Two small identical uniform spheres of density \(\rho\) and radius \(r\) are orbiting the Earth in a circular orbit of radius \(a\). Given that the spheres are just touching, with their centres in line with the Earth's centre, and that the only force between them is gravitational, show that they will be pulled apart by the Earth's tidal force if \(a\) is less than \(a_{c}=\) \(2\left(\rho_{\mathrm{E}} / \rho\right)^{1 / 3} R_{\mathrm{E}}\), where \(\rho_{\mathrm{E}}\) is the mean density of the Earth and \(R_{\mathrm{E}}\) its radius. (This is an illustration of the existence of the Roche limit, within which small planetoids would be torn apart by tidal forces. The actual limit is larger than the one found here, because the spheres themselves would be distorted by the tidal force, thus enhancing the effect. It is \(a_{\mathrm{c}}=2.45\left(\rho_{\mathrm{E}} / \rho\right)^{1 / 3} R_{\mathrm{E}} .\) For the mean density of the Moon, for example, \(\rho=3.34 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\), this gives \(\left.a_{\mathrm{c}}=2.89 R_{\mathrm{E}} \cdot\right)\)

The concept of tidal forces is pivotal to astronomy and the behavior of celestial bodies within gravitational fields. Tidal forces arise when the gravitational pull exerted by one body on another varies across its diameter. This differential force can stretch the object or create a gradient of gravitational attraction across it.

To visualize this phenomenon, let's consider our exercise that involves two spheres in a circular orbit around the Earth. As they orbit, the sphere nearest to Earth feels a stronger gravitational pull than the one farther away. This difference in force, though it may be small, can have substantial effects over time, especially when the objects are within a certain critical distance from the larger mass—the Roche limit.

Practical Example of Tidal Forces at Play

• The best-known tidal force in our daily life is the one that causes the Earth's oceans to rise and fall, creating the tides as the Moon exerts its influence upon Earth's waters.
• In our exercise, if the distance between the Earth and the two small spheres is less than a certain critical value, the tidal force will overcome the gravitational force holding the spheres together, leading to them being pulled apart.

By comprehending the delicate balance of the forces at work, one gains a deepened appreciation for how tidal forces can sculpt our universe, influencing phenomena such as the formation of rings around planets and the disintegration of bodies within celestial neighborhoods.

Gravitational force is a fundamental interaction that governs the motions of objects in the universe. From apples falling from trees to the complex orbits of planets, gravity is the invisible force that pulls objects with mass towards one another.

In the context of our exercise, understanding gravitational force helps us to comprehend how celestial bodies remain in orbit. The key equation governing gravitational attraction is Newton's law of universal gravitation, which states that the force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Key Formula for Gravitational Force

The formula provided in the exercise solution, \(F_g = G\frac{m_1m_2}{d^2}\), embodies this principle and allows us to calculate the exact force that binds the two small spheres together. It's this same force that must be overcome by tidal forces for the spheres to be pulled apart as they orbit Earth. Gravitational force not only applies to celestial bodies but also affects everyday life, influencing everything from the trajectory of a thrown ball to the structure of the universe itself.

A circular orbit is a special case of orbital motion where an object moves around another object in a path that describes a circle. This happens when the gravitational force providing the centripetal force needed to keep the object moving in its path exactly balances the inertia of the object trying to move in a straight line at a constant speed.

In our exercise, we assume the identical uniform spheres to be in a circular orbit around Earth. This assumption simplifies the mathematical treatment and helps focus on the effects of tidal forces. Nevertheless, the complexities of celestial mechanics often result in orbits that are elliptical rather than perfectly circular.

Characteristics of Circular Orbits

• Objects in circular orbits move with a constant speed, as the centripetal force required for the motion is constant.
• The force keeping the object in orbit is provided by gravity, which acts as the centripetal force.
• Any disturbance in the balance of forces, like an increase in tidal forces beyond a critical limit, can cause the orbiting object to spiral inwards or outwards, changing its orbit from circular to elliptical, or even leading to its disintegration.

Studying circular orbits offers insight into the conditions for orbital stability and helps us untangle how moons, satellites, and space debris behave around larger celestial bodies like Earth.