Show that, at radius \(r\) inside a uniform sphere of density \(\rho\), the radial force \(F_{\mathrm{r}}=-4 \pi G \rho r / 3\). If the density is zero for \(r>a\), show that $$ \Phi(r)=-2 \pi G \rho\left(a^{2}-\frac{r^{2}}{3}\right) \quad \text { for } r \leq a $$ so that the potential energy is related to the mass \(\mathcal{M}\) by $$ \mathcal{P E}=-\frac{16 \pi^{2}}{15} G \rho^{2} a^{5}=-\frac{3}{5} \frac{G \mathcal{M}^{2}}{a} . $$ Taking \(a=R_{\odot}\), the solar radius, and the mass \(\mathcal{M}=\mathcal{M}_{\odot}\), show that \(\mathcal{P E} \sim L_{\odot} \times 10^{7} \mathrm{yr}\); approximately this much energy was set free as the Sun contracted from a diffuse cloud of gas to its present size. Since the Earth is about 4.5 Gyr old, and the Sun has been shining for at least this long, it clearly has another energy source - nuclear fusion.

Gravitational force is a key concept in understanding celestial mechanics and the interactions between masses. According to Newton's law of gravitation, any two masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is given by:

\[ F = G \frac{M_1 M_2}{r^2} \] where

• \( F \) is the gravitational force,
• \( G \) is the gravitational constant,
• \( M_1 \) and \( M_2 \) are the masses,
• and \( r \) is the distance between the centers of the two masses.

For a sphere of uniform density, the force can be computed by integrating the contributions of each infinitesimal ring (or slice) of mass within the sphere. This leads to the radial force expression for a uniform sphere:

\[ F_r = -\frac{4 \pi G \rho r}{3} \] Here, \( \rho \) represents the density of the sphere, and \( r \) is the radial distance from the center.

In physics, a uniform sphere density refers to a sphere whose density, \( \rho \), is constant throughout its volume. This simplification helps in solving problems related to gravitational force and potential within the sphere. For a sphere with uniform density, the mass contained within a radius \( r \) can be calculated using the volume of the sphere segment up to \( r \):

\[ M_r = \rho \cdot \frac{4}{3} \pi r^3 \] This results in the mass being directly proportional to the volume enclosed by the radius \( r \). When combined with Newton's law of gravitation, this formulation allows us to understand the behavior of gravitational forces inside a uniform sphere. Calculations involving uniform sphere density are critical in understanding celestial bodies like stars and planets, which can often be approximated as having uniform density for theoretical studies.

Newton's law of gravitation states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This fundamental law is formulated as:

\[ F = G \frac{M_1 M_2}{r^2} \] where

• \( F \) is the gravitational force,
• \( G \) is the universal gravitational constant,
• \( M_1 \) and \( M_2 \) are the masses,
• and \( r \) is the distance between the centers of the two masses.

This inverse-square law reveals that the gravitational attraction diminishes with the square of the distance. Newton's formulation allows for the calculation of gravitational forces and potential energies and is essential for understanding the dynamics of planets, stars, and galaxies. This principle is pivotal in understanding both terrestrial and astronomical phenomena.

Solar contraction energy is the energy released as a gas cloud in space, such as the Sun, contracts under its gravity to form a more compact object. When the Sun formed, it started as a diffuse cloud of gas. As this cloud contracted under its own gravitational force, potential energy was converted into thermal energy. The gravitational potential energy available from such contraction can be estimated using the following formula:

\[ \mathcal{PE} = -\frac{3}{5} \frac{G \mathcal{M}^2}{a} \] where

• \( \mathcal{PE} \) is the potential energy,
• \( G \) is the gravitational constant,
• \( \mathcal{M} \) is the mass of the Sun,
• and \( a \) is the radius of the Sun.

Analysis of solar contraction energy helps explain how stars radiate light and energy during their formation. This process is essential for understanding stellar evolution and the transition phases that stars, including our Sun, go through during their lifespans.

Nuclear fusion in stars is the process by which stars generate energy. In the cores of stars, hydrogen nuclei fuse together to form helium and release a tremendous amount of energy. This energy generation process offsets the gravitational forces attempting to collapse the star further. The primary reactions happening within the Sun can be summarized by the proton-proton chain:

\[ 4 \,_1^1\text{H} \rightarrow \;^4_2\text{He} + 2 e^+ + 2 u_e + 26.7 \text{ MeV}\] This process transforms four hydrogen nuclei into one helium nucleus, releasing positrons, electron neutrinos, and energy. The energy produced in the core of the star travels outward and provides the heat and light that we observe. Understanding nuclear fusion is crucial because it explains the energy source of stars, including our Sun, and it accounts for their long lifespans, much longer than what could be explained by gravitational contraction alone.