the Italian "anno domini", in the year of the Lord.
In AD 525, Dionysius Exiguus decided on the start of the present calendar so that Jesus Christ was born on December AD 1.
But Jesus Christ may have been born as early as 4 B.C.
4. However, if Jesus was born Sunday, 1 March, 7 B.C., this was the year of the triple conjunction of the same two planets when in 27 May, 5 October and 1 December, Jupiter moved close to Saturn in the constellation Pisces.
The conjunctions were first calculated by the astronomer Johannes Kepler in 1603.
The first conjunction may have started the magi on their journey to Israel.
The second conjunction may have guided them.
The third conjunction in December may have pointed to the birth of Jesus.
However, there was also a conjunction of Venus and Jupiter in Leo in June 2 B.C.
In AD 314 Emperor Constantine the Great changed the date of the birth of Jesus from 1 March to 25 December, to be the same date as a pagan Sun festival.
The star seen in the east to guide the wise men is only mentioned in the Gospel according to St. Matthew.

36.51 "On the Revolutions of the Celestial Spheres", by Nicolaus Copernicus
The publication of "De revolutionibus orbium coelestium" (On the Revolutions of the Celestial Spheres), in 1543, by Nicolaus Copernicus (1473-1543), supported the heliocentric model of the universe, that the Earth is the centre of the Moon's orbit and that the motions of the planets are the result of the Earth's own rotation around its axis and of its travel in orbit.
Georg Joachim de Porris, called Rheticus (1514-1574), the author of trigonometric tables and taught by Copernicus, assisted the publication, which contradicted the geocentric "Almagest" treatise by Claudius Ptolemy (90-168), on the apparent motions of the stars and planetary paths.

36.53 Discover thrust
See diagram 36.104: Measure thrust from a balloon with a beam balance.
1. Measure thrust with a balance.
Put 50 g masses on one pan.
Firmly hold an inflated balloon over the other pan.
Allow the air to escape against the pan.
Note how many gm weight of thrust the escaping air exerts on the pan.
2. Turn on the tap and feel the thrust produced when water passes through a garden hose.
Turn the tap on more.
As the amount of water passing through the hose increases, the hose begins moving in the opposite direction to the moving water.
Attach a rotary lawn sprinkler to the hose.
Gradually turn on more water and note how the speed of the lawn sprinkler increases as the amount of water increases.
3. Large rockets may produce 300 000 to 1 000 000 kg weight of thrust.
If a rocket weighs 5 000 kg, the Earth's gravity is pulling down on this rocket with a force of 5 000 kg weight.
Before the rocket can rise, it must overcome that pull towards the centre of the Earth so the rocket's thrust must exceed 5 000 kg weight.

36.54 Discover weightlessness
See diagram 36.105: Weightless toy soldier.
1. Use string to attach a toy soldier to two arms of a framework.
Take the apparatus to a high building and drop it out of the window.
While falling, the toy soldier remains in the same position relative to the framework.
The toy soldier is not supported by either the string or the frame, but is in a weightless condition with regard to the surroundings.
To study the motion of an object we need a reference system, e.g. something relative to which it is possible to describe the location of the object at any time.
For many experiments we choose a reference system fixed to the Earth, e.g. study a falling object.
In such a reference system the Earth is at rest.
2. The weight of an object also depends on its location.
Measured in a reference system fixed to the Earth, the weight of an object is the same as the Earth's gravitational force acting on it.
This force decreases as the object moves away from the Earth and will eventually become negligible.
The weight of the object is changing under the above circumstances.
The content of matter of the object does not change, unless approaching that of light.
An astronaut whose mass on the surface of the Earth is 90 kg still has the same mass of 90 kg on the surface of the Moon, but 90 kg weight on the Earth's surface is only about 15 kg weight on the Moon's surface.
Using SI units, the mass is m kg, but the weight is mg Newton.
Since g at the Moon is about one sixth of g at the Earth, the weight of an astronaut on the Moon will be one sixth of the weight on the Earth.
3. A spaceship in orbit is still within the Earth's gravitational field.
Its weight is exactly the force required to keep the spaceship in orbit.
However, in a reference system attached to the spaceship, everything inside is weightless.

