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Exploring the Solar System

Lecture 1: Overview of the Solar System - Introduction

Professor Paul Sellin Department of Physics

University of SurreyGuildford UK

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Course Schedule

Lectures are in TB01

Computing sessions are in 32BB03

There are 2 pieces of Computing

Coursework, plus a final examination

Coursework deadlines are Friday at 4pm

Recommended Text:

‘Universe’ by Freedman and

Kaufmann

L10: Jovian Planets: Uranus and NeptuneWeek 1120 May

L9: Jovian Planets: Jupiter and SaturnWeek 1013 MaySubmit CW2 exercise

Easter Holiday

Computing L8: Terrestrial Planets: MarsWeek 98 April

ComputingL7: Terrestrial Planets: VenusComputing

Week 81 AprilGive out CW2 exercise

L6: Terrestrial Planets: MercuryWeek 725 March

L5: Formation of the Solar SystemWeek 618 MarchSubmit CW1 exercise

Computing L4: Exploration of the MoonWeek 511 March

ComputingL3: Structure of the MoonWeek 44 MarchGive out CW1 exercise

No lecture on 25th Feb - Week 3

L2: Physical Properties of the Solar systemWeek 218 Feb

L1: Introduction to the Solar SystemWeek 111 Feb

Session 2: 11-12Session 1: 10-11

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Contents

Planetary motion and basic orbital data

Titus-Bodes or planetary orbits

Kepler’s Laws

Observed orbits

The layout of the Solar System

View of the solar system to scale

Some key definitions

Sidereal rotation

Astronomical units

Rotational motion of the planets

Rotational data

Rotational measurements using Radar

Summary

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The Planets

Nine major planets have been discovered in orbit around the Sun.

In order of increasing heliocentric distance these are: Mercury, Venus, Earth and Mars (the so-called terrestrial planets); Jupiter, Saturn, Uranus, and Neptune (the so-called gas-giants or Jovian planets); and finally the small and rather odd planet Pluto, discovered in 1930.

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Planetary Moons

Earth’s satellite (the Moon) is unusually large compared to its parent body and so it is alsosometimes classed as a terrestrial planet - thus, the Earth- Moon system can be thought of as a double-planet.

Similarly, Pluto’s satellite Charon is also relatively large, and so this may also be classed as a double-planet.

Planet Sidereal Period Semi-Major Axis Eccentricity Inclination Tropical Years AU 106 km to Ecliptic

Mercury 0.241 (87.96d) 0.387 57.9 0.206 7.00°Venus 0.615 (224.70d) 0.723 108.2 0.007 3.39°Earth 1.000 (365.26d) 1.000 149.6 0.017 0.00°Mars 1.88 (686.98d) 1.524 228.0 0.093 1.85°Jupiter 11.86 5.203 778.3 0.048 1.31°Saturn 29.46 9.54 1427 0.056 2.49°Uranus 84.01 19.18 2871 0.047 0.77°Neptune 164.8 30.06 4497 0.009 1.77°Pluto 248.6 39.44 5913 0.249 17.15°

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1.1 Planetary Orbits and Motion*

The planets orbit the Sun under the influence of gravity according to Newton’s and Kepler’s laws - i.e. the planets follow elliptical paths.

Bode’s Law- The size of the semi-major axes (SMA’s) of the planetary orbits form an approximately regular pattern, discovered empirically by Titius of Wittenburg in 1766 (before Uranus. Neptune and Pluto were discovered), and publicised by Johann Bode in 1772.

The Titius-Bode Rule or Bode’s law concerns the sequence:

Dividing the numbers in this sequence by 10 gives a remarkable fit to the SMA’s of the planets, when these are expressed in terms of Astronomical Units (AU)

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The Titus-Bode sequence*

The Titius-Bode Sequence gives (only 6 planets known at the time):

and the actual planetary SMA’s are:

The fit to the six planets (known since ancient times) was good, however there seemed to be a missing planet at 2.8 AU.

This prediction was confirmed in 1801 when Giuseppe Piazzi discovered the small planet Cees in exactly the right place.

However, it was soon discovered that this was only one (albeit the largest) of numerous small bodies orbiting between 2.3 and 3.3 AU, now known as minor planets or asteroids.

It can be seen that the empirical Titius-Bode Rule works remarkably well for most of the planets. When William Herschel discovered Uranus in 1781 at 19.2 AU, nine years after Bode’s publication, the predictive power of the rule must have seemed remarkable. Shortly after its discovery, minor perturbations in the predicted orbit of Uranus suggested that an eighth planet waited to be discovered beyond Uranus - was it to be found at 38.8 AU? Calculations based on Newtonian mechanics by J.C Adams (in 1843) and independently by U.J Leverrier (in 1846), deduced the probable mass and orbit of the unknown planet, and in 1846 Johann G. Galle at the Berlin Observatory found the planet (later to be called Neptune) within 1° of its predicted position - but unfortunately not where the Titius-Bode rule said it would be. However, perturbations observed in Neptune’s orbit suggested that yet another planet existed beyond Neptune (the so-called “PlanetX’). However, predicting the orbit and mass of this planet was very difficult! Eventually, in 1930 Clyde W. Tombaugh found the planet Pluto close to the position calculated by Percival Lowell. Unfortunately this was just a fluke as Pluto is too small to perturb Neptune’s orbit. So is Planet X still out there waiting to be discovered? Has the TitiusBode Rule got any real physical basis? We just don’t know!

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Kepler’s empirical laws of planetary motion*

Keppler’s 1st Law: The orbit of each planet is an ellipse, with the Sun at one of the foci (published 1609).

Note that the Sun is one focus of a planet’s elliptical orbit, with nothing at the other focus

An ellipse can be drawn using a string of fixed length:

A series of ellipses with the same major axis, and varying eccentricities:

By 1621, Kepler had shown that these laws also applied to the Galilean satellites, with Jupiter as the primary body, and with a different value of k.

Newton, in Principia (1687), explained these empirical observations through his theory of Universal Gravitation. In this theory, an attractive force - gravity - is supposed to exist between all bodies, with a magnitude proportional to the product of the masses of the bodies, and inversely proportional to the square of their distance apart: F=G Mm I r2 A

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Conic Sections*

The ellipse is one form of conic section, which have the general expression:

where:

e is the eccentricity

p is a length ordinate:

The 4 possible conics are:

Circle:

Ellipse

Parabola

Hyperbola

Here a is the semi-major axis, and b the semi-minor axis. For an ellipse:

For a parabola, q is defined as the distance to the COM at closed approach.

The type of conic section is determined by the angle the plane makes with the horizontal.

If the plane is horizontal, the resulting curve is a circle. If the plane is at an angle which is less than the slope angle of the cone, the curve is an ellipse.

If the plane is parallel to the slope angle of the cone, the curve is a parabola. If the plane is anywhere between the slope angle of the cone and the vertical, the curve is a hyperbola.

For a 2 body planetary orbit (eg. a planet around the Sun) the motion is elliptical, hence Keppler’s first law is a result of the inverse square law of force. In such an orbit one of the masses occupies one focus of the ellipse, whilst the other focal point is empty.

Some cometary orbits are parabola, with e ~ 1 although most permanent membersof the solar system follow elliptical orbits with e<<1. Exceptions to this are Pluto (e=0.25) and Mercury (e=0.21).

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Kepler’s 2nd Law*

Keppler’s 2nd Law: The radius vector to a planet sweeps out equal areas in equal times (the Equal Areas Law).

A planet moves fastest at Perihelion and slowest at Aphelion, such that the equal areas are swept out over equal time.

The semimajor axis a is the orbiting body’s mean distance from the Sun

An elliptical orbit is defined in terms of the semimajor axis and the eccentricity e.

