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
• Period, Shape and Orientation of Planetary Orbits
• Calculated and Actual Temperature of Planetary Surfaces
• Planet Densities
• Figure of the Planets
• Moment of Inertia

Titius-Bode Law

• Some History of Titius-Bode Law
• Icarus Article on Titius-Bode Law
• Izvestiya Article on 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:

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)

Period, Shape and Orientation of Planetary Orbits

The Dead White Guys:

• Copernicus 1473-1543
• Tycho Brahe 1546-1642
• Johannes Kepler 1571-1630
• Galileo Galilei 1564-1642
• Sir Isaac Newton 1642-1727

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,

• T for a given orbiting body depends on r (actually mean r, or semi-major axis, for elliptical orbit - Kepler's 3rd Law), and does NOT depend on its mass, m, only on the mass of the primary, M
• suggests way to measure mass of a planet with natural or artificial satellites

Kepler's Laws

About Amateur Radio Satellites (OSCARs)

Calculated and Actual Temperature of Planetary Surfaces

• affected initial compositions
• affects current tectonics (cf. Earth, Venus, Mars)
• depends on distance from Sun and assumptions about planet's (moon's) surface reflectivity (bond albedo)

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

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

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

Homework Set 2

Planet Densities

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

• density of Earth is higher than density of crust (continental ~ 2700, oceanic ~ 3000)
• homework: find combined uncompressed density of Earth-Moon system
• densities yield important constraints on composition
• inner (terrestrial) planets much denser then outer (Jovian) planets
• inner planets also much smaller than outer planets (except Pluto)
   
• Density, however, depends on pressure, P, and temperature, T, according to an Equation of State. For example, for an ideal gas,

PV = nRT  (ideal gas law)

• Larger planets have higher interior pressures and vice versa, so comparisons between planets, and with materials observed at surface of Earth, must be referenced to a standard state (like STP).

• By "guessing" at EOS for planetary materials (or as gotten from seismic data), can estimate uncompressed densities, r_u. Will discuss this process later.

• Note decrease in density away from Sun - result of more volatile elements relative to refractory elements as T drops away from Sun

• Moon is exception: binary accretion, fission, capture - little iron

Jovian and Saturnian Satellites

• these multi-satellite systems have been referred to as "mini solar systems"
• do satellites obey Titius-Bode law?

Jupiter has 4 major (Galilean) satellites:

• decrease in r with distance from Jupiter suggests high T gradient in vicinity of Jupiter, at least when moons formed
• for fusion, need at least 0.08 M_Sun;  Jupiter: 0.001 M_Sun
• _{however, Jupiter still has surface heat flow significantly greater than received from Sun (gravitational energy of formation?)}

_{Saturn's Major Moons:}

• q_Saturn > q_Sun, but T-gradient must not have been too great during lunar formation

Figure of the Planets

• all large bodies (moons and planets) are nearly spherical
• implies state of hydrostatic equilibrium, i.e., fluid and deformable by gravity at some time
• planets probably never completely molten
• currently, terrestrial planets solid (e.g., for Earth, S waves in mantle)
• therefore, long-term relaxation processes active (diffusion and dislocation creep; plasticity)

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:

• which is almost exactly the observed oblateness of Earth. 
• but is this "inherited" from a time in the past when the Earth may have been molten, partially molten, or at least much softer than today?
• later we'll see that Moon has slowed Earth's rotation from a 19 hour day in the Devonian (~ 350 Ma)
• what does this say about Earth as "liquid" planet in relatively recent times?

Mars:

Discrepancy:

• Tharsis bulge ~ 3% of entire Mars mass, now at equator
• gravity data show it is supported by strong outer shell
• ~ 600 km-thick ltihosphere (c.f. 50 km for Earth)

Homework Set 3

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

"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?

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