36.70.7 Parallel rays of the Sun
BE CAREFUL!
DO NOT LOOK AT THE SUN THROUGH THE TUBE BECAUSE THE DIRECT SUN RAYS CAN DESTROY THE RETINA OF YOUR EYE.
1. To show that the sun's rays are parallel as they fall on the Earth, on a bright morning, point a piece of pipe or a cardboard tube at the Sun so that it casts a small, ring-shaped shadow.
If, at the same moment, a person 120^oeast of you, one third of the way round the world, does the same experiment, that person points the tube westward at the afternoon sun.
That tube and yours are approximately parallel.
If you point the tube at the Sun in the afternoon, and someone far to the west simultaneously does the same in the morning, that tube will be approximately parallel to your tube.
So when your globes are properly set up, people all over the world who are in sunlight can see them illuminated in just the same way.
2. You can tell from the global sundial how many hours of sunlight any latitude receives on any particular day.
Count the number of 15^o longitudinal divisions that lie within the lighted circle at the desired latitude.
Thus, at 40^onorth latitude in summer the circle may cover 225^oof longitude along the 40th parallel, representing 15 divisions or 15 hours of sunlight.
However, in winter the circle may cover only 135^o, representing nine divisions or nine hours.
When the lighted circle passes beyond either pole, that pole has 24 hours of sunlight a day, and the opposite pole is in darkness.

36.76 Constellarium
A constellarium is a simple device used in teaching the shapes of various constellations.
Experiment
1. Use a cardboard or wooden box and remove one end.
Draw the shapes of various constellations on pieces of dark-coloured cardboard large enough to cover the end of the box.
Punch holes on the diagrams where the stars are found in the constellations.
Place an electric lamp inside the box.
When the lamp is turned on and various cards are placed over the end of the box, the constellations may be seen clearly.
2. Use tin cans into which an electric lamp may be fitted.
Punch holes in the bottoms of the cans to represent the stars in various constellations.
When the lamp is placed inside a can and switched on, the light shows through the openings and the shape of the constellations may be observed.
The tin cans may be painted to prevent rusting and kept from year to year.

36.76.1 Umbrella constellarium
See diagram 36.76Bd: Northern Hemisphere, Southern Hemisphere.
An umbrella has the shape of the inside of a sphere, so it can be made into a constellarium that will illustrate portions of the sky and how they appear to move.
Use an old umbrella that is large enough for this purpose.
Northern Hemisphere:
Using chalk, mark the North Star, or Polaris, next to the centre on the inside of the umbrella.
Consult a star map, and mark the star positions for constellations with crosses.
Paste white stars made from gummed labels over the crosses or paint the stars with white paint.
Draw dotted lines to join the stars in a given constellation.
Rotate the handle of the umbrella in a counter clockwise direction so see how the stars trace a circular path about the Pole Star.
Southern Hemisphere:
South of the equator, point the umbrella towards the southern celestial pole and turn it clockwise.
Both in the Northern Hemisphere and Southern Hemisphere the stars will rise in the east and set in the west.
In the diagram above note stars and constellations marked on the umbrella.

36.77 Seasonal shift of the sky
As the Earth travels in its orbit around the Sun the constellations seem to move across the sky.
The materials required for observing the shift are a star chart and a plumb line.
Make only one set of observations and record the time.
At least one month later, repeat the same observations in the same way, at as nearly the same time of night as possible.
Compare the two observations made at the same time of night and note what change do you see in one month.
Calculate how much change would occur in one year, if the same rate continues.
Answer the same questions for the Big Dipper and North Star, if you are north of the equator or for the Southern Cross if you are south of the equator.