The perihelion distance q is the distance from the perihelion to the Sun, and similarly the aphelion distance Q:

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Orbital Period*

Circular orbital motion is derived from equating centripetal and gravitational forces:

Hence the orbital period T is proportional to r3/2

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Kepler’s 3rd Law*

Kepler’s 3rd law connects the planet’s sideral period p with the semimajor axis a:

More generally we can write:

where k is a constant which depends on the mass of the central body.

k is the same for all planets orbiting around the Sun

For calculations in our solar system it is convenient to use units of years and AU.

Hence calculate k for our solar system using Earth data:

In the special case of an object orbiting the Sun, where p is in years and a is in AU, then:

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Orbital Motion of the Planets

Direction – Most of the planets orbit counter-clockwise around the Sun seen from above the Earth’s orbit plane (North = Up). This is called the direct or prograde (as opposed to retrograde) direction

The exceptions are Venus, Uranus and Pluto which has a slow retrograde rotation

Co-planarity - All the planets’ orbital planes are virtually co-planar - i.e. the inclinations of the orbit planes with respect to the ecliptic plane (that is the plane of the Earth’s orbit) are small

Pluto is the exception, with an orbital inclination of 17° with respect to the ecliptic

All the major planets, except Pluto, are always found within the 16°-wide band of the Zodiac

Shape – Most of the planets have an orbital eccentricity of less than 0.1 (i.e. the orbits are nearly circular)

Mercury and Pluto are exceptions: Pluto is so eccentric, ranging from 29.7 AU at perihelion to 49.3 AU at aphelion that it crosses within the orbit of Neptune.

The fact that the planets move in the same direction around the Sun, and that the orbits are virtually co-planar is equivalent to saying that the angular momentum vectors of the planets’ orbital motions are very nearly aligned (as indeed they are aligned with the rotational angular momentum vector of the Sun). Any theory of the formation of the solar system must explain this observation. Unfortunately, a number of conflicting theories predict this outcome, so it is difficult to choose between them on this basis alone. However it is worth noting that the generally accepted hypothesis - that both the Sun and the planets condensed from the same swirling cloud of gas and dust - has a major problem: The rotational angular momentum of the Sun and angular momentum associated with the orbits of the planets is very different in magnitude - with the planets accounting for 97% of the Solar system’s angular momentum and yet only having a tiny fraction of the mass of the Sun (in total 0.0014 times the solar mass). How could this occur? The fact that the perihelion of Pluto lies within the orbit of Neptune, and that Pluto’s orbit is unusually eccentric, forms the basis for the conjecture that perhaps Pluto was once a satellite of Neptune and that some perturbation in the past has caused it to escape.

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Ancient view of the universe

Ancient astronomers believed the Earth to be at the center of the universe

They invented a complex system of epicycles and deferents to explain the direct and retrograde motions of the planets on the celestial sphere

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Copernicus

Nicolaus Copernicus devised the first comprehensive heliocentric model

Copernicus’s heliocentric (Sun-centered) theory simplified the general explanation of planetary motions

In a heliocentric system, the Earth is one of the planets orbiting the Sun

The sidereal period of a planet, its true orbital period, is measured with respect to the stars

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Planet’s motion viewed from Earth

A planet undergoes retrograde motion as seen from Earth when the Earth and the planet pass each other

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The Solar System to scale

4 inner planets are close to the Sun: Mercury, Venus, Earth, Mars

5 outer planets are at much greater distances: Jupiter, Saturn, Uranus, Neptune, Pluto

Pluto’s orbit is unusual, and highly eccentric

Orbital inclination of all planets except Pluto are <7°. Pluto’s inclination is 17°

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Pluto

Pluto is a special case

Smaller than any of the terrestrial planets

Intermediate average density of about 1900 kg/m3

Density suggests it is composed of a mixture of ice and rock

But is Pluto still a planet?

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Re-classification of the planets

In August 2006 the International Astronomical Union (IAU) general assembly decided that the solar system is to gain three new planets under a revised definition that spares Pluto demotion from the exclusive planetary club.

However, the 12 planets will be divided into two broad categories:

8 “classical” planets such as Earth and Jupiter

4 “dwarfs” such as Pluto and the three new members. Pluto thus survives as a planet, but outside new group of 8.

The “dwarfs” or “plutons” are initially defined as dwarf planets beyond Neptune that take more than 200 years to orbit the Sun. The other 3 are:

Ceres, hitherto regarded as the largest asteroid

Charon, once considered to be Pluto’s moon

; and a newly discovered object officially called 2003 UB313, but nicknamed Xena after the television warrior princess

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Large satellite objects

Seven large satellites are almost as big as the terrestrial planets, eg. comparable in size to Mercury

The remaining satellites of the solar system are much smaller

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Viewing a planet from the Earth

Mercury and Venus are always observed fairly near to the Sun

their orbits are smaller than the Earth’s

such planets are called inferior planets

Other visible planets such as Mars, Jupiter and Saturn, can be seen opposite the Sun

they appear high in the night sky when the Sun is far below the horizon

at this point the Earth must lie between the planet and the Sun

planets with orbits larger than the Earth are superior planets

Figure shows the configurations of the planetary orbits, viewed looking down onto the solar system from a point far above the Earth’s northern hemisphere

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Sidereal orbital period of planets*

The sidereal period is the time for a planet to complete 1 orbit, ie. the ‘true’ orbital period.

The synodic period is the time between two successive configurations, as observed from Earth. The synodic period differs from the sidereal period since Earth itself revolves around the Sun.

Consider an inferior planet (Mercury or Venus):

P is the planet’s sidereal period, S is the planet’s synodic period, E is the Earth’s sidereal period (sidereal year):

During a time S the Earth covers an angular distance D of:

In the same time the inferior planet covers a distance D’ of:

Dividing by 360 x S gives, for an inferior planet,

Note that this expression only works if the period is in years.

For periods in days, use:

SP

3651

365

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Sidereal orbital period of planets (2)*

For a superior planet the same analysis gives:

Consider Jupiter as an example:

Jupiter has an observed synodic period of 398.9 days, or 1.092 years. What is it’s sidereal period?

ie. the Earth overtakes Jupiter a little less often than once per Earth orbit, ie. at intervals of slightly greater than 1 year

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Synodic and Sideral periods

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Astronomical Units

Parsec (pc)

the distance at which 1 AU subtends an angle of 1 arcsec or the distance from which Earth would appear to be one arcsecond from the Sun

1 pc = 3.09 × 1013 km = 3.26 ly

Astronomical Unit (AU)

One AU is the average distance between Earth and the Sun

1.496 x 108 km or 92.96 million miles

Light Year (ly)

One ly is the distance light can travel in one year at a speed of about 3 x 105 km/s or 186,000 miles/s

9.46 x 1012 km or 63,240 AU

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Rotational rates and radar measurements

When microwaves (RADAR) are bounced off a rotating planet, the waves that hit the edge (or limb) of the planet that is rotating away from the observer will be red-shifted:

the reflected wave has a lower frequency () or equivalently a longer wavelength () than that transmitted.

Conversely, the wave reflected from the part rotating towards the observer will be blue-shifted - that is the reflected wave will have a higher or a shorter .

The prolate (“cigar-shaped”) nature of Mercury, coupled with its eccentric orbit have (presumably) caused the rotation rate of the planet to slow down over time until, as now, it points alternate ends of its long- axis at the Sun at each successive perihelion. This is a stable configuration or “resonance”. Venus is more complex as it is responding to tidal forces from the Sun, and (much weaker) forces from the Earth.

Such tidal forces exist between the Earth and its Moon. The Moon has also has a tri-axial figure, which, over the course of time has caused it to go into spin-orbit resonance with the Earth - to the extent that it is now in captured rotation (that is it completes one rotation for each revolution about the Earth). This is why we always see the same hemisphere of the Moon from Earth (actually somewhat more than a hemisphere due to libration). Similarly lunar tidal forces are acting on Earth, which are slowing its rotation period.

