R= 1.503 x 1011 m
Therefore: F= 7.929 x 1044 N Mass M of the Sun 1. 9891 x 1030 kg
Mass m of the Earth 5.9742 x 1024 kg
Gravitational Constant 6.6720 x 10-11 N m2 kg-2
This is a considerable amount of force! We can test the units of G by rearranging Eq .1 , to make G the subject:
G= F R2 N x m2
= = Nm 2 kg-2
Mm kg x kg
The gravitational attraction of the Sun keeps the Earth in its elliptical orbit and is inversely proportional to the square of the distances between their centres and directly proportional to their masses. This supplied the centripetal force, the Sun needed to supply in order to keep the Earth in its orbit.
The Structure of the Sun
The Sun can be divided into a number of concentric layers or shells. The visible surface I the Photosphere, a shallow shell of gas some 400 km thick from which a tiny fraction of the Sun’s light is emitted.
The solar radius measured from the centre of the Sun to the Photosphere is
696,265 km. The Sun shines as a result of nuclear reactions, releasing vast quantities of energy, which take place in the hot, dense central core, a region that extends to about 174, 066km from the centre. The central temperature is about 15 x 106 K and is sufficient to start nuclear reactions. Material outside the core is at lower temperatures, pressures and densities and nuclear reactions do not take place here at significant rates.
Energy is released in the core primarily in the form of γ- ray and X- ray radiation or high energy, short wavelength “photons”. These photons cannot travel far before being absorbed, they are re-emitted a multitude of times, and their energies decline as they move out through the solar globe. Their paths take the form of a “ random walk”, in which the direction and energy change at each absorption and emission. Photons are progressively converted from X –ray and γ- ray radiation into extreme ultraviolet and ultraviolet, finally emerging from the photosphere as visible light.
Solar Physicists have built up a convincing model of the interior of the Sun and the mechanism by which it produces its energy. In this “standard” model, the density, temperature and pressure increase sharply towards the centre of the Sun. The intensely high temperature ensures that the central matter is kept fully ionised in gaseous state, comprised mainly of protons and electrons moving about at very high velocities.
We know that the Sun has been around for a very long period of time: fossil life forms and other geological records on Earth and the Moon indicate the Sun has been shining, with about the same power as it does now, since the formation of the planetary system. Therefore the Sun is said to be in a state of equilibrium.
A state of Hydrostatic equilibrium exists, in which the Sun’s tendency to collapse under its own gravitational self-attraction is balanced by the outward pressure of the hot gas inside. Radiative equilibrium (Thermal equilibrium) on the other hand is a consequence of the general principle of conservation of energy.
If the Sun’s luminosity is 3.86 x 1026 watts, this is equivalent to it emitting
3.86 x 1026 Joules of electromagnetic energy per second.
Luminosity = Energy emitted = Joules = J s-1 = units of Power
Time taken second ( watts)
This is the same as 100 million megaton hydrogen bombs exploding each second!
It is evident that the Sun’s luminosity has not changed over the years, therefore must have an internal source that generates that much energy per second by converting other forms of energy to electromagnetic radiation.
What can be producing so much energy for so long?
It was assumed that originally the whole of the Sun had the same chemical composition as is now evident in it outer layers, with proportions (by mass) of 78% Hydrogen, 20 % Helium and 2 % other elements (principally Carbon, Nitrogen and Oxygen), and that the interior has been modified by thermonuclear reactions so that now the fraction of Hydrogen in the core is 36 %, about half the value at the surface.
The proton –proton chain
The reaction believed to be principally responsible for the production of energy in the solar core is the “proton –proton” reaction
In a series of stages:
∙ Four hydrogen nuclei ( protons ) are fused together to form one Helium nucleus, comprising two protons and two neutrons.
∙ The Helium-4, once formed, is stable because the temperature of the core is too low for the next stage of thermonuclear reaction, involving carbon, to take place.
∙ Energy is released in the core primarily in the form of γ- ray and X- ray radiation or high energy, short wavelength photons.