36.78 Time and date using the stars
Because the stars appear to rotate one full revolution in 24 hours, they can be useful in telling time, at least during those hours of darkness when the stars are visible to us.
Because the stars also make one full revolution in a year, they can be used to tell us the time of the year.
So we have not only a star clock, but also a star calendar.

36.78.1 Star calendar
See diagram 36.78BN: Northern Hemisphere.
See diagram 36.78BS: Southern Hemisphere.
The dates round the edge of the star chart for the Northern Hemisphere show when the corresponding part of the heavens is due north at midnight.
For the Southern Hemisphere the dates show the part that is due south at midnight.
Knowing this, you can easily rotate the star map so that it corresponds to what you see in the sky.
If you are north of the equator and you have to rotate the map 15^o clockwise from the midnight position, the time is 1.00 a.m.
If you have to rotate it 30^ocounter clockwise, the time is 10.00 p.m.
South of the equator it is the other way round since you are facing south.
If you have to turn the map 15^o clockwise from the midnight position it means that the time is 11.00 p.m.
The times determined this way are "Sun times" and they may differ from your local "standard time".

36.91a Photograph star trails in colour
The stars are colourful, but this is not generally known, because dark adapted eyes have low sensitivity to colour.
High speed colour film and a camera with at least an f 3.5 lens will record the red star Betelgeuse in the constellation Orion, the yellow star Capella in the constellation Auriga, and the gold star Albireo in the constellation Cygnus.
The constellation Cassiopeia contains two blue, one white, one golden, and one green star.
A good camera that can make time exposures, a rigid tripod, and fast film are all you need.
The simple star charts in this book will help you to identify the constellations.
Your local public library may have books on amateur astronomy that contain similar charts.
Dial indicators that show all the constellations overhead when the dial is set for the month, day, and hour, are also obtainable in some countries.
The Earth rotates 15^o per hour, or 1^o every 4 minutes.
To us on the Earth, it is easier to appreciate this movement by assuming that the stars move.
Furthermore, the stars appear to rotate around your celestial pole.
Each star near the pole traces a tight circle in its movement, and as the distance from the pole increases, the radius of curvature increases until the stars above the equator appear to travel in straight lines.
A star is a true point source of light and no movement of the camera can be tolerated unless you want pigtails for star images.
Mount your camera on a rigid tripod, cover the lens with a cardboard, use a long cable release to open the shutter on time or bulb, wait 3 seconds for the camera to stop moving, and then remove the cardboard from in front of the lens.
At the end of the exposure, again cover the lens with a cardboard before closing the shutter.
Film processing laboratories will probably not recognize star images for what they are and, unless you instruct them otherwise, will return your negatives unprinted.

36.92 Photograph constellations
See diagram 36.92: Old 35 mm slide used for teaching about constellations.
1. Select a constellation, set up the camera, and expose for 30 minutes with high speed black and white film, 400 ASA and a lens opening of f 11, then cover the lens for 2 minutes, open it to f 4, and throw it slightly out of focus, finally, uncover the lens for 3 more minutes.
A diffusion screen over the lens for the final exposure works just as well as throwing the lens slightly out of focus.
The resulting picture shows a constellation that appears to be plunging through space with a tail following each star.
2. Underexposed and discarded 35 mm film slides can be perforated with a pinpoint in the form of various constellations.
The slides can be projected on to a screen or used in a viewer, and students can try to identify the constellations.
The slides can also be dropped into a slot made in a mailing tube and held up to the light.

36.93 Photograph satellites
Use the same camera technique above for star trails.
The main problem is to know ahead of time where to aim the camera.
Local newspapers may publish daily the times, the degrees above the western or eastern horizon, and the direction of travel for all visible satellites.
Also, local astronomical observatories and amateur astronomical clubs may have the required data.
Satellite photography is particularly interesting when the satellite path crosses a well known constellation, or if two satellites cross within the photograph.

