Our book on the fundamentals of French grammar is now available in both printed and PDF formats. This is an excellent companion guide for beginning and intermediate students of the language. More details can be found here.
We continue our series looking at the physics of the Sun, and this month look at how the Sun’s energy is radiated away from the core (radiation opacity) and why the pressures within the Sun generated by the fusion reactions we looked at last month do not mean the star explodes (hydrostatic equilibrium).
Notre livre sur les bases de la grammaire française est maintenant disponible sous forme de livre imprimé ou en PDF. Ce livre s'adresse aux étudiants débutants et intermédiaires de la langue française. Vous pouvez trouver plus de détails ici.
Nous allons continuer à étudier la physique du soleil. Ce mois nous allons regarder comment l'énergie solaire rayonne du noyau et pourquoi le soleil n'explose pas malgré la présence de la pression interne (l'équilibre hydrostatique).
A star such as the Sun generates energy at its core through nuclear fusion reactions, and we described the principal reactions in last month’s blog. These create thermal excitation and energy and result in an increase in pressure. The higher the temperature and density of the core the higher the pressure. This pressure, or force, acts in all directions, including outwards, and if unchecked, would lead to the star expanding. The counter balance is the star’s gravity which acts towards the centre of the star.
There is clearly a pressure gradient operating. At the centre of the star the pressure is high; but this is balanced by the ‘weight’ of matter which is gravitationally acting inwards, i.e. the whole of the star’s mass. If the pressure was not sufficiently high enough then the star’s core would continue to collapse under gravitational attraction/contraction. Near the edge of the star there is only a very small proportion of the star’s mass farther out, and so the inward acting gravitational force is much smaller. However, the pressure is also much lower; if it were not the material would expand outwards.
This pressure gradient can be mathematically modelled by what is called the hydrostatic equation which relates the gravitational force, the radiation pressure and density at specific depths within the star. If the star is in a stable state then the forces balance and there would be no changing of the star’s size, the star is said to be in an equilibrium state. In this case the hydrostatic equation becomes the hydrostatic equilibrium equation. The Sun is in hydrostatic equilibrium but we will see later in this series that some stars (such as the Cepheid variable class of stars) are not, and radial pulsation results.
In the equations r is radial distance from centre of star, P is pressure, M is mass, p is density and G the constant of universal gravitation. Subscripts r refer to values at or inward of radial distance.
The fusion energy generation processes produce high energy gamma wavelength radiation. This radiation is emitted in all directions and the time it takes to be radiated away from the core depends upon the star’s opacity. Opacity, or its opposite ‘transparency’, in this context is the degree to which radiation (photons) can flow before being altered in direction or being absorbed and subsequently re-emitted. A star’s opacity is related to the mean free path of photons within the star.
The mean free path (lph) is the average distance which a photon (a quantum of radiation) will travel before interacting with an atomic particle and being absorbed or changed in direction. It can be used to determine the diffusion time of radiation from the centre of the star and is dependent upon the density of interacting particles. So, for example, no atomic particles interact with neutrinos and thus they have (very!) long mean free paths and hence very rapid (at the speed of light) diffusion from the core following production. Electrons and photons interact very readily with matter within the core and throughout their emittance and thus their mean paths are very short.
In a high density, high temperature, highly ionised plasma such as the conditions at and near the centre of the Sun, the photon mean free path is very low. This is because there is a large number of free electrons and both scattering (deflection by electrons) and atomic absorption (primarily by elements heavier than hydrogen) are prevalent. In lower temperature and partly/non-ionised gas, photon scattering and atomic absorption still occurs, but as the nett effect of these is reduced the mean-free path (the star’s opacity) is commensurately lower. (For further study, the atomic processes are Thomson, Rayleigh and Compton scattering, together with bound-bound and bound-free absorption).
The mean free path and the density of the matter which the photon is transiting are related by the relationship: (lph) = 1 /kp where k is opacity (measured in m2 / kg) and p is density (of interact-able particles). The opacity (and thus the mean free path) is also wavelength/frequency dependent but this aspect is for the reader’s further study rather than here! On average, for the Sun as a whole, has a value of about 4mm. Photon energy produced within the core will take of the order of 10 to 20 thousand years to ‘escape’.
So, if we ‘personalise’ this to a particular photon, our radiation photon looking to escape from the Sun has a hard time through being battered (scattering) and absorbed then re-emitted (assimilated then subsequent escape). As our photon slowly but surely gets scattered and travels further away from the core, its path gets incrementally easier until eventually it escapes from the solar photosphere. Travelling at the speed of light it will then, so long as its direction of travel is towards us takes just a further 8 minutes 19 seconds to travel the 150 million km to arrive at the Earth. And gives us the sunshine we all rather appreciate!
We will see how the radiation and convection processes interact within the sun, how convection becomes dominant in outer layers of the Sun and produces the easily observable Solar granulation
More details on this topic at undergraduate level
The Sun – Shining light on the Solar System
Neil Taylor. Observatoire Solaire
An excellent graduate level description of the processes, and more, we have mentioned here
Hale Bradt. Cambridge University Press.
A highly recommended advanced graduate/post-graduate level exposition.
Atomic Astrophysics and Spectroscopy
Anil Pradham and Sultana Nahar. Cambridge University Press.