The Earth as a Planet

It is important to look at the Earth in its context as only one of the nine planets which make up the solar system.

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

Let's start by looking at some patterns in the solar system:

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"):

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

r_n = 0.4 + 0.3 x 2ⁿ

• The Titius-Bode law was used to help discover Ceres, a 1000 km asteroid, in 1802, and Uranus in 1781.
• Check satellites of Saturn and Jupiter
• Law has never been given a scientific foundation, and may be "chance," combined with the fact that the outer planets are larger (more on why later), and hence "take up more room."
• May have hindered models of Solar System formation as an arbitrary constraint (Brahic, Formation of Planetary Systems)

Orbital, Rotational Consistencies

1. All of the planets have nearly circular orbits around the sun.

2. All of the planets orbit the sun in a counterclockwise direction when viewed from the "North."

3. All but two of the planets spin on their own axes in a counterclockwise direction.

4. The sun spins on its own axis in a counterclockwise direction.

Nebular theory for the origin of the solar system (Laplace, Kant)

• Solar system begins as a nebular cloud with some slow rotation. It consists of about 98% H, about 2% He, and small amounts of heavier elements
• Gravity causes the cloud to collapse or condense
• Conservation of angular momentum causes the cloud to spin faster and faster.
• In the plane of rotation, centrifugal force partly counteracts gravity; perpendicular to this plane, gravity forces the cloud to take on the shape of a disk. (spinning pizza dough)
• Approximately one half of the nebular cloud’s mass collapses to form the Sun.
• Due to tremendous heat and pressure, fusion kicks off in the Sun.
• The Sun goes through a T. Tauri stage, blowing most of the rest of the nebular cloud’s mass out of the solar system.
• Within the remaining nebular cloud are local concentrations of mass and rotation (eddies); these condense to form the planets.

An object bigger than Pluto has been found in the outer solar system by Mike Brown (Caltech), Chad Trujillo (Gemini Observatory), and David Rabinowitz (Yale University). It is possible, perhaps likely, that it will eventually be considered to be our solar system's tenth planet. For more info, see NASA's press release and the discoverer's web site. Its temporary designation is 2003UB313; an official name will be given in due course (more). Brown et al have now also spotted a moon orbiting this object.

Reminder to self: homework neatness; textbook availability

IODP Drillhole

Inner (Terrestrial) planets vs. outer (Jovian) planets

• High temperatures near the Sun allowed only the most refractory elements could condense. (Refractory literally means “capable of enduring high temperature,” i.e., elements with a high condensation and melting point.)
• Includes Fe and silicate minerals.
• Farther from Sun, lighter elements could condense, e.g., water (ice), ammonia ice, etc.
• Larger outer planets had gravity to retain H, He (but also Fe, silicates)
• Terrestrial planets:
• Smaller
• Denser
• Primarily Fe (and Ni) and silicates
• Jovian planets:
• Larger
• Less dense
• Primarily H, He, with icy moons

A sidereal day is 23 hours 56 minutes and 4.09 seconds long. si·de·re·al [sahy-deer-ee-uhl] adjective 1. determined by or from the stars: sidereal time. 2. of or pertaining to the stars.

"Jupiter is so big that all the other planets in our Solar System could fit inside Jupiter (if it were hollow). "

Tycho Brahe (1546-1601)

• Danish astronomer
• Compiled data on planetary orbits
• Used astrolabe (no telescopes!)
• [Galileo (1564 - 1642) introduced telescope to astronomy in 1609]

Kepler’s Laws (1571-1630)

Danish astronomer, Brahe’s assistant

Developed Kepler’s 3 laws:

LAW 1: The orbit of a planet/comet about the Sun is an ellipse with the Sun's center of mass at one focus

This is the equation for an ellipse:

LAW 2: A line joining a planet/comet and the Sun sweeps out equal areas in equal intervals of time

LAW 3: The squares of the periods of the planets are proportional to the cubes of their semimajor axes:

Sir Isaac Newton (1643-1727)

From Kepler’s laws (not an apple) Newton conceived of Universal Law of Gravitation:

For example, consider planet in circular orbit around Sun:

(Equation 1) - to be used in homework to find period of orbit as a function of orbit radius, etc.

Mass of the Earth

From Newton's Universal Law of Gravitation, it would seem apparent that we could find the mass of the Earth. Take M to be Earth's mass, and m to be a "test mass." Since F = ma, we can divide both sides of the equation below by m to get the acceleration of gravity, a_g:

Assuming our test mass is at the surface of the Earth, r is the radius of the Earth. In 200 B.C., Eratosthenes used shadows of a vertical stick at two different latitudes to determine that the radius of the Earth was 250,000 stadia. Unfortunately, we do not know which one of the various measurements used in antiquity is represented by the stadia of Eratosthenes. According to the researches of Lepsius, however, the stadium in question represented 180 meters, giving a radius of the Earth of 7,160 km. Today we know the mean radius of the Earth is roughly 6,371 km. Measuring the acceleration of an object just requires accurate measure of time (Galileo used water clocks). [Galileo and the Leaning Tower of Pisa]  So, we can find GM, but not M independently!

