Gravitazione Universale e Meccanica Celeste: Teoria, Derivazioni e Applicazioni

Two lines of reasoning meet in this lecture. Kepler found, from years of careful observation, three rules that the planets obey. Newton showed that one force, falling as the inverse square of distance, explains all three. We will build that force, test it, and then derive every orbit it allows. Start with the observations. Here is one planetary orbit, and here is the Sun. The first law says the orbit is an ellipse, and the Sun sits not at its centre but at one focus. The second law concerns speed. Join the Sun to the planet. In one interval of time near the Sun, the line sweeps this green sector. Now let the planet travel round to the far side. There, in the same interval, it sweeps this thin sector, and the two areas are exactly equal. Far from the Sun the planet moves slowly; near it, quickly. The third law compares different planets. The square of the period, divided by the cube of the semi-major axis, is the same number for all of them. These three laws describe the motion; they do not explain it. Newton's argument is cleanest for a nearly circular orbit of radius r. The planet moves uniformly, so the only acceleration is centripetal, pointing at the Sun. So the force is m omega squared r. Writing omega as two pi over the period, the force becomes m times four pi squared over T squared, times r. Now use the third law: T squared equals k r cubed. Substitute, and the period disappears in a telling way. The force on the planet is proportional to its mass and falls as one over r squared. The third law of dynamics finishes the job. The planet pulls the Sun with an equal and opposite force, so the same expression must also be proportional to the mass M of the Sun. The constant four pi squared over k must therefore be G times M. This is the law of universal gravitation. Every pair of masses attracts along the line joining them, with a force G m M over r squared, and G is one universal constant. Look closely at one step we took for granted. On the left of the second law of dynamics sits the inertial mass, the resistance to changes of velocity. In the law of gravitation sits the gravitational mass, the charge that sources and feels the attraction. Dividing, the acceleration carries the ratio m g over m i. Only if these two masses are equal, for every material, does every body fall with the same acceleration, G M over r squared. That equality is the principle of equivalence. It can be tested with a plumb line. On the rotating Earth the bob feels gravity, proportional to m g, and a centrifugal pull, proportional to m i. If the ratio differed between materials, plumb lines of lead and of wood would hang at different angles. They do not, to better than one part in ten trillion in modern torsion experiments. With the two masses equal, a circular orbit follows at once. Gravity supplies the centripetal acceleration, v squared over r, so the orbital speed is the square root of G M over r. Closer orbits are faster. Measuring G is hard, because gravity between laboratory masses is tiny. Cavendish used a torsion balance. A light rod carrying two small masses m hangs from a thin fibre, and a mirror on the fibre reflects a light beam onto a distant scale, magnifying the smallest twist. Two large masses M are brought close, one on each side. Each attracts its neighbour with a force G m M over r squared, and the rod twists until the restoring torque of the fibre balances the torque of the two forces. The torque of the pair is two b times F, with b the lever arm. The fibre answers with kappa theta, where kappa is its calibrated torsion constant. Equating the two and solving gives G, about six point six seven times ten to the minus eleven, in newton metre squared per kilogram squared.