36.94 "Twinkle, twinkle little star"
Twinkling star (stellar scintillation, astronomical scintillation), is caused by refraction of star light when moving though different density layers in the atmosphere.
Stars close to the horizon twinkle more, because the light from them has passed through more atmosphere.
The turbulence in Earth's atmosphere affects the path of light reaching our eyes.
Sirius has the clearest twinkle effect, because it is so bright and it appears low on the horizon in the northern hemisphere.
Planets appear bigger than stars, so any twinkling effect is not usually noticeable.
"Twinkle, twinkle little star" is the first line of a poem published in 1806, "The Star" by Jane Taylor, of Ongar, England.

36.106 Satellite launcher
See diagram 36.106: Simulated satellite launcher.
Materials required are a bucket, a football, a coat hanger, or other suitable wire, sinker or weight, a piece of string and a test-tube or a cap of some sort.
Place the ball securely in the bucket.
Bend the wire so that about 30 cm of it is straight and the rest is curved into a circular base as shown in the sketch.
Using masking tape, secure the circular portion on the ball, allowing the straight, 30 to stand upright in the centre of the top of the ball.
Attach the sinker or weight to the string.
Fasten the other end of the string to the test-tube or cap with tape.
Invert the cap on top of the upright wire, see diagram.
Explain that the ball represents the Earth, and the sinker represents the satellite.
All that it takes to set the sinker into motion in any direction is the tap of a finger.
Let the students find out what happens when the satellite is launched in the following different ways:
1. With a slight tap, push the sinker up and away from the surface of the ball, as shown in the figure.
The sinker moves up and then falls back to the starting point.
This is how an object travels when it is projected at low speed straight up from the Earth.
2. With a slight tap, push the sinker of f the surface of the ball at an angle.
Show by a diagram what happens.
The sinker moves away from the ball and then falls back at some distance from the starting point.
The distance spanned depends upon the angle of launching and upon the forcefulness of the tap.
3. With a stronger tap, push the sinker of f the surface of the ball at an angle.
Make a diagram of the orbit.
The sinker moves away from the ball, circles it, and lands.
Evidently, a complete orbit passes through the starting point of the orbit.

36.107 Kepler's laws of planetary motion
(Johann Kepler 1571-1630)
1. Each planet orbits in an elliptical path, with the Sun at one focus.
2. A line joining the Sun and any planet sweeps out equal areas in equal time intervals.
3. The squares of the orbital periods of the planets are proportional to the cube of their mean distances from the Sun.
The ratio r^{3 /}T²is the same for all planets, where r is the mean orbit radius, and T is the period of revolution.
For all satellites of the Sun, r³/ T² = 3.3 × 10^-18m³/ sec².

36.40 Astrology and the zodiac
(Greek zōion animal, because most of the zodiac signs are animals.)
1. The ecliptic is divided into 12 equal sections of 30^o, each containing a constellation, a sign of the zodiac.
On or near 21 March each year the Sun moves into 0^o of Aries, first point of Aries, which defines the start of the tropical year of 365.242 194 mean solar days.
The timetable for the Sun passing through the 12 signs of the zodiac as follows, may vary plus or minus 1 day depending on leap years:
Table 36.40

Constellation	Period
Aries (Ram)	21 March to 20 April
Taurus (Bull)	21 April to 20 May
Gemini (Twins)	21 May to 21 June
Cancer (Crab)	22 June to 23 July
Leo (Lion)	24 July to 23 August
Virgo (Virgin)	24 August to 23 September
Libra (Scales, balance)	24 September to 23 October
Scorpius (Scorpion)	24 October to 22 November
Sagittarius (Archer)	23 November to 22 December
Capricorn (Goat)	23 December to 20 January
Aquarius (Water carrier)	21 January to 19 February
Pisces (Fish)	20 February to 20 March

