Johannes Kepler (1571-1630)

• German Mathematician and Astronomer

• Passionately convinced of the ‘rightness’ of the Copernican view.

• Set out to ‘prove’ it!

Kepler’s Life Work

• Kepler sought a unifying principle to explain the motion of the planets without the need for epicycles.

• Wanted to work with the best observational Astronomer: Tycho Brahe.

• ‘Obtained’ Brahe’s data after his death.• Eventually ‘discovered’ that ellipses would

dramatically simplify the mathematics!No more circles!

Kepler’s First Law

The orbital paths of planets are elliptical(not circular) with the Sun at one ‘focus’.

Properties of ‘conic sections’ known since

Euclid

Definition on an Ellipse

‘Locus’ of points produced by this practical

geometric construction.

Property of an Ellipse

• Major and Minor axes

• Two Foci.• The Sun is at

one focus, the other is not physically significant.

Eccentricity

• The eccentricity of an ellipse is the ratio:Distance from the center to a focusLength of the semi-major axis

• If e = 0 → a circle, e = 1 → a line • The semi-major axis, a, is the average

distance between the planet and the Sun.• Perihelion = a(1- e) = ‘closest approach’• Aphelion = a(1+ e) = ‘greatest distance’.

e =

Elliptical Terms Visualised

Comments on Elliptical Motion

• Elliptical motion: No small achievement!• Challenged the Authority of Aristotle.• Except for Mercury (and Pluto)

eccentricity is so small cannot easily distinguish it from circular motion.

• Hence Ptolemaic and Copernican models did a pretty descent job (for those days).

• Galileo did not like ellipses!

Kepler’s Second Law

• “An imaginary line connecting the Sun to any planet sweeps out equal areas in equal intervals of time.”

• Planets therefore have different speeds at perihelion and aphelion.

• Challenges Aristotle’s insistence that planets have a constant or uniform speed.

Kepler’s Second Law Visualised

Red arcs all take the same timefor equal areas A, B, C.

Kepler’s Third Law

• Laws (1) and (2) published in 1609, based on a long study of the motion of Mars.

• An appeal to simplicity in mathematics.• During next 10 years extended to all

known planets and devised 3rd law.

• “The square of the planet’s orbital period is proportional to the cube of its semi-major axis.” or: P2/a3 = Constant

Orbital Properties of the PlanetsPlanet Semi-Major

Axis, aPeriod,

PEccentricity,

eP2/a3

Mercury 0.387 0.241 0.206 1.002Venus 0.723 0.615 0.007 1.001Earth 1.000 1.000 0.017 1.000Mars 1.524 1.881 0.093 1.000

Jupiter 5.203 11.86 0.048 0.999Saturn 9.537 29.42 0.054 0.998Uranus 19.19 83.75 0.047 0.993Neptune 30.07 163.7 0.009 0.986

Pluto 39.48 248.0 0.249 0.999

Further Notes…

• Period is ‘Sidereal period’• “Astronomical Unit” = semi-major axis

of the Earth’s orbit (or average Sun-Earth distance)

• Deviations in P2/a3 for Uranus and Neptune is mutual gravitational effect.

• Predictive Law ….applies to all planets.

Incidentally: Kepler andThe Star of Bethlehem

• In 1604 Kepler observed a brilliant conjunction (a close apparent mutual approach) of Jupiter, Saturn and Mars.

• He calculated that this type of conjunction takes place every 805 years.

• Hence previously occurred in AD 799 and in (February) 6 BC.

• Now regarded as the ‘Star’ of Bethlehem.

The Size of the Solar System• Kepler’s 3rd Law (P2/a3 = Constant) only gives

the relative scale of the solar system:Orbital periods known in terms of Earth years.Semi-major axis known in terms of that of the Earth.

• Need to determine the actual length of the Astronomical unit (A.U.)……..how?

• Need reliable measure of the parallax when the planet is closest to us… (hence biggest parallax)

Reminder:Parallax

• Consider a planet as seen against the background stars (very far away).

• View from A and B are different –the planet moves with respect to the background stars

• Apparent angular displacement is Parallax.

Early Attempts…

• Mars in direct ‘opposition’ – Earth directly between Mars and the Sun.

• Tycho Brahe tried in 1582…Twice per day for Earth diameter baseline (before Dawn and at Sunset)

• He ‘claimed’ Success! However….• 0.012o was too small for his instruments!

Try again!• In 1672 Cassini measured Mars in

‘opposition’ at Paris, while his assistant, Richer, did the same in South America.

• Concluded that Mars was ~4000 Earth diameters away (at opposition).

• Hence AU ~87 million miles(Modern day value : 93 million miles)

• Fluke - as large experimental uncertainty!

Halley’s Solar Transits

• November 7 1677 observed the ‘transit of Mercury’across the face of the Sun.

• Realised these rare events could be used to determine the A.U.

Halley’s Predictions

• In 1716 Halley predicted the next transits of Venus would occur on: 6th June 1761 and 3rd June 1769 (He never lived to see it.)