Newton's law is written for point masses, yet planets are large. The shell theorem tells us how an extended sphere attracts. Take a thin uniform shell of mass M and radius R, and a point P at distance r from its centre. Slice the shell into rings around the line to P. The ring at angle theta has radius R sine theta, and all of it lies at the same distance s from P. Its share of the mass is M over two, times sine theta d theta. The ring contributes a potential minus G d M over s. The law of cosines links s to theta, and differentiating it gives s d s equal to R r sine theta d theta, which turns the integral over angles into an integral over s. Everything cancels except a plain integral of d s, running from the nearest point of the shell to the farthest, so the potential is minus G M over two R r, times the length of that range. Outside, the range runs from r minus R to r plus R, a length two R. The potential is minus G M over r: exactly that of a point mass at the centre. Now move P inside. The range runs from R minus r to R plus r, a length two r, and the potential becomes minus G M over R, the same everywhere. A constant potential means zero field. Inside a uniform shell there is no gravity at all. Any spherically symmetric body is a nest of such shells. At radius r, the shells outside pull nothing, and the shells inside act as a point mass. So g of r is G times the mass enclosed, over r squared. Consider a body with a hollow core of radius R one, a shell of density rho two out to R two, and an outer layer of density rho three out to R three. The enclosed mass is the integral of four pi r squared rho over the radius. Follow the field outward. In the hollow core it is exactly zero. Through the first shell it grows as r cubed minus R one cubed, over r squared. Through the second it keeps rising, but more gently, because rho three is smaller. Beyond the surface the whole mass acts from the centre, and the field falls as one over r squared. The potential energy is minus m times the integral of g from r out to infinity: continuous everywhere, and flat inside the cavity, where the field vanishes. Here is a striking consequence. Treat the Earth as a uniform sphere and drill a straight frictionless tunnel along a diameter. A body at distance x from the centre feels only the mass inside radius x, which grows as x cubed. So the force grows linearly, minus G M m over R cubed, times x. That is a spring law. The body oscillates harmonically about the centre, with angular frequency the square root of g over R. Putting in numbers, the period is about eighty-four minutes. Drop a stone in at one end and it reaches the other end some forty-two minutes later, at rest, then falls back. Curiously, this is also the period of a satellite skimming the surface. Energy answers the next question: how fast must a body leave the surface never to return? Take zero potential energy at infinity. The minimum launch arrives there with nothing to spare, so its total energy is zero. Solving, the escape speed is the square root of two G M over R, or the square root of two g R: about eleven point two kilometres per second for the Earth. It is exactly root two times the speed of a circular orbit grazing the surface. The same circular orbit reasoning weighs stars. Two stars of equal mass circle their common centre of mass, each at two hundred and twenty kilometres per second, with a period of fourteen point four days. Each star sits at radius r, but the other is two r away. So gravity, G M squared over two r squared, supplies the centripetal force M v squared over r. The radius comes from the motion itself: r is v T over two pi. Solving, M equals four r v squared over G, about one point three times ten to the thirty-two kilograms, some sixty-four solar masses each. This is how astronomers weigh stars they will never visit.

To find every possible orbit we need a better description of motion under a central force. Use polar coordinates centred on the source: a distance r and an angle theta, with unit vectors u r pointing outward and u theta pointing along increasing angle. The unit vectors turn as the point moves. The derivative of u r is theta dot u theta, and the derivative of u theta is minus theta dot u r. With these, the velocity has a radial part r dot, and a transverse part r theta dot. Differentiate once more and collect terms. The radial acceleration is r double dot minus r theta dot squared, the second piece being the familiar centripetal term. The transverse acceleration is one over r times the time derivative of r squared theta dot. Now two bodies, m one and m two, attract each other. Their centre of mass moves uniformly, so we watch only the separation vector r, from one body to the other. Subtracting the two equations of motion gives a single one, for a fictitious particle of reduced mass mu, attracted to a fixed centre by the force minus k over r squared, with k equal to G m one m two. The force is central, so it exerts no torque, and the angular momentum L equals mu r squared theta dot, constant. The transverse acceleration vanishes, and the area swept per unit time, L over two mu, is constant. That is the second law of Kepler, now derived. The energy is conserved too. Kinetic energy splits into radial motion and a centrifugal term, L squared over two mu r squared, which acts like a repulsive barrier added to the gravitational potential minus k over r. The idea of Binet is to describe the orbit by r as a function of theta, and to use u equals one over r. The chain rule and constant L turn time derivatives into angle derivatives: r dot becomes minus L over mu, times d u by d theta. Differentiating again, r double dot is minus L squared over mu squared, u squared, times the second derivative of u. Insert this into the radial equation, and every power of u lines up. The result is astonishingly simple: the second derivative of u, plus u, equals the constant mu k over L squared. It is the equation of a harmonic oscillator in the angle, shifted by a constant. Its general solution is the constant plus a cosine. Choosing the orientation, u equals mu k over L squared, times one minus epsilon cosine theta. Invert it, and r of theta is D over one minus epsilon cosine theta, with D equal to L squared over mu k. This is the polar equation of a conic with its focus at the centre of force. The first law of Kepler appears as a theorem, and epsilon, the eccentricity, decides the shape. Evaluate the energy at closest approach, and the eccentricity turns out to be the square root of one plus two E L squared over mu k squared. The sign of the energy now classifies every orbit. At the minimum allowed energy, epsilon is zero: a circle. For negative energy, epsilon lies below one: a closed ellipse, a bound orbit like those of the planets. At exactly zero energy, epsilon equals one: a parabola, the borderline case of a body that just escapes. For positive energy, epsilon exceeds one: a hyperbola, an open path that swings once past the centre and leaves forever, like a comet or a probe on a flyby. The ellipse also yields the third law, now exactly. Its area is pi a b, swept at the constant rate L over two mu, so the period is pi a b divided by that rate. Square it, and use the geometric fact that b squared equals a times D. The angular momentum cancels, and T squared equals four pi squared a cubed over G times the total mass. The constant of the third law depends, slightly, on the mass of the planet itself. Add a third body and this beautiful solvability is lost. Poincaré showed that the three-body problem has no general solution in closed form, so the Sun, Earth and Moon are followed numerically, advancing positions and velocities in small time steps.

Back on Earth, the measured g is not one fixed number. With height h, the distance to the centre grows, and expanding to first order, g falls by the fraction two h over R. Climb one kilometre and g drops by about three parts in ten thousand. Gravimeters detect far smaller changes, enough to map dense rock underground. Latitude matters too, because the Earth rotates. A point at latitude lambda circles the axis at distance R cosine lambda, and in the rotating frame it feels a centrifugal acceleration directed away from the axis. Its component along the vertical reduces the effective g by omega squared R cosine squared lambda. At the equator that is about three hundredths of a metre per second squared, a third of a percent; the flattening of the Earth adds a comparable effect. Tides arise not from the pull of the Moon itself, but from its variation across the Earth. The whole Earth falls toward the Moon with the acceleration at its centre. The near side is pulled a little more, the far side a little less. Subtract the central acceleration and expand for r much smaller than d. The difference is two G M r over d cubed: tidal effects fall as the inverse cube of distance, like the field of a dipole, because they are a difference of two nearly equal pulls. On the sides, the pull converges slightly inward. The result is two bulges, one facing the Moon and one opposite, and as the Earth turns beneath them each coast sees two high tides a day, about twelve hours apart. Now compare the Sun. Its mass is twenty-seven million times that of the Moon, but it is about three hundred and ninety times farther away, and the cube of that ratio wins. The solar tide is about forty-six percent of the lunar one: roughly half. At new and full moon, the syzygies, Sun, Earth and Moon line up, the two tides add, and we get spring tides. At the quarter moons they partly cancel. What we observe on a coast depends on geography. Each basin has its own natural period, so resonance amplifies or suppresses the forcing: barely half a metre at the Po delta, fifteen metres at Mont Saint-Michel. The rock itself is not rigid. The solid Earth rises and falls twice a day, and at CERN the twenty-seven kilometre ring of the collider stretches by about a millimetre, enough to shift the beam energy, so physicists correct for the Moon. One last equilibrium. Between the Sun and the Earth lies a point where a small body can orbit the Sun in step with the Earth. The pull of the Earth partly cancels that of the Sun, so the body needs less centripetal force than a free orbit at that radius would. Require the net pull to equal omega squared times the distance from the Sun, with the angular velocity of the Earth itself. Expanding to first order in x over d, the distance from the Earth is d times the cube root of M E over three M S. That is about one and a half million kilometres, one percent of the way to the Sun. Solar observatories are parked there, with an uninterrupted view of the Sun. Finally, a manoeuvre that uses everything at once. A ship circles a planet of radius R at radius two R. It releases a capsule at point A, which must land at B, diametrically opposite, on the surface. The free path of the capsule is half an ellipse, with A as apogee and B as perigee. Angular momentum fixes the speeds: v A times two R equals v B times R, so the capsule is twice as fast at B. Energy conservation between A and B adds a second equation. Substituting v B equals two v A, the capsule must leave A with speed the square root of G M over three R. But the ship moves at the square root of G M over two R, which is faster. So the capsule must be thrown backward, against the direction of the orbit, with a relative speed of about point one three root G M over R. Braking, not pushing, brings it down. Released this way, it falls along the ellipse and meets the surface at B with speed twice the square root of G M over three R. Every step used only the two conservation laws that the orbits of Kepler taught us to trust.