Mass-radius relationship is derived by applying the Virial theorem by combining the total translational kinetic energy K and potential energy:
Equation 1: Virial theorem
We assume that each electron occupies a spherical space with radius r:
Equation 2: Radius that an electron occupies in a spherical space
Where A and Z are the mass number and atomic number respectively,and – Atomic mass unit and planetary mass respectively and is a planetary radius.Now we combine the total electrostatic energy and the total gravitational potential energy of the planet:
Equation 3: Total electrostatic energy
Where is electrostatic energy per electron, – a number of atoms in planet and is a constant:
Equation 4: Total gravitational potential energy
Where is a constant and G = 6.674×10?8 cm3?g?1?s?2 – gravitational constant.
Assuming each electron has its own spherical space and is not affected by electrostatic repulsion by other electrons, therefore using Virial theorem mass-radius relationship can be determined:
Equation 5: Virial theorem
Equation 6: Mass-radius relationship for planets with different chemical composition 1a
Where is a constant.
Pressure in planetary interiors
Now an analytical approximation of the pressure within the cores of planets can be devised. The interiors of planets are subject to extreme weight of various materials pressing down on the planet from above. To illustrate this, a pictorial representation of Earth’s interior:
Figure 1: Various layers of the Earth’s interior 3
We can see that there are multiple layers of the earth’s structure, each contributing to the pressure.
Referring to the figure below the pressure at the centre of a planet can be obtained:
Figure 2: A cross-section of a planet illustrating the methodology of deriving interior pressure of a planet 3
Assuming that the planet of radius Rp is perfectly spherical and an infinitesimal change in mass is dM, the associated pressure changes due to gravity is caused by the shaded area:
Equation 7: Expression for the pressure caused by the mass of the matter in shaded area
During this approximation, the density assumed does not vary with the radial distance from the core. Each layer within a planetary core is comprised of varying elements, therefore the planetary structure is inhomogeneous. This assumption can be made because the density of each layer does not vary drastically.
Equation 8: Pressure at the centre of a planet
Crystal Structures and Phase diagrams
The crystal structure is a periodic arrangement of atoms and molecules forming an ordered microscopic structure. Below is a pictorial representation of such atomic arrangement, such as crystalline solid:
Figure 3: Example of a crystal structure of a metal 4
Figure 3 depicts 3 structures various solids can obtain, e.g. Iron is an example of cubic body centred structure, silver – cubic face cantered. There are many factors that contribute to a specific structure such as valence electrons or bonding.
Phase diagrams arise from transitions of the four states of matter: liquid, solid, gas and plasma. This type of diagram illustrates how pressure and temperature affect the phases of mater and usually can be assigned a triple and a critical point. A general phase diagram depicted below: