Lecture 5: Overview of the Solar System, Matter in Thermodynamc Equilibrium

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There are systematic trends of properties within the Solar System that are clues about the planet formation process:

1) Commonality of rotation and revolution directions

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2) When viewed edge-on, the Solar System is quite flat with only the orbits of Mercury and Pluto deviating much from the plane of the ecliptic [but note that comets have a nearly spherical distribution around the orbits of the planets]

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3) Physical properties of the planets vary in a systematic fashion as a function of distance from the Sun [but note that Pluto is somewhat of an exception to this]

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(multiply these values by 1000 to get densities in the MKS system which uses kg/m3 for density). Recall that

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The Solar System also contains a host of small (but scientifically significant!) bodies:

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Measuring Planets and their Characteristics

What does it take to characterize a planet or moon?

knowledge of its

    - size

    - density

    - surface temperature

    - atmosphere (or lack thereof!)

    - chemical composition and internal structure

    - surface structures

    - magnetic field

    - rotation rate

For planets in the Solar System we can measure their sizes directly (Pluto's diameter was measured during a stellar occultation).

For planets with moons we can use Kepler's Third Law to measure the masses (we could use satellites as artificial moons to do the others). Densities follow from knowing the size and mass.

How can we measure the sizes of objects -- say asteroids -- which are too small to be seen as extended objects?

    -- stellar occultations (can't count on this!)

    -- compare the amount of sunlight it reflects with the amount of blackbody radiation it produces (if we could count on an asteroid to be perfectly black we could just use

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but then we wouldn't have known about the object from visible light observations).

We can measure the reflected and emitted fluxes with the reflected being seen at visible wavelengths and the emitted being detected at ~10,000 nm in the infrared. The ratios of these fluxes are

                

so we can determine A.

A = albedo, the fraction of incoming light reflected by the object. Typically in the range of 4-50% for Solar System objects.

By combining A with our knowledge of the distance of the asteroid from the Sun, we can compute its diameter even though it is too small to measure directly (D=distance between asteroid and Earth,d=distance between asteroid and Sun):

So for example, an asteroid is 2.6 AU from the Sun and 1.9 AU from the Earth when we observe it. We use a measurement in the infrared to learn that the albedo is 10%. We measure a reflected flux of 1x10-16 watts/m2. We can compute the radius:

 

Measuring Surface Temperatures

Planetary surface temperatures can be measured using Wien's Law:

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but this must be used with care as absorptions in a planet's atmosphere can distort the blackbody output. For example, the output of Jupiter peaks at ~30 microns (or 30,000nm) so its T~5.1x106nm K/30,000nm ~150K.

Existence of  Atmospheres around Planets and Moons

Most planetary atmospheres were discovered by seeing clouds and changes indicative of atmospheres. The atmosphere of Saturn's moon Titan was suspected from spectroscopy and confirmed by the Voyager fly-by.

Several conditions need to be met if a planet or moon is to have an atmosphere:

    First, the requisite gases must have been present at the object's formation or some chemical reaction must be capable of generating such gases (release of CO2 by volcanoes and reaction of acidic rain with limestone are examples).

Second, the object must be warm enough for the gas to remain gaseous and not freeze on the object's surface.

Third, the object's gravitational field must be strong enough to retain a gas given the object's mass and temperature.

Consider the Moon: First two conditions are met by analogy with Earth. What about the third?

Need to consider the Moon's escape velocity as compared to the velocities of possible atmospheric constituents:

Recall:

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The escape velocity is a property of the body you're trying to escape from and is independent of the size or mass of the object trying to escape.

The escape velocity needs to be compared with the average velocities of atoms and molecules which is a function of the surface temperature:

   

This average can be computed by using the Maxwell-Boltzmann distribution which describes the distribution of velocities in a gas if the gas is in thermodynamic equilibrium which it will be if collisions between the gas molecules occur often enough.

