Planetary properties

3.3 Planetary properties

3.3.1 Planetary masses

It is possible to fi nd the mass of a planet if: (1) It has one or more natural satellites in orbit around it, as in the case for the

planets Earth, Mars, Jupiter, Saturn, Uranus and Neptune and the dwarf planets Pluto (Charon) and Eris (Dysnomia).

(2) It has acquired an artifi cial satellite as in the case of the Magellan space- craft in orbit about Venus. (3) It has been passed by an artifi cial satellite as was the case when Mariner

10 fl ew by Mercury. It has thus been possible to calculate the mass of all the Solar System planets and

two of its dwarf planets. In September 2007, the Dawn Mission spacecraft was launched to visit both Ceres and a second body within the asteroid belt, Vesta, so we will then have masses for all currently known planets and dwarf planets.

The method is essentially that used in Chapter 2 to calculate the mass of the Sun using the period and semi-major axis of the Earth’s orbit. Let us take Mars as an example.

Calculating the mass of mars

Mars has a satellite, Phobos, which orbits Mars with a period of 7 h 39.2 min (27 552 s) in an almost circular orbit having a semi-major axis of 9377.2 km or

6 m. The gravitational force between Mars and Phobos is given by MmG/a 2 , where M

is the mass of Mars, m is the mass of Phobos, a is the semi-major axis of Phobos’s orbit and G the universal constant of gravitation.

This force must equal that resulting from the centripetal acceleration, ω 2 , or v 2 /a on Phobos of mass m:

MmG /a 2 2 /a

Giving:

2 a /G

80 Introduction to Astronomy and Cosmology

2 a 3 /GP 2

with units of kilograms, seconds and metres. So

M 2 6 ) 3 4 ) 2 ] kg

23 kg

Having completed this calculation as this text was being written, the author was

23 kg. The calculation used the slight approximation that the orbit of Phobos was circu-

lar so perfect accuracy should not have been expected.

3.3.2 Planetary densities

From the angular size of a planet and its distance one can calculate the diameter of

a planet and hence its volume so, given its mass, one can thus calculate its density. It is interesting to use Saturn as an example. Saturn is not a sphere, but an oblate spheroid having a greater equatorial than polar radius. We will use an ‘average’

23 m 3

26 kg, this gives a density of ∼662 kg m , which is a little less than the accepted value of 687 kg m . You will note that this is less

than that of water (1000 kg m at 4°C)!

3.3.3 Rotation periods

For some planets, such as Mars, Jupiter and Saturn, one can observe the rotation of a marking on the surface or in the atmosphere – such as the ‘red spot’ which lies in the atmosphere of Jupiter.

The surface of Mercury is very indistinct as seen from Earth and Venus is cloud covered! In these two cases, planetary radars have been able to measure the rota-

Our Solar System 2 – The Planets

the Sun and a continuous radio frequency is refl ected from Venus when a radar transmitter lies on the line between the centre of the Earth, through the centre of Venus to the centre of the Sun. At this time (and this time only) the motion of Venus will be across the line of sight and there will be no Doppler shift in the returned echo. (To be totally accurate, there will be a very small Doppler shift called the transverse Doppler shift due to a prediction of special relativity.)

The centre frequency of the returned echo will be at precisely the same fre- quency as that transmitted and, if Venus is not rotating, all the returned energy will be at this frequency. Suppose, however, Venus is rotating. One limb will be coming towards us relative to the centre, whilst the other will be moving away from us. The echoes from the limbs will thus be Doppler shifted above and below the centre frequency so that the returned echo is ‘broadened’ in frequency. The greater the broadening, the greater the rotation rate of the planet.

Radar observations made in the 1960s showed that Venus had a very slow rota- tion rate, taking 243.01 days to rotate once round its axis – 18.3 days longer that it takes to orbit the Sun! Even more surprising, it rotates in the opposite direction to that expected. Looking down from above the Solar System all the planets rotate in an anticlockwise direction. The spin of most planets is also in an anticlockwise direction, but Venus (along with Uranus and Pluto) rotates in a clockwise direction, in the opposite sense to its orbital motion, and the rotation is said to be retrograde.

3.3.4 Planetary temperatures

There are three ways that we can measure or estimate the surface temperature of a planet:

(1) In the case of Venus and Mars, spacecraft on the surface have made direct measurements. (2) The temperature of Mercury was estimated from the intensity of its radio emission assuming it to act as a black body. In a similar way, the tempera- tures of the outer planets can be estimated from their infrared emission.