2. The zodiac is the circular band of stars seen along the same path as the Earth's orbit around the Sun.
It is a belt on the celestial sphere 8^o on either side of the ecliptic, forming a background to the motion of the Sun, Moon and planets.
In twelve groups, these stars make up the twelve signs of the zodiac, each 30^o long.
They are named after the constellations identified during the time of the ancient Greek astronomers.
Astrologers believe that the positions of heavenly bodies when you were born influence what you are so they match zodiac signs with human characteristics.
The ascendant is a point on the ecliptic, i.e. degree of the zodiac rising above the eastern horizon just as a particular event occurs, especially the birth of a child.
This point changes as the Earth rotates on its axis.
The "house of the ascendant" is defined as 5 degrees of the zodiac above down to 25 degrees of the zodiac below the point.
Any planet "within the house" is called "lord of the ascendant" and is supposed to influence the life of the child.
So a person gaining in influence or prosperity is said to be "in the ascendant".
3. Some traits associated with signs of the zodiac:
Aquarius: erratic, detached, honest
(The "age of Aquarius" is supposed to be a time of freedom, including sexual freedom, and general brotherhood.)
Aries: aggressive, courageous, self-motivating, impulsive, dynamic, selfish, irascible.
Aries was the first constellation of the zodiac, but the vernal equinox, the point at which the Sun crosses the celestial equator from south to north, also called the spring equinox and the first point of Aries is now moved into the area of Pisces, because of precession causing the movement westwards by one seventh of a second of arc daily.
Cancer: persistent, possessive, moody, cautious,
Capricorn: resent interference, patient, careful, fatalistic,
Leo: leadership ability, generous, egotistical, patronizing,
Libra: fair minded, diplomatic, urbane, indecisive, during 24 September to 23 October the day and night periods are about equal, i.e. have equal "weight", so are "balanced".
Pisces: creative, changeable, emotional, intuitive,
Sagittarius: friendly, optimistic, enthusiastic, restless,
Scorpio: subtle, determined, possessive, compulsive,
Taurus: determined, practical, unemotional, inflexible,
Virgo: modest, diligent, reliable, fussy.
4. List which of the traits in the list describe yourself and a friend.
Then ask the friend to make a similar list.
Note how many traits in the list were according to the astrological prediction.