(See Fig 2)
Fig. 2
Hydrogen nuclei (protons) are converted to give Helium (42 He an atom containing two neutrons and two protons). The simplest route involves only three steps (A-B-C)
Two protons combine to give a deuterium nucleus (a proton and a neutron), which emits a positron and a neutrino (A); the deuterium then combines with a proton yielding a nucleus of helium-3 which emits a photon (B); finally the Helium-3 nucleus Formed in the same way), producing a nucleus of ordinary Helium-4 plus two extra protons (C). However, several variations in this chain are possible (A-B-D-E-F, or A-B-D-H-I). In these rarer alternative routes, involving Beryllium-7, Lithium-7 and Boron-8, neutrinos with different energies are emitted. The neutrinos given off by Boron-8 (H) should be detectable by the Brookhaven experiment.
The mass of the end product is 0.7 % less than the components that went to assemble it: this small percentage loss of mass is converted into energy, as we find out from the relationship between mass (m) and energy (E) derived from the well-known Special Theory of Relativity.
Einstein’s Special Theory of Relativity
According to the special theory of relativity, the work done on a body increases the kinetic energy of that body, not only by increasing the speed of the body but also by increasing its mass. This suggests that mass and energy are not entirely independent of each other. In fact, the special theory of relativity requires that mass and energy are inter-convertible at a ‘ rate of exchange’ given by the expression:
E = m c 2 where c = speed of light in a vacuum
2.9979 x 108 ms-1
Since c is a very large number, a very large amount of energy can be released by the destruction of small quantities of matter. In order to sustain the present luminosity of the Sun just over 3.86 x 1026 tonnes of matter must be converted into energy every second by this process. This theory can be applied to concepts of momentum, recoil and kinetic energy and confirms the law of conservation of energy: that mass and energy are each conserved.
The Brookhaven experiment
The first step in the proton-proton reaction releases neutrinos. Neutrinos are a common type of “particle” (symbol ν) with a zero electrical charge and zero rest mass. Neutrinos travel at the speed of light and very rarely interact with matter, passing through solid objects almost as if they were transparent. Thus neutrinos created at the Sun’s core are able to emerge from the surface in a fraction of a second. The neutrinos produced in the p-p reaction turn out to be rather low energy, usually less than 0.42 MeV. In the comparatively rare alternative routes of the reaction chain higher energy neutrinos are released.
The rate of their production depends sensitively on the internal temperature of the solar core, so that, if it were possible to measure the flux of neutrinos emerging from the Sun, this would provide a measure of internal temperature. However, since neutrinos can penetrate the Sun so easily, it is evident that they are externally difficult to capture and measure.
Nevertheless, in 1964 a neutrino “telescope” was established by the Brookhaven National Laboratory capable o detecting solar neutrinos. The instrument consisted of some 400, 000, litres of perchloroethylene (C2 Cl4) - dry – cleaning fluid-containing large amounts of Chlorine in a tank located at the bottom of a mine in South Dakota, at a depth of some 1, 500 m. The isotope Chlorine – 37, which comprises about 25 % of the Chlorine content of the dry – cleaning fluid, is effective at capturing the energetic neutrinos. In the process it is converted into radioactive Argon (and an electron is given off). The radioactive Argon is flushed from the tank by means of Helium, and collected in a charcoal trap cooled to the temperature of liquid Nitrogen (77 K). The decay of the Argon is registered on the particle detectors and the number of Argon atoms is measured. Thus the flux of neutrinos entering the detector can be determined.
The chance of an individual Chlorine-37 atom capturing a neutrino is very tiny indeed: if standard theory is correct, a given Chlorine –37 atom in the detector should capture a neutrino once in about 1028 years – far in excess the estimated age of the Universe. However, the container holds a very large number of atoms. Neutrino astronomers have invented a unit of measurement known as the Solar Neutrino Unit (SNU), where 1 SNU = 10-36 neutrino captures per second per target atom.
According to the standard model, the detector should measure between 6 and 7 SNU, equivalent to about 1 neutrino capture a day. In the fifteen or so years that the experiment has been operating it has failed to detect anything like the expected number of neutrinos. The measured flux is 2.2 SNU, about one third of the predicted value.
