The Sun

2.2 The Sun

Our Sun is certainly a typical star; its structure and the nuclear fusion process by which it generates its energy are applicable to the vast majority of stars. However, it is not, as is often stated, an average star. As we will see in Chapter 6, there are seven stellar types: ranging from the hottest to the coolest these are Types O, B, A,

F, G, K, and M. Our Sun is a Type G star and thus appears, if anything, to be below average. This ignores the fact that there are far more cool stars than hot ones. Our Sun is of Type G2 (at the hotter end of the G type stars), and ∼95% of all stars are

smaller and cooler than the Sun. Our Sun is well above average!

2.2.1 Overall properties of the Sun

Observations from Earth, coupled with the basic laws of physics, enable us to measure the main properties of the Sun.

Diameter

The Sun subtends an angle of ~30 arcmin in the sky. Given its distance from us of ∼150 000 000 km one can directly estimate the Sun’s diameter:

D ⫽ Rθ (With θ in radians and R is the Earth-Sun distance.)

44 Introduction to Astronomy and Cosmology

θ ⫽ 30/(60 ⫻ 57.3) rad ⫽ 8.7 ⫻ 10 ⫺3 rad

So

D ⫽ 1.5 ⫻ 10 8 ⫻ 8.7 ⫻ 10 ⫺3 km ⫽ 1 308 900 km

This is reasonably close to the accurate value of 1 391 978 km. The Sun’s radius is thus ∼700 000 km.

(The distance of the Earth from the Sun varies from a minimum of 147 085 800 km when closest to the Sun on January 3 to 152 104 980 km when furthest from the Sun on July 4. To derive a precise value one would measure the observed angular diameter when at a known distance from the Sun.)

Mass

The mass of the Sun can be derived using the same methods that Newton employed with the Moon. Equating the force attracting the Earth to the Sun given by Newton’s Law of Gravitation with that due to centripetal acceleration we get:

MmG /R 2 ⫽ mv 2 /R

where M is the mass of the Sun, m is the mass of the Earth, v is the velocity of the Earth around the Sun, G is the universal constant of gravitation and R is the (assumed constant) distance of the Earth from the Sun.

The mass of the Earth, m, cancels out, giving:

M ⫽v 2 R /G

But v ⫽ 2πR/P where P is the period of the Earth’s orbit, so substituting: M ⫽ 4π 2 R 3 /GP 2

⫽ 4 ⫻ (3.14159) 2 ⫻ (1.496 ⫻ 10 11 ) 3 /6.67 ⫻ 10 ⫺11 ⫻ (3.156 ⫻ 10 7 ) 2 kg

⫽ 1.99 ⫻ 10 30 kg

The mass of the Sun is 2 ⫻ 10 30 kg.

Our Solar System 1 – The Sun

Density

From the radius and mass of the Sun one can easily derive its average density: Volume ⫽ 4/3πr 3

⫽ 4/3π(700 000 000) 3 ⫽ 1.4 ⫻ 10 27 m 3

So the density is:

⫽ M/V

⫽ 2 ⫻ 10 30 /1.4 ⫻ 10 27 kg m ⫺3

⫽ 1428 kg m ⫺3

This is about 40% greater than that of water and about 26% that of the Earth.

2.2.2 The Sun’s total energy output

It might surprise you that any of you could almost certainly derive a value of the Sun’s energy output good to a factor of 10 simply by knowing the distance of the Sun and using your own experience!

Let us assume one has a value, e, for the Sun’s energy falling on 1 m 2 of the Earth’s atmosphere. This is just 1 m 2 of a spherical surface centred on the Sun which has an area of:

A ⫽ 4π(1.5 ⫻ 10 11 ) 2 m 2

where 1.5 ⫻ 10 11 m is the distance of the Earth from the Sun. As all of the Sun’s energy must pass through this surface, the Sun’s total energy output is simply A ⫻ e.