(3) We can calculate a nominal temperature on the assumption that a

planet acts like a black body and will radiate away the energy that it receives from the Sun. (There must be an equilibrium between the energy absorbed from the Sun and that emitted by a planet.)

We will consider this last method in more detail and carry out a calculation to derive the surface temperature of the Earth.

We know that, above the atmosphere, 1368 W of solar energy (the solar constant) fall on the Earth per square metre. Figure 3.4 shows that the Earth will

82 Introduction to Astronomy and Cosmology

Figure 3.4 The Earth’s eff ective area for absorption of solar energy.

intercept this radiation over an area given by the Earth’s cross-section. If SC is the value of the solar constant, then the total energy that will fall on the Earth is given by:

πR 2 SC

If we assume that the planet acts as a black body, then the energy emitted by it is given by the Stefan–Boltzmann Law and equals:

4πR 2 σT 4

In equilibrium, these energies must be equal:

This is not far off the actual average surface temperature, but should it be?

The Earth is, on average, about 50% cloud covered and absorbs only ∼77% of the incident solar radiation from the Sun. Taking this into account by reducing

the incident energy by 0.77, T Earth would only be ∼260 K. However, the Greenhouse

Effect produced by the carbon dioxide, methane and water vapour in the atmo- sphere prevents the Earth radiating away as much energy as would a perfect black body, so increasing the Earth’s temperature. The two effects roughly cancel out giving us an average temperature for T Earth of ∼288 K. It is worth pointing out that without the greenhouse gases in our atmosphere our planet would be uninhabitable.

Our Solar System 2 – The Planets

Greenhouse gases absorb infrared radiation emitted by the Earth and then re-emit it in random directions – so much of the infrared radiation will thus be directed back at the Earth!

3.3.5 Global warming

The major constituents of the atmosphere, nitrogen, N 2 , and oxygen, O 2 , are not greenhouse gases. This is because diatomic molecules such as these nei- ther absorb nor emit infrared radiation. Carbon dioxide is the main greenhouse gas in the atmosphere. Over aeons of time its percentage in the atmosphere has remained stable but, unfortunately, the burning of fossil fuels (which have stored carbon within them) is rapidly increasing the amount of carbon dioxide in the atmosphere and this is almost certainly a major contribution to the fact

that our Earth’s temperature is increasing – termed global warming or climate

change . Water vapour is a naturally occurring greenhouse gas and actually accounts for the largest percentage of the greenhouse effect, somewhere between 36% and 66%. The amount of water vapour in the air from locality to locality is very vari- able but overall, human activity does not directly affect water vapour concentra- tions (except near irrigated fi elds for example) and its effects on the Earth’s climate are remaining stable.

However, the amounts of two further greenhouse gases are now also increasing: (1) Methane is 20 times more effi cient at retaining heat than carbon dioxide

and we are adding up to 500 million t of methane into the atmosphere per year from livestock, coal mining, drilling for oil and natural gas, rice cultivation, and garbage decaying in landfi lls.

(2) Each year, 7–13 million t of nitrous oxide is added to the atmosphere from the use of nitrogen based fertilizers, the disposing of human and animal waste in sewage treatment plants and automobile exhausts.

An increase in the Earth’s average temperature of more than 2 degrees could begin to have very harmful consequences for the human race, which explains why the problem is being treated so seriously.

3.3.6 Albedo

As the example of the Earth has shown, the actual temperature of a planet is affected by how much of the Sun’s incident energy is refl ected back into space – called the albedo of a planet – and the effects of greenhouse gases, if any.

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The Earth has an albedo of ∼0.37, meaning that it refl ects ∼37% of the Sun’s energy and so will absorb 63%. Venus has an albedo of ∼0.7 (published values vary from 0.65 to 0.84) so that it only absorbs 30% of the incident solar energy, but its carbon dioxide atmosphere is so thick that its surface temperature is raised signifi cantly. Mars has an albedo of 0.15 so absorbs much of the incident solar energy but its thin carbon dioxide atmosphere (about 1/100th that of the Earth) is unable to trap much heat so it is now too cold for carbon/water based life forms to survive on the surface. However, in the past, when giant volcanoes were emit- ting vast amounts of gas into the atmosphere (including water vapour, carbon dioxide and methane) its temperature would have been signifi cantly higher and life could, perhaps, have arisen there.