36.5.0 Transit of Venus 2012
See diagram 36.26: Transit of Venus across the Sun.
1. Jeremiah Horrocks was the first person to successfully predict and observe a transit of Venus in 1639.
Observations of the transits of Venus became scientifically important when in 1716 Edmund Halley proposed that observations from different locations on the Earth could be used to determine the distance between the Sun and the Earth (called the Astronomical Unit), and the scale of the solar system could be subsequently determined by applying Kepler's third law of planetary motion.
At that time there was great uncertainty about the size of the solar system and by extension the size of the universe.
Many scientific expeditions were sent around the globe to observe the 1761 transit and there was great competition between nations to solve what was called "the noblest problem in astronomy".
Unfortunately difficulties in accurately timing the transit led to conflicting results, and consequently an even greater effort was mounted for the 1769 transit.
Combined results from the two transits produced a better understanding of the solar distance, but further refinement was undertaken in later transits.
The transits of 1761 and 1769 and later at 1874 and 1882 provided a way of measuring the distance between the Earth and the Sun.
This was the key distance that astronomers needed to work out the scale of the Solar System and to establish the distances to the nearest stars.
The idea was to time the instants when Venus just appeared to touch the inside edge of the Sun at the beginning and at the end of the transit.
If the timing could be done accurately astronomers could compare observations from widely separated places and determine the distance by simple trigonometry.
The 1769 transit has a vital historical connection to Australia.
Lieutenant James Cook was dispatched to Tahiti on HMS Endeavour to observe the transit.
Captain James Cook, who observed the 1769 transit from the Pacific island of Tahiti, was despondent, because his times differed slightly from those of the two other observers with him.
He was not to know that observers elsewhere in the world had experienced similar problems and the observations from Tahiti were better than most.
After completing the necessary observations in Tahiti, Cook opened sealed orders to search for "Terra Australis Incognita" or the "Unknown Southern Land".
After a successful observation he was directed to search for the "great south land" thought to exist in the South Pacific Ocean and following that search he discovered and charted the east coast of Australia.
Scientific expeditions were spread across Australia to observe the 1874 transit and again for the 1882 transits, with many of these successfully recording the event.
2. What is a Transit?
See diagram 36.52.1: Transit of Venus.
A Transit of a planet occurs when the planet passes directly between the Earth and the Sun so that as seen from the Earth, the planet appears to pass across the face of the Sun.
Transits can only occur with planets whose orbit is between that of the Earth and the Sun; that is, Mercury and Venus.
A transit of a planet is similar to a solar eclipse, but the planet appears to be much smaller that the Moon so it cannot cover the Sun and looks like a small black disc slowly crossing the Sun.
3. How often do Transits occur?
Transits of Mercury occur quite regularly (with about 13 each century) but they are difficult to observe due to the very small apparent size of Mercury.
Transits of Venus are much rarer and are more interesting due to the larger apparent size of Venus and due to their historical connections.
Transits of Venus occur in a pattern that repeats every 243 years with pairs of transits eight years apart separated by gaps of 121.5 years and 105.5 years.
Venus and the Earth are aligned in the same direction out from the Sun about every 584 days (this is called in conjunction), however a transit does not occur each time, because Venus's orbit is usually above or below the Sun in the sky.
Since the phenomena was first recognized there have only been six transits of Venus - 1639, 1761, 1769, 1874, 1882 and the most recent one in 2004.
The 6th June 2012 transit is our last opportunity to observe a transit of Venus, as the next event occurs on 11th December 2117.
4. The Transit of 6th June 2012
See diagram 36.52.2: The Transit of 6th June 2012.
The latest transit of Venus occurred on Wednesday 6 June in Australia, 5 June in the USA).
As the following transit was not until 2017, this was the last opportunity for many to see one of the rarest and most famous astronomical events.
Australia and New Zealand was among the best places from which to view the 2012 transit as, clouds permitting, it was visible from beginning to end from most of the two countries.
From Sydney, the transit began at 8: 16 am and ended at 2: 44 pm AEST with similar times elsewhere in Australia, and New Zealand, after allowing for different time zones.
From Perth, the transit was already in progress at sunrise.
The entire transit was visible from New Guinea, Japan, Korea and the eastern parts of China and the Russian Federation.
It was fully visible from Hawaii and Alaska, while from the rest of the USA the transit was still be in progress at sunset.
From Europe (apart from parts of Spain and Portugal), the Middle East, eastern parts of Africa, India and Indonesia the transit was already be in progress at sunrise.
For the transit of 6th June 2012, Venus took about six and a half hours to travel across the face of the Sun.
Venus must be above the horizon for the transit to be visible.
The predicted path of Venus across the Sun's disc is shown in the diagrams above for locations on the east coast of Australia.
Venus travelled in a straight line across the Sun.
However, because the Sun appears to rotate as it crosses the sky, Venus appeared to move in an inverted "U" shape when viewed from Australia.
Timing for the transit is given in terms of "contacts".

