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# Gravitational Field Strength

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Modern physics is defined mainly in terms of fields, which are physical entities that extend in space and time. These objects are the usual sources of non-contact forces and allow us to describe the dynamics of almost every system we know of.

British-born scientist Isaac Newton already figured that gravity is a field that exists because of the presence of mass. Furthermore, he realised that it was always an attractive force. Let’s take a look at the definition of gravitational field strength:

The gravitational field strength is the measure of the intensity of the gravitational field that has mass as a source and attracts other masses.

Gravitational field strength is generated by masses, and it gives rise to an attractive force that weakens with distance.

## 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.

When considering more complex phenomena, such as black holes, galaxies, deviation of light, we need more fundamental theories such as General Relativity, developed by Albert Einstein.

Recall Newton’s law of gravitation. Its formula is

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 er 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

### 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.

### The gravitational field strength on Earth

Important to know! The value of the gravitational field strength on Earth is 9.81m/s2 or N/kg.

Newton’s law of gravitation, Wikimedia Commons

## 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 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.

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?

For instance, if we take two charges of 1 coulomb separated by 1 meter, we get a certain electrostatic force. If we do the same with two bodies of 1 kilogram each, we get another number for the gravitational force. The value is, essentially, the value of the constant in front of each of the formulae. It turns out that the constant for gravitation G is smaller than the constant for electromagnetism k (8.988 ⋅ 109Nm2/C2), so gravity is a weaker force.

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.

The four fundamental forces are gravity, electromagnetism, strong force, and weak force.

## 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.

• Earth. The radius of Earth is approximately 6371km. Its mass is about 5.972 ⋅ 1024kg. Applying the equation gives us a surface gravitational field strength of 9.81m/s2.
• Moon. The radius of the Moon is approximately 1737km. Its mass is about 7.348 ⋅ 1022kg. Applying the equation yields a surface gravitational field strength of 1.62m/s2.
• Mars. The radius of Mars is about 3390km. Its mass is approximately 6.39 ⋅ 1023kg. Applying the equation gives us a surface gravitational field strength of 3.72m/s2.
• Jupiter. The radius of Jupiter is about 69.911km, and its mass is around 1.898 ⋅ 1027kg. Applying the equation yields a surface gravitational field strength of 24.79m/s2.
• Sun. The radius of the Sun is approximately 696.340km, and its mass is about 1.989 ⋅ 1030kg. Applying the equation gives us a surface gravitational field strength of 273.60m/s2.

## Gravitational Field Strength - Key takeaways

• Gravity is a field and its strength in its classical model can be measured and modelled by the mathematical theory developed by Isaac Newton.
• Although there are more fundamental theories, Newton formulated the first rigorous approach to understanding gravitational field strength. It is valid only for certain circumstances (not including very massive objects, small distances, or very high speeds).
• