The Solar System

Solar System

The Solar System consists of a single G2 class yellow dwarf star, Sol or Helios (the Sun) orbited by two gas giant planets (Jupiter and Saturn), two ice giants (Uranus and Neptune), four terrestrial planets (Mercury, Venus, Earth, Mars) and a number of dwarf planets.

Orbital Dynamics

Kepler was the first to realize that the planets in the Solar System follow elliptical orbits. The ellipses are a family of shapes, resembling flattened circles, with an eccentricity, e, between 0 and 1. (A circle has e = 0). Projectiles follow parabolic trajectories and if launched with the escape speed needed to escape the planet's surface it will follow an orbit with an eccentricity = 1.

Strictly, an ellipse is the locus (collection) of points for which the sum of the distances to two foci in the ellipse to the point is a constant.

If c is the distance from the center of the ellipse to each focus (F1 and F2 being the foci) and A is the largest radius (semimajor axis) and B the smallest radius (semiminor axis) of the ellipse then it can be shown that c2 = A2 - B2 and e = c/A. Some ellipses are illustrated below:

The ellipse

In the Solar System orbital eccentricities, e, of planet-sized objects range from 0.00678 for Venus (almost perfectly circular) to 0.2488 for Pluto to 0.4361 for Eris. Thus, they all orbit on ellipses intermediate between the circle and the first ellipse illustrated above. Note that since the Sun occupies one focus, each planet will have a point of closest approach to the Sun, its perihelion, and a point of furthest approach, its apihelion. Mercury and the dwarf planets have appreciably elliptical orbits. Earth has an orbital eccentricity of 0.0167, so its orbit would appear near circular, and yet this degree of eccentricity has a significant impact on the seasons with its perihelion of about 147 million km and its apihelion of about 152 million km.

Comets generally have orbital eccentricities between 0.5 and 1. A comet that originates outside the Solar System, swings around the Sun once and then leaves the Solar System is following a hyperbolic orbit with eccentricity > 1.

Kepler devised his three laws of orbital motion:

1. The law of ellipses - the orbit of each planet is an ellipse with the Sun at one focus.

2. The law of areas - the radius vector of a planet sweeps out equal areas in equal intervals of time. (The radius vector is an imaginary arrow joining the focus with the Sun to the planet).

3. The harmonic law - the squares of the sidereal periods of the planets are proportional to the cubes of the semimajor axes (mean radii) of their orbits. That is:

P2 = ka3

Where: P is the sidereal period (the time taken for a planet to complete one orbit from the point of view of an observer outside the Solar System), a is the semimajor axis and k is a constant dependent on the system. It can be shown that the average radius of an elliptical orbit (mean distance between the planet and the Sun) is equal to the semimajor axis. Isaac Newton later derived Kepler's laws from mechanical principles.

Kepler's constant k can be calculated by equating the centripetal and gravitational forces by assuming a circular orbit. For uniform motion in a circle, the acceleration (directed towards the center of the circle) that brings about the constant change in direction (acceleration means a change in speed and/or a change in direction) is the centripetal acceleration caused by the centripetal force. This is due to the attractive gravitational force between the planet and Sun, so these two forces must be equal (they are different mathematical expressions for the same force).

Kepler's constant

Here G = 6.674 x 10-11 m3kg-1s-2 is the universal Gravitational constant and M the mass of the central object (Sun). We have used T for period, rather than P, as this is often used for circular motion, and r instead of a.

Kepler's constant as we have defined it has units of m-3s2, but sometimes it is defined as the reciprocal 1/k with units of m3s-2. For objects orbiting the Sun, Kepler's constant therefore has the same value (k = 2.97 x 10-19 m-3s2, or 1/k = 7.54 x 10-6 AU3s-2 which agrees well with the empirical value of 7.50 x 10-6 AU3s-2 for the Earth, obtained from measured values of P and a) but for other systems, such as the moon orbiting Jupiter, k has a different value (k = 3.12 x 10-16 m-3s2). Calculated and measured values agree well, despite the fact we assumed circular orbits.

Measured values of k are all approximately the same for the planets orbiting the Sun, as expected, with minor differences, but that of Mercury differs significantly with 1/k = 7.64 x 10-6 AU3s-2, but Mercury is so close to the Sun that its orbit can not be precisely explained used Newtonian gravity (as we have applied here) but requires a relativistic correction; its orbit is also more elliptical.

What is a Planet?

According to the IAU (International Astronomical Union) of planet Earth, a planet is defined as follows:

A planet is a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit.

This is a problematic definition. First of all it excludes exoplanets, planets orbiting other stars - it is a solar-centric definition. Also 'cleared the neighbourhood around its orbit' is an imprecise and unquantifiable statement. Planets may have other bodies sharing their orbit (in Lagrangian points where the gravitational forces are balanced to prevent the larger object ejecting the smaller objects from its orbit).

The IAU further defines 'dwarf planets' as a separate class of objects, rather than a subclass of planet, as follows:

A "dwarf planet" is a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, (c) has not cleared the neighbourhood around its orbit, and (d) is not a satellite.