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Radar (2)*

The amount of wavelength shift () depends upon:

the magnitude of the radial velocity (Vr)

the speed of light (c).

This assumes the planet’s rotation is at non-relativistic speeds!

‘L’

‘S’There is an overall shift from all reflected waves from the planet, due to the orbital velocity of the planet around the sun. This is common to all the reflected waves, and can be factored out.

Waves reflected from the sub-radar point show this ‘mean’ due to the planet’s orbit alone

Waves reflected from the limbs show an additional due to the planets rotational radial velocity

A wave reflected at ‘L’ travels an extra distance 2d compared to those from the sub-radar point ‘S’:

RADAR energy will penetrate the atmospheres of planets, but be reflected from their solid surfaces -thus the solid surface of Venus can be investigated even though it cannot be directly observed from Earth because of the cloudiness of Venus’ thick atmosphere.

Similarly, RADAR is not affected by the glare of the Sun (provided that the Sun does not fall within the angle of view of the RADAR antenna), and so Mercury can be probed from Earth.

The Moon, Mars and even some asteroids can be similarly investigated from Earth. In all cases features on the surface provide scattering points which can be followed as the planet rotates.

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Radar (3)*

If f is the frequency shift between the waves reflected at ‘S’ and ‘L’, then the doppler effect gives:

where v0 is the rotational velocity component parallel to the line of sight to Earth, at the point where the wave is reflected. This component is perpendicular to the tangent of the planet’s surface.

The equatorial rotational velocity v is related to v0 by

where y can be calculated by pythagoras:

So the maximum shift in measured wavelength is from waves reflected from the edge of the planet, where y ~ R, and so v ~ v0

The rotational period T is then:

and

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Rotation of Mars*

Calculate the maximum spread in frequency ftot for a 400 MHz radar wave reflected off Mars, given that the rotation period of Mars is 24hrs 37mins.

The radius of Mars is R = 3390 km

The maximum component of the rotational velocity v0 is given by:

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Rotation of Venus and Mercury

The sidereal rotation period of Mercury was found to be exactly two- thirds of its sidereal orbital period - that is it completes three rotations for every six revolutions around the Sun. Thus, Mercury is in a 3:2 sidereal lock on the Sun.

In the case of Venus, it was found that its synodic rotation period (that is its rotation period with respect to the Earth) at 146 days was approximately one-quarter of its synodic orbital period. Thus, Venus would appear to be in an approximate 4:1 synodic resonance with its sister planet the Earth.

Both Mercury and Venus are prolate and tidal forces have caused these resonances.

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Rotation of the gas giants

The gas-giants: Jupiter, Saturn, Uranus and Neptune only show the top of their atmospheres, and so we follow their rotation rates by noting the motion of cloud features, etc.

Optical doppler techniques, similar to Radar, are used to observe the Doppler-shifted radiation from their limbs - e.g. by measuring the small red- and blue-shifts of spectral lines associated with gases in their atmospheres.

The atmospheres of these planets rotate faster at the Equator than they do at the poles, therefore no single rotation rate can be unambiguously defined, although sometimes the rotation rate of the magnetic field associated with the planet is given.

In the case of Pluto (a solid body), brightness fluctuations associated with differences in the albedo of surface features allows a rotation rate to be inferred.

Presumably, as with Earth, the atmospheres of the gas-giants are in rotation with respect to any solid-body that might lie deep inside the planet. If, as with Earth, the magnetic field is co-rotational with this solid body, then rotation-rate of the field will give information on the deep interior of each of the gas-giants.

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Planetary rotation data*

Some definitions:

Oblateness - This is a measure of shape of a planet, and quantifies the “flattening” of the poles and the “bulging” of the Equator. It is defined as follows:

where: re is the equatorial radius, and rp is the polar radius of the planet

The Earth (as typical for a “solid” body) has a small oblateness of 1/298.3, whereas the gas-giants are all very oblate due to the fluidity of their visible surfaces (i.e. the top of their atmospheres) and their relatively rapid rotation rates.

