Fundamental Principles: Radioactive Decay, Planetary Motion, and Material Properties

Radioactive Decay: Principles and Equations

If the decay rate is equal to λ, the probability that a given nucleus will decay in a small time interval dt is λdt. Therefore, if at any time we have P parent nuclei, the number that decay in the next moment is P(λdt).

The rate of change of parent nuclei (dP/dt) is proportional to the number of parent nuclei present:

dP/dt = -λP

This differential equation can be solved as follows:

• Separate variables: dP/P = -λdt
• Integrate both sides: ∫(dP/P) = -λ∫dt
• Resulting in: ln(P) = -λt + C

The boundary condition is found because we know that when t = 0, P = P₀ (the initial number of parent nuclei). Therefore, C = ln(P₀), and we can write the equation as:

ln(P) = ln(P₀) - λt

Rearranging this equation:

• ln(P) - ln(P₀) = -λt
• ln(P/P₀) = -λt
• Exponentiating both sides: P/P₀ = e^-λt
• Thus, the number of parent nuclei at time t is: P = P₀e^-λt

As parent nuclides diminish, the number of daughter nuclides (D) increases. By definition, D is the difference between the initial parent nuclei (P₀) and the current parent nuclei (P):

D = P₀ - P = P₀(1 - e^-λt)

If P₀ is unknown, we can reformulate this by eliminating P₀. From P = P₀e^-λt, we have P₀ = P/e^-λt = Pe^λt. Substituting this into the equation for D:

D = P(e^λt - 1)

This equation can be solved with respect to time (t):

t = (1/λ) ln(D/P + 1)

Kepler's Laws of Planetary Motion

Johannes Kepler formulated three fundamental laws describing the motion of planets around the Sun:

1. Law of Ellipses: The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. Law of Equal Areas: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. Law of Harmonies: The square of the orbital period (T) of a planet is proportional to the cube of the semi-major axis (a) of its orbit (T² ∝ a³).

Key Concepts in Geophysics and Material Science

Seismic Velocity

Seismic velocity refers to how fast elastic waves travel through a material. Seismic velocities vary significantly in different rock types. Therefore, seismic surveys are more effective at distinguishing geological structures (boundaries where rock types change) rather than precisely determining specific rock types.

Bulk Modulus (K)

The Bulk Modulus (K, often denoted as Y in some contexts) is defined as the ratio of the change in pressure (dP) to the relative change in volume (dV/V). Mathematically, K = -dP / (dV/V). It generally indicates how well a material can resist compression.

Elasticity

Elasticity is a material property describing its ability to return to its original shape and size after the deforming force (stress) is removed. These changes are reversible.

Plasticity

Plasticity is a material property describing its tendency to deform permanently after a certain stress is applied. These changes are non-reversible.

Viscosity

Viscosity describes a material's resistance to slow deformation by shear or tensile stress. Viscous materials are often described as thick or sticky (e.g., honey). High viscosity implies significant friction between the molecules or particles within the fluid.

Strain

Strain is the relative change in the shape or size of a material or object due to external forces. Its SI unit is dimensionless because it represents a relative change.

Poisson's Ratio (ν)

Poisson's Ratio (ν) is the ratio between the decrease in width (transverse strain) and the increase in length (axial strain) when a material is stretched or compressed. More formally, it is the ratio of transverse strain to axial strain.

Isostasy

Isostasy is the gravitational equilibrium between the Earth's crust and mantle. It results in the crust "floating" at a certain elevation, much like an iceberg in water. This elevation depends on the thickness and density of the crust.