Again, a solar-centric definition. It is also something of a semantic anomaly: a 'dwarf planet' is not a 'planet' apparently. I prefer hierarchical definitions, for example, a Holm Oak is, like a Pedunculate Oak, a species of oak and therefore in the genus Quercus.

I prefer a definition that is not solar-centric and which is not based on orbital dynamics but based on the fundamental physical properties of the object itself (after all, this is how stars are classified). I define a planet as follows:

A planet is an object that has pulled itself into a spheroid due to its self-gravity but is not massive enough to undergo nuclear fission.

One might (or might not) like to add that a planet should be at or close to hydrostatic equilibrium and so is not undergoing significant expansion or contraction. Hydrostatic pressure is the pressure (force or weight per unit area) exerted by a (stationary) fluid, such as the hydrostatic pressure at the bottom of the Earth's ocean due to the mass of water and air above. Hydrostatic equilibrium occurs when an object has been compressed to a high enough density that its internal pressure (increasing towards the center) balances the force of gravity attempting to compress the object.

Such objects typically have a diameter of approximately 1000 km or above and those that have been studied invariably have significant and interesting geology. What about moons? Many moons are small irregular objects, but some are large enough to count as planets by the above criterion. Simple, planet-sized moons are satellites that can be considered as secondary planets (planets orbiting other or primary planets).

With a simple definition as this, we can always add smaller subdivisions such as gas giant (or supergiant), ice giant (or giant), terrestrial (or medium planets) and dwarf planets. A rogue planet might be a planet that has been ejected from a star system and is currently not orbiting a star. With a proper hierarchical classification more and more subdivisions can be added as required as other star systems are explored (expanding the types of known planets). Dwarf planets likely to have crowded orbits, due to their inability to remove other objects because of their small size and low gravity, but some may well occupy relatively empty orbits.

Our definition is a geological or planetary science based definition rather than an astronomical one based on orbital dynamics and is not restricted to the Solar System: it is a universal definition. it is also an acceptable definition: science is not dictated to by political establishments, though organizations like the IAU can of course make recommendations, scientists are not bound by them. Statements such as 'Pluto is no longer a planet' are simply not true. Stating that 'the IAU no longer classifies Pluto as a planet' is true.

Dwarf Planets

The Solar System contains the following confirmed (primary) dwarf planets:

  1. Pluto
  2. Eris
  3. Ceres

and the following probably dwarf planets:

  1. Makemake
  2. Haumea

and a number of other candidates such as:

  1. Orcus
  2. Gonggong
  3. Sedna
  4. Quaoar

confirmation of which requires good estimates of their mass or diameter (and preferably confirmation of their shape).

The following moons are equivalent in size and form to dwarf planets and can be considered secondary planets or planetary moons:

  1. The Moon
  2. Jupiter system: Ganymede, Callisto, IO, Europa
  3. Saturn system: Titan, Rhea, Iapetus, Dione, Tethys, Enceladus, Mimas
  4. Uranus system: Titania, Oberon, Umbriel, Ariel, Miranda
  5. Neptune system: Triton
  6. Pluto system: Charon

To this list we may need to add Dysnomia, Eris' moon. That gives us at least 22 dwarf planets in total. It would not be surprising if most planets in the galaxy were dwarf planets.

One of the reasons for the IAU reclassifying Pluto as a 'dwarf planet' was due to a 'panic' that many similar objects might be found (this seems irrational to me) but this fear has not yet been realized. It was also once thought that such small objects would be geologically uninteresting, inert lumps of rock and/or ice. Nothing could be further from the truth.

Ceres

Above: Ceres (NASA/JPL, Dawn mission) is a dwarf planet located in the asteroid belt at 2.56 to 2.98 AU from the Sun (it has not 'cleared its orbit') accounting for about one-quarter of the mass of the asteroid belt; orbital eccentricity, e = 0.0760; Ceres orbits the Sun once every 4.6 earth years. It is been suggested that Ceres may have occupied a different position originally, being scattered into the asteroid belt when Mars migrated outwards. If so, then did it originally occupy a cleared orbit?

Ceres is the smallest known primary planet in the Solar System. With a radius of about 470 km it just has sufficient mass to have pulled itself into a spheroid, albeit a slightly bumpy one. An astronaut of mass 100 kg would weigh a mere 2.9 N (effectively less than a kg), however the escape velocity is about 0.5 km/s so it should be safe for our astronaut to run and jump around on this world without launching themselves into space (even if they could run faster on this low g world).

Ceres may look like an unexciting rock but that conceals some interesting geological phenomena.Bright surface deposits of crystallized hydrated slats (such as MgSO4.6H2O and salts of sodium and ammonia) suggests there is a subsurface reservoir of brine (saltwater) and the world may have had an early surface ocean, before surface water evaporated into space or froze.