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The m above is the mass of the gas particle under consideration.

Let vthermal denote this most probable velocity and then

           

setting vthermal = vescape gives a condition where the average atom or molecule could escape.

           

If you compute Tesc for the Moon for the types of molecules that constitute the Earth's atmosphere, you will discover that they could all escape from the Moon due to its lower surface gravity and somewhat higher surface temperature. It is also important to keep in mind that most of a gas will escape even if only a small fraction (say 5%) has a velocity higher than the escape velocity -- this assumes that there are no new sources of the gas from within the planet and that you are interested in times long after the planet has formed.

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Common Surface Features in the Solar System

The commonest surface features are craters.

Are  craters due to volcanoes or impacts?

How massive a meteor is required to produce 10 km diameter crater?

Information from explosions on Earth:

1 megaton of TNT = 4.2x1015 joules

and produces a crater 1 km in diameter

Study of explosions reveals that

                    Dcrater = S E1/3

where E = energy of the explosion and S is a scaling factor that we must determine from experiment:

Compute S:

                    1 km = S(4.2x1015joules)1/3

                    S= 6.2x10-6 km/joules1/3

To produce a 10 km crater:

                    10km=6.2x10-6 km/joules1/3E1/3

                    E = 4.2x1018 joules

Energy for producing a crater on an object is the kinetic energy of the colliding object :

                    E = 1/2 mv2  m=mass of object v=velocity

Take v=30 km/sec, the orbital velocity around the Sun for the

Earth and Moon (choose appropriate to system under study)

                    E=1/2 m(3x104m/sec)2

                           1/2m(3x104m/sec)2 =4.2x1018 joules

                    m = 9.3x109 kg

Assume that a meteor has a density of 5000 kg/m3,

           

so the existence of many craters this size and substantially larger tells us a lot about the size distribution of impacting objects (thankfully there are not many, new, large craters!).

Chemical Composition and Internal Structure

   -- surface composition can be deduced using spectroscopy to see what wavelengths of sunlight are absorbed rather than reflected

    -- internal composition may need to be deduced from density and models of a planet's structure

     -- existence and strength of a magnetic field can give important clues about a planet's internal structure

Comparison of Magnetic Fields for Terrestrial Planets and the Moon

Object Magnetic Field Strength Density (kg/m3) Radius (km) Rotation rate (days)
Mercury .011 5430 2440 58.6
Venus <.001 5240 6052 243
Earth 1 5520 6378 1
Moon .0001 3340 1738 27.3
Mars .00125 3930 3394 1.026

What can we conclude from this chart?

A planet's magnetic field strength depends on

    1. Rotation rate
    2. Density
    3. Radius

This can be placed into a coherent model via dynamo theory which suggests that the magnetic field is generated by electrical currents in a rotating, conductive fluid in the planet's core.

Tides and the Roche Limit

Tides in the Earth's ocean are caused by the differing gravitational force of the Moon on the front and back sides of the Earth:

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The gravitational field of the Sun can also contribute to tides in the oceans:

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The Earth also causes tides on the Moon. Since the Moon is relatively far from the Earth, these tides only cause friction that transfer angular momentum between the Earth and the Moon. In other cases, the tidal forces can become so great that a moon could be torn apart by such forces:

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Consider a planet of mass M being orbited by a moon of mass m and radius R at a distance dplanet  from the planet. The moon is much smaller than its distance from the planet (R<<dplanet):

   

This disrupting tidal force must be compared to the ability of gravity to hold the moon together:

           

In the outer Solar System, densities of planets and moons are similar so

                

so r above becomes

       

This value of dplanet is called the Roche Limit an expresses how close an object held together by self-gravity can approach a more massive object before being ripped apart by the differential tidal force. A more exact calculation which takes into account a varying density throughout the moon yields

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Note that the densities of many objects in the outer Solar System are so close to the same value that the density term is often set equal to 1.