Equatorlal Inclination (Obliquity) - The equatorial planes of the planets are inclined by different amounts with respect to their orbital planes.

The rotation axes of the Moon, Mercury and Jupiter are all virtually aligned with their revolution axes, whereas the obliquities of the Earth, Mars, Saturn and Neptune are all around 25°.

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Planetary rotation data (2)

Planetary Rotation Data:

Planet: Sidereal Oblateness Obliquity Rotation Period

Mercury 58.6 days 0 0.00 Venus 243 days 0 177.4°Earth 23h 56m 4.ls (= 1 sidereal day) 0.0034 23.5°Moon 27.3 days 0.0006 6.7°Mars 24h 37m 22.6s 0.0052 25.2°Jupiter 9h 50m - 9h 55m 0.062 3.1°Saturn 10h 14m - 10h 38m 0.096 26.7°Uranus 17h 0.06 98°Neptune 17h 50m 0.02 29°Pluto 6.4 days ? 65°

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Orientation of Uranus

Uranus is unusual in that its rotation lies virtually within its orbit plane (its obliquity is 98°), so that currently one of its poles points towards the Sun. In twenty years or so, its equator will point sunwards, and twenty one years later the other pole will point sunwards, etc.

Most other planets have obliquities of less than 90° - i.e. they rotate from west to east - the direct or prograde direction.

Uranus is virtually tilted onto its “side”. This leads some to speculate that Uranus has suffered some major calamity in its past. Has some passing object so perturbed Uranus that its rotational axis has been flipped over?

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Retrograde rotation of Venus

Venus rotates slowly in a retrograde direction with a solar day of 117 Earth days and a rotation period of 243 Earth days. There are approximately two Venusian solar days in a Venusian year.

Although Venus is rotating the “wrong way”, its rotation axis is relatively perpendicular with respect to its orbital plane - just as with all the other major planets - except Uranus.

Venus is also unusual in that, as with Uranus, it rotates from east to west - i.e. in the retrograde

direction. Its obliquity is 177° - you could think of this as being almost upside down!

The rotation rate of Venus was first measured during the 1961 conjunction, observed by radar from a 26 m antenna at Goldstone, California, the Jodrell Bank Radio Observatory in the UK, and the Sovietdeep space facility in Eupatoria, Crimea. Accuracy was refined at each subsequent conjunction, primarily from measurements made from Goldstone and Evpatoriia. The fact that rotation was retrograde was not confirmed until 1964.

When Venus is near inferior conjunction so that it is passing close to the Earth, because of its faster angular velocity it appears to an observer on Earth to move from east to west on the celestial sphere from one night to the next. This apparent motion is called retrograde motion because it is the opposite of the apparent west to east motion that Venus displays most of the time. This effect is quite different to Venus’ retrograde rotation.

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Key Definitions

The sidereal period is the time for a planet to complete 1 orbit, ie. the ‘true’ orbital period.

The synodic period is the time between two successive configurations, as observed from Earth. The synodic period differs from the sidereal period since Earth itself revolves around the Sun.

inferior planets are those with their orbits are smaller than the Earth’s

superior planets are those with their orbits are larger than the Earth’s

inferior conjunction, when an inferior planet is directly between the Sun and the Earth

superior conjunction, when an inferior planet is directly behind a line between the Sun and the Earth

One Astromical Unit (AU) is the average distance between Earth and the Sun (1.496 x 108 km)

One Light Year (ly) is the distance light can travel in one year (9.46 x 1012 km or 63,240 AU)

One parsec (pc) is the distance that 1 AU subtends an angle of 1 arcsec, or the distance from which Earth would appear to be one arcsecond from the Sun (1 pc = 3.09×1013 km = 3.26 ly