The Planets

This is an introduction to the Solar System, its formation, and composition, with special emphasis on the "terrestrial" or "inner" planets. I take a kind of historical approach, noting the patterns and regularities observed, for example, by Tycho Brahe, described by Johannes Kepler, and explained by Sir Isaac Newton. Laplace and even the philosopher Immanuel Kant figure into shaping our modern-day notions of the origin and composition of the solar system.

"Early attempts to explain the origin of this system include the nebular hypothesis of the German philosopher Immanuel Kant and the French astronomer and mathematician Pierre Simon de Laplace, according to which a cloud of gas broke into rings that condensed to form planets." - Encarta, http://encarta.msn.com/encyclopedia_761557663/Solar_System.html

Homework Set 1



Titius-Bode Law

Taking the point of view of a first-time visitor, one of the first things you would notice about the Solar System is that the spacing between the planets' orbits consistently increases as you move away from the Sun (with one exception). Furthermore, it's not a linear increase, so we need essentially two figures, at different scales, to represent the solar system (pictures courtesy of "The Nine Planets"):

Inner:

Outer:

Titius-Bode's law: Distance, r, of the nth planet from the Sun (in A.U.s) is given by:

rn = 0.4 + 0.3 x 2n

Planet n rn Actual
Mercury -infinity, -1 0.4, 0.55 0.39
Venus 0 0.7 0.72
Earth 1 1.0 1.0
Mars 2 1.6 1.52
Asteroids 3 2.8 -
Jupiter 4 5.2 5.2
Saturn 5 10.0 9.6
Uranus 6 19.6 19.2
Neptune 7 38.8 30.1
Pluto 8 77.2 39.4

Period, Shape and Orientation of Planetary Orbits

The Dead White Guys:

  1. all planets have direct orbits, i.e., CCW as viewed from N; Sun spins CCW also
  2. all planets spin CCW, except Venus (slow retrograde motion), Pluto, Uranus (on its side)
  3. path of body orbiting a point mass or spherical mass (Sun) forms conical sections: ellipses, hyperbolae, circles (extension of Kepler's 1st Law: ellipses with Sun as focus):

    [cool graphing site!]
  4. most orbits nearly circular, with eccentricity < 0.1, except Mercury (0.21) and Pluto (0.25)
  5. orbital periods increase with distance from Sun in a regular way

eccentricity, e, = (a2 - b2)/a2 where a, b semi-major, semi-minor axes

Assuming stable, circular orbit, Kepler's 3rd Law derives from:

centripetal force = gravitational force

Expressing angular velocity in terms of period,

Kepler's Laws

About Amateur Radio Satellites (OSCARs)


Calculated and Actual Temperature of Planetary Surfaces


Planet, radius R, distance r from Sun, receives:

As a result of Ercv, planet heats up and re-radiates energy; reaches steady state (heat received equals heat radiated, Erad).

Black-body Radiator: idealized surface for which relationship between T and radiated energy may be derived from thermodynamic (statistical mechanics) first principles:

The total surface of a planet is so total radiated energy is:

Setting received and radiated energies equal:

Example: Earth - r = 149.6 x 106 km = 1.496 x 1011 m  so  T = 278o K = 5o C  (very close)

  Actual Temperatures (oK) Calculated
Planet Day Night Mean
Mercury 700 100 452 444
Venus 721-731 732 730 323
Earth 277-310 260-283 281 276
Moon 380 100 280 276
Mars 240 190 215 223
Jupiter 120-150 - 120 121
Saturn 120-160 - 88 90
Uranus 50-110 - 59 63
Neptune 50-110 - 48 50
Pluto - - 37 44

Homework Set 2


Planet Densities

Once the mass and radius of a planet are determined, bulk density can be estimated:

Planet r (kg/m3) ru (kg/m3)
Mercury 5420 5300
Venus 5250 3900
Earth 5520 4000
(Moon) 3340 3340
Mars 3940 3700
(Asteroids) 3710 3710
Jupiter 1310 N.A.
Saturn 690 N.A.
Uranus 1190 N.A.
Neptune 1660 N.A.
Pluto 2080(?) ?

PV = nRT  (ideal gas law)

Jovian and Saturnian Satellites

Jupiter has 4 major (Galilean) satellites:

Galilean Moon r 103 km from Jupiter
Io 3550 412
Europa 3040 670
Ganymede 1930 1,070
Callisto 1810 1,880

Saturn's Major Moons:

Moon r 103 km from Saturn
Mimas 1400 185
Enceladus 1200 238
Tethys 1210 295
Dione 1430 377
Rhea 1330 527
Titan 1880 1,222
Iapetus 1160 3,560

Figure of the Planets

Oblateness:  Because of rotation, planets tend to be larger at equator (even if solid, but elastic).

For a body in hydrostatic equilibrium, oblateness is approximately equal to non-dimensional centripetal acceleration, , where

For a spherically symmetric body, as we'll show later,

As an example, consider Earth:


Mars:

Discrepancy:

Homework Set 3


Moment of Inertia

Moment of inertia for some ideal bodies

Ideal Body (in order of increasing central concentration)  I, moment of inertia
planet, mass m, in orbit of radius r 1.0 mr2
ring, radius R, mass m, spinning about sym. axis 1.0 mR2
hollow sphere, radius R 2/3 mR2
homogeneous sphere, radius R 2/5 mR2
sphere, core radius 1/2R, core density = 2 x mantle density 0.367 mR2
mass concentrated on axis 0.0 mR2

"Real" Planets

Body I/mr2
Moon 0.391
Mars 0.365
Earth 0.3307
Neptune 0.29
Jupiter 0.26
Uranus 0.23
Saturn 0.20
Sun 0.06

Angular Momentum

Angular momentum of a system remains constant unless external torque acts on the system.

Moment of Inertia of 2-Part Planet

Moment of inertia for planet spinning on its own axis, given above.  For body orbiting primary, consider body a point mass, so I = mr2


Albedo is the fraction of light that is reflected by a body or surface. It is commonly used in astronomy to describe the reflective properties of planets, satellites, and asteroids.

Albedo is usually differentiated into two general types: normal albedo and bond albedo. Normal albedo, also called normal reflectance, is a measure of a surface's relative brightness when illuminated and observed vertically. The normal albedo of snow, for example, is nearly 1.0, whereas that of charcoal is about 0.04. Investigators frequently rely on observations of normal albedo to determine the surface compositions of satellites and asteroids. The albedo, diameter, and distance of such objects together determine their brightness. If the asteroids Ceres and Vesta, for example, could be observed at the same distance, Vesta would be the brighter of the two by roughly 10 percent. Though Vesta's diameter measures less than half that of Ceres, Vesta appears brighter because its albedo is about 0.35, whereas that of Ceres is only 0.09.

Bond albedo, defined as the fraction of the total incident solar radiation reflected by a planet back to space, is a measure of the planet's energy balance. (It is so named for the American astronomer George P. Bond, who in 1861 published a comparison of the brightness of the Sun, the Moon, and Jupiter.) The value of bond albedo is dependent on the spectrum of the incident radiation because such albedo is defined over the entire range of wavelengths. Earth-orbiting satellites have been used to measure the Earth's bond albedo. The most recent values obtained are approximately 0.33. The Moon, which has a very tenuous atmosphere and no clouds, has an albedo of 0.12. By contrast, that of Venus, which is covered by dense clouds, is 0.76.
[http://zebu.uoregon.edu/~js/glossary/albedo.html]


Copyright 2004, Judson L. Ahern