If the temperature of the Sun’ s core were abut 10% lower than implied by the standard model it would explain the low flux of neutrinos; on the other hand, a lower core temperature would reduce the Sun’s luminosity to well below the observed value.
The life expectancy of the Sun
The life expectancy of a luminous object such as he Sun can be thought of as the length of time it can continue to emit light with a certain luminosity before it runs out of fuel. Stars consume fuel at a constant rate:
Therefore: t = amount of fuel
Rate of fuel consumption
The amount of fuel is proportional to the mass M of the Sun (1. 9891 x 1030 kg) which is proportional to E : the total energy content of the Sun (the total amount of energy that can be extracted from its fuel supply which is 1 x 1047 Joules). The rate at which the fuel “burns” is proportional to the luminosity L. The life expectancy tx is just the rate of energy emitted and can be given by:
t x = E = 1 x 1047 Joules = 8.214 x 1012 years
L 3.86 x 1026 watts
If all the Sun’s mass were converted into energy there would be about 1047 Joules available, over a 100 times more than enough to keep the Sun burning for 5 to 10 billion years! In fact if less than 1 % of the Sun’s mass were converted into energy, it would suffice to keep it burning for billions of years. We can conclude that the conversion of mass to energy (Einstein’s Theory of Special Relativity) is the energy source of the Sun.
The Future of our Sun
In 1979 evidence was put forward indicating that the Sun was shrinking. From an analysis of the Solar diameter measurements made at the Royal Observatory, Greenwich, over a period of about 120 years from 1836 to 1954, it was suggested that the diameter was decreasing by about 0.1 % per century, as a result of the periodic expansion and contraction of the Sun.
Our Sun is expected to become a red giant in five billion years or so, the hydrogen inside it will get depleted, and the outer region will swell up and redden. The star will become much bigger than its present size. As the Sun will age it will become much smaller and smaller, and may begin to pulsate, that is, it will alternately become bigger and smaller.
At this stage the mass of the Sun becomes less than about three times the present size. The temperature reached in the core after the completion of helium burning will not be high enough to ignite any further thermonuclear reactions. The Sun will, however become unstable and shed the outer layers of gas, which make up approximately half its mass. The core of the Sun will continue to collapse into a smaller and smaller space, and become correspondingly more dense, until the electrons (which are responsible for most of the volume of matter) are packed closely enough together to generate an effect known as Fermi pressure.
This pressure then prevents any further collapse. The eventual result will be a very hot star that is very small, typically about one percent of the diameter of the present size of the Sun, and which has a very high density, typically one tonne cm- 3. Here the Sun is said to be at white dwarf stage. White dwarfs are not very bright stars and are difficult to see, even using large telescopes. They gradually cool, becoming progressively dimmer and are classed as variable stars.
There is, however an upper limit to the size of a white dwarf. In 1930, Subramanyan Chandrasekhar calculated that if a white dwarf has a mass greater than 1.44 times the present mass of the Sun – a mass now known as the Chandrasekhar limit – even the Fermi pressure of electrons will not be able to support the immense gravitational pressure exerted on the core. The electrons will combine with protons in the nuclei of atoms – producing neutrons – and the collapse will then continue until these neutrons are themselves tightly packed. This final collapse of the Sun will be very rapid- taking less than a second- and will be accompanied by an externally rapid rise in temperature.
A red giant that is massive enough to collapse beyond the stage of a white dwarf does not do so immediately after the completion of its helium burning. One or more of a series of further thermonuclear reactions first occur, each of which produces equilibrium and prevents further collapse until its fuel is exhausted. Each successive reaction is ignited however only if gravitational collapse of the core on completion of the previous thermonuclear reaction produces a sufficiently large increase in temperature.
When the fuel for the Sun’s final thermonuclear reaction is exhausted, it’s collapsing core exceeds Chandrasekhar limit. The collapse will, therefore, continue until the neutrons of which the core is now exclusively made are themselves compressed as tightly as they will go. The combined effect of the shock wave which is produced when the very rapid final collapse is suddenly brought to a halt and the intense radiation pressure from the immensely hot core, then causes the Sun to explode. The result is known as a supernova and may, for a few days, emit as much radiation as a complete galaxy of stable stars before becoming a rather less prominent nebula.
