Gravitational Field Strength

The gravitational field strength equation

Historically, there has not been a unique description of gravity. Due to experimentation, we know that Newton’s expression works on planets, stars (etc.) and their surroundings.

Recall Newton’s law of gravitation. Its formula is

\[\vec{Z} = G \cdot \frac{M}{r^2} \cdot \vec{e}_r\]

where the vector Z is the field strength sourced by the mass M, G is the universal constant of gravitation, r is the radial distance measured from the centre of the mass of the source body, and the vector e_r is the radial unit vector going towards it. If we want to obtain the force a body with mass m experiences under the influence of the field Z, we can simply calculate it as

\[\vec{F} = m \cdot \vec{Z}\]

The gravitational field strength unit

Concerning units and values, we find that the force of gravity is measured in Newtons [N = kg⋅m/s²]. As a result, the field strength is measured in m/s², i.e. it is an acceleration. The mass is usually measured in kilograms and the distance in meters. This gives us the units of the universal gravitational constant G, which are Nm²/kg² = m³/s²⋅kg. The value of G is 6.674 ⋅ 10^-11m³/s²⋅kg.

Gravitational potential energy, on the other hand, is measured in Joules.

The gravitational field strength on Earth

Important to know! The value of the gravitational field strength on Earth varies over height however near the Earth's surface is 9.81m/s² or N/kg.

What are the main features of gravitational field strength?

The main features of the gravitational field include

• The symmetry from the description of any of the two bodies.
• The radial symmetry.
• The specific value the universal constant for gravitation takes.

Understanding these characteristics is important, even for current scientists, to develop better models for gravity that reproduce the basic aspects of Newton’s gravity.

Reciprocity of the bodies

One of the most important consequences of Newton’s expression for the gravitational field strength is the reciprocity of the masses. This is consistent with Newton’s third law of motion, which states: if a body exerts a force on another body, the latter exerts the same force with opposite direction on the first.

The reciprocity is deeper than it seems since it states that a fundamental feature of the gravitational field strength is that it is equivalent to describing the gravity interactions from the perspective of one body or the other. This seems trivial but has deep implications concerning, for instance, general relativity.

Radial dependence and orientation

One of the main features of Newton’s expression for the gravitational field strength is the radial quadratic dependence. It turns out that in three-dimensional space, this is the right dependence to achieve an infinite range of field strength reaching any part of the space. Any other dependence would not allow it to have an infinite range or cause physical inconsistencies.

Additionally, this spherical dependence is joined by a spherical radial symmetry in the direction of the field strength. This not only ensures an attractive character but is also consistent with isotropy: there is no special direction in three-dimensional space. The way to put all directions on equal footing is to impose spherical symmetry, which leads to the radial dependence and the radial vector.

Value of the universal constant of gravitation

The universal constant of gravitation or Cavendish constant measures the intensity of the gravitational field strength. Of course, the intensity of the field will depend on the characteristics for each case, but it is a measure in the following sense: if we set all variables to one (with appropriate units), what number do we get?

In fact, out of the four fundamental forces (gravity, electromagnetism, strong force, and weak force), the gravitational field strength is the weakest one. It is also the only one acting relevantly at interplanetary scales.

Examples of gravitational field strength

Here are some examples of calculations of gravitational field strengths to get a better understanding of how it operates in various astronomical objects.