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ASTRONOMY 9: HISTORY OF COSMOLOGY

Handout #16

J. E. Baker
UC Berkeley, Spring 2000

The Netownian Synthesis

1.

Rene Descartes (1596-1650, France)

``I think therefore I am.'' Mind/matter, existence of God and the soul
Application of algebra to geometry $\Rightarrow$ Cartesian geometry
Realizes circular motion is ``unnatural'' (forced)
Denied possibility of Void
Matter infinitely divisible, filling all space
Denied that action could operate at a distance (gravity)
Forces must operate through direct physical contact
``Vortices'' carry planets around the Sun
Theory influential in France even 100 yrs after Newton!
Seeds of the ``mechanical universe'', man as ``master'' of nature
Remote God sets machine in motion but need not intervene

2.

Sir Isaac Newton (1642-1727, England)

Culmination of scientific revolution; laid foundations of rational scientific world-view until 20th century!
1642: Born Christmas day, father died earlier
1646: Mother departs and leaves Isaac with grandmother
1665: Cambridge graduation
1665-1667: Home due to plague: invents calculus (also Leibniz), seminal work on mechanics, theory of color, extension of gravity to planets!
1669: Lucasian professorship at Cambridge
1671: Invents reflecting telescope
1687: Publishes masterful Principia (Mathematical Principles of Natural Philosophy) in Latin
1693: Temporary insanity
1704: Publishes Opticks
Fragmentary state of physical knowledge before Newton
- Kepler's 3 laws of planetary motion
- Galileo's ideas of inertial motion and free fall on Earth
- Two different physics: terrestrial and celestial
Background definitions for Newton's Laws of motion
- Acceleration: change in velocity (speed or direction) [m/s²]
- Force: push/pull, strength of the agent of change in motion [Newtons (metric), pounds (imperial)]
- Mass: measure of inertia, resistance to change in motion [kg]
- Absolute space and time: static background arena, completely independent of observers
Three laws of motion

1.
Objects continue in state of rest or uniform straight-line motion unless acted on by a force
2.
F=ma: change in motion is proportional to the force (and in the same direction)
3.
To every action there is an equal and opposite reaction
Story of the apple: the Moon is falling! Example of firing projectile from a high mountain
Celestial physics the same as terrestrial!
Universal law of gravitation
- Force points towards centers (of mass)
- Distance to Moon is $60\times$ radius of Earth
- Distance ``fallen'' by Moon in 1 sec is 1/3600th of distance fallen by object at Earth's surface
- $F\propto 1/r^2$ , $F\propto M_1 M_2 \Rightarrow F_{\mathrm{grav}} = GM_1M_2/r^2$
- $G\approx 6.7\times 10^{-11}$ m³ kg^-1 s^-2 (measured at Cavendish, 1798)
- Spherical masses can be treated as if all mass is in a point at the center!
- Note: weight is not the same as mass!
  - Weight is the force of gravity on an object: F = mg
  - g is the gravitational acceleration: g = GM/r²
  - At Earth's surface, $g \approx 10$ m/s²
  - So weight depends on where you are, mass doesn't!
- Problem of action at a distance
  - What is gravitation?
  - Newton: ``I feign no hypotheses...''
With these laws, Newton can derive Kepler's 3 laws: cosmology understood from simple physical principles!
- Kepler 1: Newton gave geometrical proof that under central inverse-square force, orbital path is elliptical
- Kepler 2: Geometrical proof for any central force
- Kepler 3: Follows from F=ma and F=GMm/r²:
  - Take special case of circular orbit (Newton did ellipses too!) with $M\gg m$
  - For uniform circular motion, a=v²/r
  - $GMm/r^2 = mv^2/r \Rightarrow v^2 = GM/r$
  - $v=2\pi r/P \Rightarrow 4\pi^2 r^2/P^2 = GM/r$
  - $P^2 = 4\pi^2 r^3 / GM \Rightarrow P^2 \propto r^3$
Comets: also moving in elliptical orbits determined by Newton's laws
- Halley: guessed comet of 1682 same as 1531, 1607
- Newton computed orbit, correctly predicted return in 12/1758
- Spectacular confirmation of Newton's ideas
Correct, quantitative explanation for tides
- Gravitational attraction of Moon greater on side of Earth closest to Moon, smaller on side farther away
- Water pulled by difference in these forces relative to force on the whole Earth
Earth is oblate (bulges at equator) due to rotation
Computes 26,000-year precession of equinoxes
- Moon's gravitational force on oblate Earth tries to tip Earth's axis
- Earth's axis turns like that of a spinning top
Arguments with Hooke over nature of light: ``standing on the shoulders of giants'' (Hooke was very short)
Strong interest in alchemy, theology, and magic
Viewed Universe as manifestation of God's power
- Noted planets perturb each others orbits
- Thought perturbations add up, solar system unstable
  - Note: Paths of two bodies interacting can be solved exactly, three or more cannot
  - Stability of the solar system over billions of years is a hard problem even today!
  - Chaos: positions after long time are extremely sensitive to starting conditions
- God required to constantly intervene to keep things in order
- Also necessary to explain gravity: magical action at a distance
Newton's cosmology
- A finite Newtonian universe would collapse!
- So Universe must be infinite, homogeneous
- God's hand again required for stability
- Implication of Newtonian physics: Mechanistic, deterministic cosmos: if all forces, positions and velocities are known, the future (and past) can be determined exactly (in principle)--room for free will?

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Up: Astronomy 9 Lecture Notes

jonathan baker
2000-03-01