The problem of understanding the life cycle of the Sun involves the more general problem of the evolution of stars. If the temperatures of a large number of stars are plotted against their luminosities several interesting features emerge in the resulting Hertzsprung Russell diagram (see Fig 3).
The Sun belongs to the “Main Sequence” group, which lies in a narrow band sloping from the upper left of the graph to the lower right: in other words, the hotter they are the brighter they are. There is a cluster of relatively cool, red stars in the top right quarter of the Hertzsprung Russell diagram whose absolute magnitudes indicate that they are very bright. Since a cool star radiates energy at a relatively low rate per unit area, these stars must be very large. They have diameters in the range 10 – 100 times greater than the Sun and are know as red giants.
The absolute magnitude of these stars, as shown in the bottom left of the H-R diagram, indicates that they are rather dim. Since a hot star radiates energy at a relatively high rate per unit area, these stars must be very small. They reach about one-thirtieth the diameter of the present size of the Sun.
“Red Giants” are the cooler stars, very much larger than the main sequence stars, and although they have the same temperature, they are more luminous. “White Dwarfs” are very hot but small stars, therefore their luminosities are very low. These three types of star represent different stages through which a star like the Sun will pass during the course of its life.
For a star like the Sun, the main sequence stage is thought to last for about 1010 years. Since the estimated age of the Sun is about 4.6 x 109 years, it should remain on the main sequence for a further 5 or 6 x 109 years.
Conclusion
Understanding of the solar interior is in a state of flux. No one seriously doubts that the Sun shines by means of thermonuclear reactions, but the precise mechanism is open to doubt. The Sun has been shown to be variable although only on a relatively minor scale, and its variability is sufficient to exert climatological effects on Earth. The comfortable view of a steady and unchanging Sun has been replaced by a slightly confused picture of a somewhat inconstant Sun. No-one can be confident that the mystery has been solved. Revolutionary changes of opinion may lie only just around the corner.
Bibliography
Active references:
1 Table of Physical and Chemical Constants and some Mathematical functions by G.W.C Kaye and T. H Laby (14th edition)
2 “Cosmology” Cambridge Modular Sciences (1995) by Bryan Miller ( University
Press)
3 Relativity and Quantum Physics (1995) by Roger Muncaster (Stanley Thorne
Publishers)
4 Solar System by Peter Cattermole (Fellow of the International Union of Planetologists)
Iain Nicolson (Senior Lecturer in Astronomy, Fellow of the Royal Astronomical
Society)
5 On line references:
http: // www. seds.org/nineplanets/nineplanets/ol.html
http: // www.nasa.gov/features/planets/sun/sun.html
6 Another source available online: www.space.com/scienceastronomy/solarsystem/sun-ez.html
7 http:/ www.solar views.com/eng/sun.html
8 Other consulted references:
http://
http:// system.com
“Our Wonderful Universe” By Dinesh Goswami ( National Book Trust, India)
Bibliography
Active references:
1 Table of Physical and Chemical Constants and some Mathematical functions by G.W.C Kaye and T. H Laby (14th edition)
2 “Cosmology” Cambridge Modular Sciences (1995) by Bryan Miller ( University
Press)
3 Relativity and Quantum Physics (1995) by Roger Muncaster (Stanley Thorne
Publishers)
4 Solar System by Peter Cattermole (Fellow of the International Union of Planetologists)
Iain Nicolson (Senior Lecturer in Astronomy, Fellow of the Royal Astronomical
Society)
5 On line references:
http: // www. seds.org/nineplanets/nineplanets/ol.html
http: // www.nasa.gov/features/planets/sun/sun.html
6 Another source available online: www.space.com/scienceastronomy/solarsystem/sun-ez.html
7 http:/ www.solar views.com/eng/sun.html
8 Other consulted references:
http://
http:// system.com
“Our Wonderful Universe” By Dinesh Goswami ( National Book Trust, India)