5. Viewing the transit
It needed to be emphasized that looking at the Sun is highly dangerous.
Serious and irreparable eye damage occurred from viewing the Sun with the unaided eye or, even worse, through binoculars or a telescope.
Viewing the Transit of Venus across the Sun required special equipment.
For those without an alternate safe viewing option or if there is bad weather on the day, a live video of the whole event was broadcast.
Coverage will started from 8 am, Eastern Australian Time (GMT +10 hours) and concluded six and a half hours later after Venus had completed its transit.
The event was covered by high quality specialist telescopes in Australia to ensure the best possibility of getting a live feed.
6. Timing of the transit and the "black drop"
See diagram 36.52.3: "Black drop" (Diagram from The University of Queensland, School of Mathematics and Physics).
It is now possible to predict the timing of the transit contacts with great precision, because we now have accurate information on the distance to the planets and the Sun.
When early astronomers were trying to measure the distance to the Sun, it was necessary to time the transit contacts as accurately as possible.
However they had great difficulty with this, because of what has become known as the "black drop" effect.
This effect occurs immediately after second contact and again immediately before third contact when Venus appears to be connected to the Sun's limb (edge) by a narrow dark zone.
The black drop effect is thought to be due to image blurring from atmospheric distortion and equipment diffraction coupled with solar limb darkening.
7. The scientific value of the Transit of Venus
The original benefit of observing transits of Venus was to assist in determining the Astronomical Unit.
Later on, transits were used to examine Venus's atmosphere using spectroscopy.
Currently there is a great deal of scientific effort directed towards the search for exoplanets (planets outside the Solar System) and planetary transits across distant stars are the main method used to search for them.
The 2004 and 2012 transits of Venus are providing a valuable benchmark and comparison with a known planet transiting a known star.
[1 AU Astronomical unit (approximately the average distance from the Sun to Earth) = 149, 600, 000 km],

36.4.7 Measurements using the Sun
Safety
Be careful!
Danger of the Sun's rays, risk to eyesight in unprotected viewing due to intense light and IR
Never look at the Sun with eyes, magnifying glass, sunglasses, binoculars, telescope, burnt glass.
Never stay in the Sun if your shadow is shorter than you.
1. Compare the lengths and directions of the shadow at noon to other times, shadow of a Sundial gnomon, pencil gnomon, hole gnomon (rectangular piece of metal with a round hole in it).
2. Join two positions of the shadows over 15 minutes of a 2 metre long stake to show east west direction.
3. Focus the Sun's rays, focus the light from the Sun on a screen, focus the heat from the Sun to burn paper.
4. Area of sun's radiation on tropical or polar regions
5. Use refraction through water to separate colours in the Sun's rays to form separate colours on a screen.
6. Use a pinhole camera Sun viewer to view an image of the sun.
7. Use binoculars to view an image of the Sun.
8. Use the Earth orbit machine and see the seasons in the Southern Hemisphere and Northern Hemisphere.
Table 36.25 Distance and diameter of planets
Planet............Distance (AU)....................Diameter (AU)
Mercury...........58 (0.4)...............................4 100 (0.4)
Venus.............108 (0.7).............................12 000 (1.0)
Earth..............150 (1.0).............................13 000 (1.0)
Mars...............228 (1.5)..............................6 100 (0.5)
Jupiter.............778 (5.2)..........................140 000 (11.2)
Saturn...........1 420 (9.5)..........................120 000 (9.5)
Uranus..........2 870 (19.2)..........................50 000 (3.7)
Neptune........4 490 (30.1)..........................53 000 (7.1)
Pluto.............5 900 (39.5)............................2 700 (0.2)

Table 36.3.5a Rising and setting Moon times

Phase	Rising time	Time in eastern sky	Time highest in sky	Time in western sky	Setting time
New Moon	Sunrise	Morning	Noon	Afternoon	Sunset
Waxing crescent	After sunrise	Morning	After noon	Afternoon	After sunset
First quarter	Noon	Afternoon	Sunset	Evening	Midnight
Waxing gibbous	Afternoon	Sunset	Night, before midnight	Midnight	Night, after midnight
Full Moon	Sunset	Night, before midnight	Midnight	Night, after midnight	Sunrise
Waning gibbous	Night, before midnight	Midnight	Night, after midnight	Sunrise	Morning
Third quarter	Midnight	Night, after midnight	Sunrise	Morning	Noon
Waning crescent	Before sunrise	Morning	Before noon	Afternoon	Before sunset

Table 36.42 Nautical measurements

1 fathom	1.83 m	6 feet
1 shackle	27.43 m	15 fathoms (anchor chains)
1 cable	185.32 m	608 feet
1 nautical mile	1.85 km	1 minute of arc (in English Channel)
1 knot (kt)	0.51444 ms^-1	1 nautical mile per hour