The surface of Ceres is carbon-rich (20% carbon), about 30% water ice and also contains salts, ammonia-rich clays and tholins. Volatile materials, such as sulphur and graphite, are thought to alternately sublime and freeze on Ceres and weathering is thought to release sulphur dioxide gas. There is possibly cryovolcanism on Ceres, with Ahuna Mons a candidate cryovolcano. Outgasing presumably occurs since there is a very thin atmosphere of water vapor. Worthy of note is the large impact Occator Crater (92 km in diameter).

Ceres is  40 to 50% water by volume and 73% rock by mass. It may have a small rocky core, possibly highly porous, or a core of condrules (rocks that accreted to one-another when ceres formed). If the latter, then Ceres may be considered a protoplanet that failed to develop to maturity due to a lack of material. The mantle possibly consists of ice and mud (hydrated silicates) and may have a thick layer of brine and mud. The crust apparently consists largely of hydrated minerals.

Orbits within the Solar System

Orbits

Above: the 10 innermost orbits from Mercury's (1) to Pluto's (10). Orbit 5 is occupied by Ceres (dashed circles indicate the approximate extent of the asteroid belt).

Orbits

Above:outer orbits of the Solar System; 10a Pluto, 10b Haumea, 10c Makemake, 10d Eris. The orbits of these objects pass through the Kuiper Belt (indicated by the shaded ring). The Kuiper Belt is essentially another asteroid belt, consisting of protoplanetary building blocks that likely failed to form a large planet due to disruption by Neptune's gravitational field.

Haumea orbits at 34.65 to 51.59 AU with an eccentricity, e, of 0.1964. It has a radius of about 780 km and is a peculiar ovoid shape with its long axis about twice the length as its short axis. This is due in part  to its rapid rotation, spinning on its axis about once every 4 earth hours, causing its equator to bulge considerably. However, its surface apparently does not conform to hydrostatic equilibrium for a fluid (for one thing its three axes all have different lengths when we might expect the equatorial radius to be constant) and it has been suggested that its interior consists of condrules or similar large particles. It has two small satellites (Hi'iaka and Namaka). It has a high albedo (surface reflectivity) although estimates vary considerably indicating surface ices, and possibly a high density (again estimates vary) and a ring.

Makemake has a radius of about 720 km and a very red surface, indicating methane. It has an eccentric orbit (e = 0.1613) ranging from 38.1 and 52.76 AU. It has one small satellite, MK2.

Orbital Resonance, Gaps and Rings

In orbital resonance the gravitational field of one object influences the orbital positions of nearby objects. Any aspect of the orbits may be affected, but in mean motion orbital resonance the orbital periods resonate and become commensurable, that is the ratio of orbital periods becomes a simple fraction. For example, Pluto is in a 2:3 orbital resonance with Neptune, meaning for every two orbits of Pluto, Neptune completes three orbits. This means that whenever Pluto crosses Neptune's orbit, Neptune is far away and so the arrangement is stable.

Orbital resonances may stabilize or destabilize orbits. For example, gaps in the main Asteroid Belt occur at 1:4, 1:3, 2:5, 1:2, ... orbital resonances with Jupiter's orbital period. These are known as the Kirkwood Gaps. Asteroids in these orbits would come into frequent alignment with Jupiter and experience gravitational torque from Jupiter's immense gravitational field increasing the eccentricity of their orbits and causing them to cross the orbits of Mars and Earth where they may collide and be removed/destroyed.

More eccentric orbits result whenever a body is perturbed by the gravitational field of a third object nearby. Elliptical orbits have the same energy as circular orbits where the radius of the circular orbit equals the semimajor axis of the elliptical orbit. Thus they are equivalent solutions to Newton's equations of gravitation. Objects in elliptical orbits travel more slowly when far from the Sun and speed up when closer to the Sun. Although the total orbital energy is constant at all points on the orbit, when a planet nears perihelion it loses gravitational potential energy (potential energy is stored energy) which is converted into kinetic energy (energy of motion) causing the planet to speed up,and when it approaches aphelion this kinetic energy is converted back into gravitational potential energy. Whenever an object is pushed or pulled into a different orbit, changing the orbital energy, then it tends to adopt a more eccentric orbit probably due to the external force being applied most strongly at one point in the orbit.

Similarly, gaps in the rings of Saturn are due to resonances with Saturn's moons. Haumea is a dwarf planet with a ring which is possibly in a 1:3 spin resonance with Haumea (the ring completing one rotation for every three rotations of Haumea). More exact measurements are needed to ascertain how precise this apparent resonance is.

Haumea

Above: Haumea has an estimated oblateness of around 0.5 to 0.6 (measurements currently lack verifiable precision) making this dwarf planet the most oblate planet in the Solar System. Knowledge of the surface is limited, but some measurements such it may be highly reflective due to surface ices and a red spot, due to methane, has been detected. In this model the simulated surface reflects this information within artistic bounds.

The orbit of Haumea appears to be in a weak 7:12 resonance with Neptune's. However, weak resonances could be coincidental: precision measurements are required. Makemake is far enough from Neptune not to have its orbit significantly perturbed by it. Eris is in an eccentric and inclined (tilted, in this case to 44o from the plane of the Solar System) and is thought to have been perturbed by Neptune and scattered to a more remote orbit.