What is the value of e? First imagine that you were standing in a desert directly under the Sun in a cloudless sky. Then instead, imagine standing in a room under

a 100 W bulb, a 1000 W infrared heat lamp or a 10 kW arc lamp. Which might be comparable? I suspect that you would pick the heat lamp. So a ‘guestimate’ of the power from the Sun falling on 1 m 2 might well be 1000 W. This would actually be quite close as the measured value is 1370 W. The Sun’s total energy output is thus:

E ⫽ 1370 ⫻ 4π(1.5 ⫻ 10 11 ) 2 W ⫽ 3.86 ⫻ 10 26 W

46 Introduction to Astronomy and Cosmology

2.2.3 Black body radiation and the sun’s surface temperature

In order to estimate the surface temperature of the Sun it is necessary to know about the concept of black body radiation which arose out of quantum mechanics in the early years of the last century. A black body , a term introduced by Gustav Kirchhoff in 1860, is an object that absorbs all electromagnetic radiation that falls onto it – it neither refl ects any radiation nor allows any radiation to be trans- mitted through it. One way of approximating a black body is to make a cavity with a very small aperture whose interior surface is matt black. Any radiation that enters the aperture will almost certainly be absorbed within the cavity. If the cavity were heated to some temperature then the small aperture will emit

electromagnetic radiation and this is said to be black body radiation. Another name for this radiation is, not surprisingly, cavity radiation . As the radiation inside the cavity will be in thermal equilibrium with its walls, this will be a source of thermal radiation.

Thermal radiation has the property that both the emitted power and the wavelength at which the electromagnetic radiation is a maximum are directly related to its effective temperature; in the case of a cavity, this is its internal temperature. The radiation is then said to have a black body spectrum (Figure 2.3). Below about 700 K (430°C) black bodies produce very little radiation at visible wavelengths and appear black to our eyes – though we could sense the infrared radiation emitted at temperatures somewhat below this. Above this temperature

Our Solar System 1 – The Sun

black bodies will emit enough radiation at visible wavelengths for us to see a colour which passes through red, orange, yellow, and white to blue as the temperature increases. As the temperature increases further the peak wavelength moves into the ultraviolet, but there is still considerable energy in the blue part of the spec- trum so that the object appears blue.

The problem of calculating the form of the spectral curve for black body radiation was fi nally solved in 1901 by Max Planck and is known as Planck’s law of black body radiation. Planck had found a mathematical formula that fi tted the experimental data, but to fi nd a physical interpretation for this formula he had to invoke the principle of quantization: ‘photons’ within the cavity could only have certain allowed energies. In 1905, Einstein proposed that electromagnetic radiation was quantized in order to explain the photoelectric effect and was later awarded the Nobel Prize in Physics for this insight. There are two laws which relate to electromagnetic radiation which follows a black body spectrum.

Wien’s Law

The fi rst law is Wien’s displacement law which states that the wavelength at which the spectrum peaks is inversely proportional to the surface temperature. (In other words, the greater the temperature, the shorter the peak wavelength.) The peak wavelength, λ max , is thus given by a constant, Wien’s constant, divided by the

temperature. It value is 2.897 ⫻ 10 ⫺3 m K so with λ in metres and T in kelvin:

max ⫺3 ⫽ 2.897 ⫻ 10 /T

or, with λ in nanometres:

max ⫽ 2.897 ⫻ 10 /T

Wien’s Law enables the temperature of inaccessible objects, such as a lava fl ow or the interior of a blast furnace (an excellent black body as it really is a cavity), to be measured and, as we will see, allows us to estimate the surface temperature of the Sun.

Stefan–Boltzmann Law

The second law relating to black body radiation is called the Stefan–Boltzmann Law, which is often known simply as Stefan’s Law. It was discovered experimentally by Jožef Stefan in 1879 and derived theoretically from thermodynamic principles

48 Introduction to Astronomy and Cosmology

by Ludwig Boltzmann in 1884. The Law states that the total energy radiated per unit surface area of a black body per unit time is directly proportional to the fourth power of the black body’s absolute temperature. The constant of

proportionality, σ, is called the Stefan–Boltzmann constant or Stefan’s constant, and has the value:

σ ⫽ 5.67 ⫻ 10 ⫺8 Wm ⫺2 K ⫺4

Thus the emitted power across the electromagnetic spectrum of a body of surface area A and temperature T is given by:

E ⫽ σAT 4 W

Real objects do not, in general, act as perfect black bodies but in the majority of astronomical applications we fi rst assume that the object (such as a star or a planet) does act as a black body and then make a suitable correction if

necessary. An interesting astronomical result that came out of the radiation laws was related to the planet Mercury. It was long thought that Mercury’s rotation period was the same at its orbital period of 88 days. If this was the case, one face would

be locked to the Sun (like the Moon to the Earth) and would thus be very hot, but the face away from the Sun would be very cold – close to absolute zero. However, observations made at wavelengths of 11.3 and 1.9 cm in the radio part of the spectrum indicated that the supposed dark side surface temperature must be in the region of 250 K. This observation was later confi rmed when radar observations showed that Mercury’s rotation period was not 87.97 days, but 68.65 days, so all parts of the surface do, at some time, face the Sun.