• Next possible opportunity would be:• 9th December 1874 and 6th Dec 1882• Next chance was that: 8th June 2004!• Why so rare? Venus’ orbital plane is

slightly (3.4o) inclined to that of the ecliptic.

Halley’s Quest.

• Halley realised that the transit of Venuswould give much better accuracy for A.U.

• Observers need to measure time when Venus enters and exits the Sun’s disc.

• Need accurate clock and telescope (quadrant)• Needed observers all over the world.

(unprecedented, non-political collaboration)

• Correlate results to determine the A.U.

The Calculation – in principle

• Assume orbit is circular (for simplicity)• Earth-Sun average distance is 1 AU.• Venus’s orbit has ‘radius’ of ~ 0.7AU.• Therefore, at closest approach, Venus is

only 0.3AU from Earth.• From parallax measurement, with known

baseline, can determine distance to Venus.• Hence find AU.

Halley’s Legacy

• 150 observations of Venus’s transits worldwide.

• Even so, experimental problems persisted…Optical distortions, atmospheric turbulence

• A.U. found to be 91 million miles.Modern determination using radar methods gives A.U. = 149,597,870km!

Images of the Transit of Venus: 8th June 2004

The Transit of Venus

The Atmosphere of Venus – illuminated by the Sun.

Venus Transit: Close –up views

Quest for Simplicity

• Kepler’s Laws discovered empirically.• Based upon observational evidence.• What is the underlying reason the laws to

work? What ‘forces’ are involved?• This question was addressed by

Isaac Newton….

Isaac Newton (1642-1727)

• By the time he was 25, he had ‘discovered’ the laws of motion –including gravity.

• Only published them 20 years later at the prompting of Halley!

The Title Page of Newton’s

Principia, 1686

“The Mathematical Principles of

Natural Philosophy”

One of the most influential books in

Physics

Newton’s First Law: Inertia

• “Every body continues in a state of rest, or in a state of uniform motion in a straight line (‘inertia’), unless it is compelled to change state by an external force acing on it.”

• Aristotle thought (wrongly) the natural state was at ‘rest’, and to move required a force.

• Newton: “Uniform motion requires no force!”• Observation needs a frictionless environment.

Newton’s Second Law: Mass

• Changing speed or direction implies an “acceleration”.

“The resultant acceleration (a) is directly proportional to the applied force (F)”.

The constant of proportionality being the mass (m) of the body.

F = ma

Newton’s Second Law - continued

• The greater the force, the greater the acceleration of the body.

• For a constant force, the smaller the mass of an object, the larger its acceleration.

• SI unit of force is the ‘Newton’.1 Newton of force is needed to make a1 kilogram object accelerate at the rate of1 meter per second, every second.

Newton’s Third Law

• To every ‘action there is an equal and opposite ‘reaction’.

• Forces do not occur in isolation. If A pulls on B, then B necessarily exerts a force on A too.

• (eg: jet planes, bullet recoil, hammer…)

Gravity

• Newton hypothesised that any object with mass exerted an attractive gravitational forceon other ‘massive’ objects.

• To Newton this force was ‘action at a distance’. (no physical link between bodies)

• Now regarded as a property of space – a “force field” that influences ‘massive’ objects.

Newton’s Law of Gravity“Every particle of matter in the universe attracts every other particle of matter with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.”

221

rmGmF = G = “Gravitational constant”

= 6.67x10-11 Nm2/kg2

Inverse-Square Law

Inverse square force rapidly weakens with distance from the source……but never

quite reaches zero!

Gravity and ‘Circular’ Motion

Direction of force is along

the line between the two bodies.

If motion was circular,

speed would be constant.

Equations for Planetary Motion

• Mutual attraction between Sun and planets.

• Force for circular motion can be shown to be:

• Assume circular motion: forces must be equal.

2rMGMF PS=

rvMF P

2=

Algebra!

rvM

rMGM PPS

2

2 =Hence:

Simplifies to: rvGM S2= (1)

Pr

periodorbitalncecircumferev π2

==Planet’s speed v is :

Substitute (2) in (1):

(2)

2

324P

rGMSπ

=

Kepler’s Third law!

• Rearrange equation:2

324P

rGMSπ

=

SGMrP 2

3

2 4π=3

22 4 r

GMP

S ⎥⎥⎦

⎤

⎢⎢⎣

⎡=

π or = constant

If the G value is known (Cavendish - 1793) then the mass of the Sun can be found!

Mass of Sun ~2.0x1030kg, Earth ~6.0x1024kg

Principia: From Ballistics to Satellites

A – Vertical motionB, C Parabolic motionE – Circular motionD, F Elliptical Motion

Conclusions

• Newton’s laws of motion combined with that of gravity explains motion of all objects.

All earth-bound objectsAll heavenly bodies

• Without an exact inverse square law the universe would be very different!

Are Newton’s Laws Always Valid?

• NO! Two exceptions:• When the speed of the motion approaches

the speed of light: “Relativity”• When the dimensions become very small -

the world of atoms: “Quantum Mechanics”

The motions of ‘everyday’ objects follow Newtonian Mechanics.