The Cavendish Experiment - "Measuring the Mass of the Earth" - 1798

Henry Cavendish (1731-1810) developed an apparatus for experimentally determining the value of G involved a light, rigid rod which was 6-feet long. Two small lead spheres were attached to the ends of the rod and the rod was suspended by a wire. The angle of rotation yields the amount of torsional force. A diagram of the apparatus is shown below:

[How were the masses of the spheres determined?] It was not until Cavendish determined that G had a value of 6.75 x 10^-11 N m²/kg² that the mass of the Earth was known! Today, the currently accepted value of G is 6.67259 x 10^-11 N m²/kg². And this results, as you will show in homework, in:

• Commercial Cavendish Apparatus
• "Big G" vs. "time"

This is perhaps the most important constraint on the composition of the Earth.  Any model for the composition and structure of the Earth must result in this mass. Is the Earth made of green cheese? Well, does (density of green cheese) x (volume of Earth) = M??

Bulk Density of the Earth

Once the mass of the Earth was established, its bulk, or average, density could be determined:

We know that the average density of continental crust is about 2670 kg/m³ and the average density of oceanic crust is about 3000 kg/m³ . Furthermore, from meteorites, among other things, we know the density of mantle material (peridotite) is about 3300 kg/m³. This would seem to suggest that the deeper interior of the Earth must have a very great density if the average is to be 5540 kg/m³. However, this density cannot be used to compare with the previously mentioned densities because they are measured at STP. The Earth's bulk density is "inflated" because of the tremendous pressures in the interior of the Earth. Although we will discuss later how it is done, the uncompressed bulk density of the Earth is about 4000 kg/m³. Still, the argument holds that the interior of the Earth must have a density significantly higher than mantle density.

Planet Densities

By observing the orbital radius and period of a natural or artificial satellite about a planet, its mass can be determined. Its radius is determined by astronomical observation. Once the mass and radius of a planet are determined, bulk density can be estimated:

Jovian and Saturnian Satellites

Jupiter has 4 major (Galilean) satellites; this multi-satellite system has been referred to as a "mini solar system"

Calculated and Actual Temperature of Planetary Surfaces

• affected initial compositions
• affects current tectonics (cf. Earth, Venus, Mars)
• affects current "climatic" conditions, ergo existence of water, life, etc.
• depends on distance from Sun and assumptions about planet's (moon's) surface reflectivity (bond albedo)

Energy received by a planet is proportional to cross-sectional area of planet, pR²:

As a result of E_rcv, planet heats up and re-radiates energy; reaches steady state (heat received equals heat radiated, E_rad).   Now, let's assume planets re-radiate that energy over their entire surface (because they are rotating, entire surface "shares" this energy, and that planets can be assumed to be perfect black-body-radiators.

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 = 278^o K = 5^o C  (very close)

• Moon, Mercury, Mars, Earth are close to "predicted"
• Venus is much hotter, due to ?
• outer planets (Jovian planets) only roughly approximate black bodies

For more information, see this site and this one

"Linear" vs. Rotational Mechanics

Moment of Inertia

• determined from deviations of satellite's orbit from conical section, or precession of equinoxes
• mass is 0th moment of density:
  
• center of mass found from 1st
• moment of inertia is second moment of density

Moment of inertia for some ideal bodies

Precession of the Equinoxes

Animation of Precession of Equinoxes

• change in position of Earth's spin axis ("seasons") w.r.t "fixed stars"
• caused by torque of Sun on Earth's equatorial bulge
• allows estimation of C-A (C: I about spin axis;  A: I perpendicular to that)
• period: 26,000 years
• thus, in 13,000 years, winter will be in July, if calendar is fixed to position of Earth and Sun w.r.t. fixed star reference frame
• one part of Milankovitch Cycles

• this (combined with spherical harmonic analysis of Earth's gravity) is how we determine moment of inertia of Earth, kMa², where k is the gyrational constant (0.33078)
• measure of central condensation of mass (second moment of mass)
• important constraint on radial distribution of mass in Earth for Earth models (e.g., A&Z's PREM)

 "Real" Planets

• Moon may have homogeneous distribution (note non-uniqueness), small Fe core at most
• Mars has some central concentration, Fe core
• Earth's I consistent with large Fe core (I very well known; very important constraint on density models, as is M!)
• Gaseous giants, and Sun, have r "inflated" by light gasses, thus reducing I/mr²
• value for Mercury?

Milankovitch Cycles

• precession of equinoxes (26,000 year period)
• change in inclination of the ecliptic (41,000 year period)
• change in orbital eccentricity (96,000 year period)
• observed in sedimentary record
• may be responsible for timing of Ice Age (Pleistocene) glaciations

Copyright J. L. Ahern 2007