Both the peak wavelength and total power output of a black body are related to its surface temperature. Assuming that the Sun’s visible surface acts as a black body, this then gives us two ways of estimating its surface temperature.

The peak of the visible spectrum is at a wavelength of ∼500 nm (0.5 ⫻ 10 ⫺6 m). Using Wien’s displacement law, temperature is given by:

T ⫽ 2.9 ⫻ 10 6 /λ peak K (where λ peak is in nanometres) ⫽ 2.9 ⫻ 10 6 /500 K

⫽ 5800 K

Using the total solar power output and the radius of the Sun (from the diameter as determined above) we can also use the Stefan–Boltzmann Law:

Our Solar System 1 – The Sun

E ⫽ σAT 4 ⫽ 5.671 ⫻ 10 ⫺8 ⫻ 4 ⫻ π ⫻ (6.95 ⫻ 10 8 ) 2 ⫻T 4 So

These agree quite well, but are a little higher than the accepted value of 5780 K.

So, in summary, the main properties of the Sun are: Diameter ⫽ 1391978 km

Mass ⫽ 2 ⫻ 10 30 kg Density ⫽ 1400 kgm ⫺3 Luminosity ⫽ 3.86 ⫻ 10 26 W Surface Temperature ⫽ 5780 K

2.2.4 The Fraunhofer lines in the solar spectrum and the composition of the sun

In 1666, Isaac Newton, using a prism, showed that sunlight is composed of all the colours of the spectrum and in 1804, William Wollaston observed that there appeared to be some gaps in the spectrum that looked like dark lines. Later, in 1911, Joseph Fraunhofer mapped many of these lines with reasonable accuracy. They have thus become known as Fraunhofer lines (Figure 2.4) . They represent wavelengths where there is a lessening of the observed solar emission and are thus called absorption lines. Later, Gustav Kirchoff and Robert Bunsen found that the wavelengths of the absorption lines seen in the Sun corresponded to those of the emission lines observed when the atoms of a particular element are excited. This can be achieved by sprinkling a compound of the element into a Bunsen burner fl ame, when, for example, salt gives an orange colour due to a close pair of emission lines called the sodium D lines.

Figure 2.4 The solar spectrum showing the Fraunhofer lines. The peak intensity is in the yellow part of the spectrum close to the strong pair of sodium D lines in the centre of the spectrum.

50 Introduction to Astronomy and Cosmology

Before long, Fraunhofer lines corresponding to all the known elements had been found in the Sun’s spectrum except for one set of lines. It was realised that there must be an element in the Sun’s atmosphere that had not then been discovered on Earth. It was thus called helium after ‘Helios’ the Greek name for the Sun. How are these lines formed? The photosphere will emit a continuous (almost black body) spectrum. The photons will then pass through the Sun’s upper atmosphere, the chromosphere, where atoms can absorb photons that correspond to transi- tions between their energy levels. Thus the lines, called absorption lines, will be at just the same wavelengths as the emission lines that we can observe on Earth.

This is where many books stop, but it cannot be quite as simple as this. Within a short while all the atoms would be in their upper energy states and the absorption would stop! In a steady state situation, the atoms must eventually return to their original states and may do so by emitting the very same wavelengths as had been absorbed – which would imply that there would be no net absorption! The atoms will then emit in random directions so only a small percentage of the re-radiated emission from the line of sight atoms will come in our direction. However, atoms over the whole face of the Sun will also be emitting their photons in random

directions. Their emission towards us will partly balance the emission that we do not receive from the atoms in our line of sight. There is a second mechanism, called collisional de-excitation, which can prevent re-emission. Before the atoms have a chance to re-emit a photon they may interact with another atom in the atmosphere and the excitation energy is turned into kinetic energy of the atoms, so heating up the Sun’s atmosphere.

As we will see in later chapters, observations of the lines in the spectra of stars and galaxies play a critical role in our understanding of both our Galaxy and the Universe. From the analysis of the solar spectrum it is possible to estimate the composition of the majority of the Sun’s interior as the outer layers are ‘mixed’ by convec- tive currents as will be described later. The composition is about 71% by mass of

hydrogen (91.2% in number of atoms), 27.1% by mass of helium (8.7% in number of atoms), oxygen makes up 0.97% (0.078% in number of atoms), and carbon 0.40% (0.043% in number of atoms). The small remainder then comprises all the other atoms detected in the Sun’s spectrum. In the core there will be less hydrogen and more helium due to the nuclear fusion processes to be described below.