Gravitational fields

At the heart of celestial mechanics is a simple but profound idea from Isaac Newton: every object with mass pulls on every other object with mass. This pull is called the gravitational force. Newton's Law of Universal Gravitation states that this force is directly proportional to the product of the two masses (the more massive the objects, the stronger the pull) and inversely proportional to the square of the distance between their centers (the force gets weaker very quickly as the objects move apart). This is known as an 'inverse square law'. The force acts along the line connecting the centers of the two objects. For spherical objects like planets and stars, we can treat all their mass as being concentrated at a single point at their center.

A mass creates a gravitational field in the space around it. This field is a region where another mass will experience a gravitational force. We quantify the strength of this field using the concept of gravitational field strength, symbol 'g'. It is defined as the gravitational force experienced per unit mass at a particular point. So, g = F/m. Since force is a vector, field strength is also a vector, and its direction is the same as the direction of the force – always towards the mass creating the field. By combining this definition with Newton's Law of Gravitation, we can derive a formula for the field strength created by a point or spherical mass M: g = GM/r^2. The units of 'g' are N kg^-1, which are equivalent to m s^-2, meaning gravitational field strength is numerically equal to the acceleration of free fall at that point.

What happens when you are influenced by more than one gravitational field, for example, at a point between the Earth and the Moon? The Principle of Superposition provides the answer. It states that the total gravitational field strength at any point is the vector sum of the individual field strengths from all masses present. If the fields from two masses point in the same direction, you add their magnitudes. If they point in opposite directions (like at a point on the line between two planets), you subtract the smaller magnitude from the larger one. This principle allows us to find points in space where the net gravitational field is zero. Such a point is called a 'null point' or 'neutral point'.

To understand energy in a gravitational field, we must first define a zero point. In physics, we define the gravitational potential energy (GPE) of a mass as zero when it is at an infinite distance from the source mass. To bring the mass from infinity to a point 'r' in the field, the gravitational force does work on the mass. Because the force is attractive, the mass 'falls' towards the source, and its GPE decreases from zero. Therefore, the gravitational potential energy of any mass in a gravitational field is always negative. We also define a related quantity, gravitational potential (V_g), as the gravitational potential energy per unit mass (V_g = E_p / m). It represents the work done per unit mass to bring a test mass from infinity to that point. Like GPE, gravitational potential is also always negative and is a scalar quantity, which makes energy calculations much simpler than vector force calculations.

An equipotential surface is a surface where the gravitational potential is the same at every point. For a single spherical mass, these surfaces are concentric spheres. Since potential is constant everywhere on an equipotential, the change in potential (ΔV_g) for any movement along the surface is zero. As the work done is mΔV_g, this means no work is done when moving a mass along an equipotential surface. There is a direct link between field strength and potential: the gravitational field strength 'g' is equal to the negative of the potential gradient. This means g = -ΔV_g/Δr. In simpler terms, the field is strongest where the equipotential surfaces are closest together. A key property is that gravitational field lines are always perpendicular to equipotential surfaces and point in the direction of decreasing potential (i.e., towards more negative values).

For a satellite to maintain a stable circular orbit around a planet, it must have a specific velocity. The force keeping it in this circular path is the gravitational force from the planet. This gravitational force provides the necessary centripetal force. By equating the formula for gravitational force with the formula for centripetal force (F = mv^2/r), we can derive expressions for the satellite's orbital speed and its period (the time for one full orbit). The derivation shows that the orbital speed 'v' depends only on the mass of the central body 'M' and the orbital radius 'r', not the mass of the satellite 'm'. A consequence of this is Kepler's Third Law, which states that the square of the orbital period (T^2) is directly proportional to the cube of the orbital radius (r^3).

An orbiting satellite possesses both kinetic energy (KE) due to its motion and gravitational potential energy (GPE) due to its position in the field. The total energy is the sum of these two. By using the formulas for orbital speed and GPE, we can show that the total energy of a satellite in a circular orbit is always negative and is given by E_total = -GMm/(2r). The negative sign is significant: it means the satellite is 'bound' to the planet and cannot escape without an input of energy. To escape the planet's gravitational pull completely (i.e., to reach infinity), an object must be given enough kinetic energy to overcome its negative GPE. The minimum speed required to do this from a point 'r' is called the escape velocity. This occurs when the object's initial kinetic energy is exactly equal to the magnitude of its initial GPE. This gives the formula v_esc = sqrt(2GM/r).

For satellites in Low Earth Orbit (LEO), the Earth's atmosphere, though extremely thin, still exerts a small frictional force known as atmospheric drag. This drag force acts in the opposite direction to the satellite's velocity, doing negative work on it. This causes the satellite's total energy to decrease (become more negative). Here comes the paradox: since the total energy is E_total = -GMm/(2r), a decrease in E_total means the orbital radius 'r' must also decrease. The satellite moves to a lower orbit. As the satellite drops to a lower orbit, its gravitational potential energy decreases significantly, while its kinetic energy increases (since v = sqrt(GM/r), a smaller 'r' means a larger 'v'). The decrease in GPE is twice the increase in KE, with the difference being dissipated as heat by the drag force. So, counter-intuitively, atmospheric drag causes a satellite to speed up as it spirals inwards towards the Earth, eventually burning up in the denser lower atmosphere.