Gravitation and Satellites

The story says an apple fell in front of Isaac Newton, but the real discovery was a question: if the Earth pulls the apple down, does the same pull reach the Moon? Newton answered yes. The Moon is also falling towards the Earth all the time; it just keeps missing it because of its sideways speed. In his book Principia (1687) he stated the Universal Law of Gravitation: every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

In symbols: F = G m₁ m₂ / r². Read it as a sentence: the masses sit on top (more mass, more pull) and the distance sits below, squared (more distance, much less pull). Here G is the universal gravitational constant. For spheres, r is measured between their centres. The force on the Earth by the Moon and the force on the Moon by the Earth are equal and opposite (Newton's third law), even though the Moon accelerates much more because it is lighter.

Gravitation explains the motion of planets around the Sun, the Moon going around the Earth, the fall of objects, and the tides in the sea. The Moon is close, so its pull on the ocean water is strong. On the side facing the Moon the water bulges, and on the opposite side it bulges too, so most coasts see two high tides and two low tides a day. Because the Moon also moves on in its orbit, the tide arrives about 50 minutes later each day.

Drop a stone and a feather in a tube from which the air has been pumped out, and both reach the bottom together. The Earth gives every freely falling body the same acceleration, called the acceleration due to gravity, g. Its value near the Earth's surface is about 9.8 m/s², directed towards the centre of the Earth. The reason is hidden in the formula. The Earth pulls a body of mass m with force GMm/R², and this force also equals mg. The m cancels, so g = GM/R² and it does not depend on the mass of the falling body. In 1971 astronaut David Scott showed this on the Moon by dropping a hammer and a feather together; they landed at the same time.

Why would g be zero at the equator if the Earth spun faster? The spin takes away Rω² from g at the equator. Today Rω² is only about 0.034 m/s², but if the Earth turned about 17 times faster (a day of roughly 1 hour 24 minutes), Rω² would equal g and a body at the equator would be weightless. The poles are unaffected, because there the distance from the axis is zero.

People use 'weight' for the number on a bathroom scale, but in physics weight and mass are different things. Mass is the quantity of matter in a body. It is the same on the Earth, on the Moon and in deep space. Weight is the force with which a planet pulls the body, so W = mg. Since g changes from place to place, weight changes too. Radhika has a mass of 60 kg. On the Earth her weight is 60 × 9.8 = 588 N. On the Moon, where g is about 1.62 m/s², it is about 97 N. Her mass is still 60 kg in both places.

Now the important idea behind weightlessness. A weighing machine does not measure gravity directly. It measures the push of the floor on you (the normal reaction), and you feel your 'weight' because of that push. If the floor falls away with you, the push disappears and you feel weightless, even though gravity is still acting. That is exactly what happens in a lift in free fall and in a satellite.

Why do astronauts float? At the height of the International Space Station (about 400 km), gravity is still about 8.7 m/s², nearly 90 per cent of the surface value. Astronauts float because they and the station are in continuous free fall around the Earth, so nothing pushes on them. Weightlessness means 'no apparent weight', not 'no gravity'.

For centuries people believed that planets move in perfect circles. The German astronomer Johannes Kepler studied years of careful observations of Mars made by the Danish astronomer Tycho Brahe and found that circles simply did not fit. His three laws, published in 1609 (first two) and 1619 (third), describe how planets move around the Sun. Later Newton showed that all three follow from his law of gravitation, which is why they are taught in this chapter.

Why does the planet speed up near the Sun? Gravity pulls along the line to the Sun, so it gives no twist about the Sun. Therefore the planet's angular momentum (mvr) stays constant. When r becomes small near the Sun, v must become large. This is the same idea as a skater who pulls in her arms and spins faster. So the second law is really conservation of angular momentum.

Because Earth's orbit is only slightly elliptical, the Earth is closest to the Sun (perihelion) in early January, about 147 million km away, and farthest (aphelion) in early July, about 152 million km away. This small difference does not cause the seasons; the tilt of the Earth's axis does. Comets such as Halley's follow very long ellipses, racing near the Sun and crawling when far away. Halley's comet returns about every 76 years; it was last seen in 1986 and is due again in 2061.

Imagine standing on a very tall mountain and throwing a ball sideways. Throw it slowly and it lands nearby. Throw it harder and it lands farther away. Newton imagined a cannon so powerful that the ball's path curves down at exactly the same rate as the Earth's surface curves away; the ball then never lands and keeps going around the Earth. That is a satellite, and the sideways speed needed is the orbital velocity. Throw it even faster and it leaves the Earth for ever; that minimum speed is the escape velocity.

Why does a satellite not need an engine to keep going? In orbit, gravity is the only force and it points towards the Earth's centre, so it just keeps turning the satellite's path into a circle: GMm/r² = mv²/r. The mass m cancels, so every satellite at the same height, a tiny cube-sat or the heavy space station, moves at the same speed. A higher orbit has a smaller speed and a longer period. The shortest possible period, just above the surface, is about 84 minutes, so no Earth satellite can complete an orbit faster than that.

Escape velocity explains why bodies have atmospheres or not. The Moon's escape velocity is only about 2.4 km/s, and gas molecules at ordinary temperatures move fast enough to slip away over time, so the Moon has almost no atmosphere. The Earth holds on to the heavier gases, but very light hydrogen and helium leak slowly into space. The idea even gives us the black hole: if the escape velocity of a collapsed star were greater than the speed of light, 3 × 10⁸ m/s, nothing, not even light, could leave it. For the Sun to become a black hole, it would have to be squeezed to a radius of about 3 km.

A satellite is a smaller body that goes around a bigger one. The Moon is the Earth's only natural satellite. A satellite made by humans and launched into orbit is an artificial satellite. The first one was Sputnik 1 (USSR, 4 October 1957). India's first satellite was Aryabhata, launched on 19 April 1975. Satellites are carried up by rockets, which work on Newton's third law: the hot gases rush down and the rocket is pushed up. Launching towards the east helps, because the Earth's own spin already gives a launch site near the equator a speed of about 0.46 km/s.

Why exactly 36,000 km? For a satellite to stay over one spot its period must equal the Earth's spin period, about 24 hours. Kepler's third law (T² ∝ r³) links the period to the orbit's radius, and only one radius gives 24 hours: about 42,164 km from the Earth's centre, which is 35,786 km above the surface. It must also be above the equator, because only there can a circular orbit keep pace with the ground below it.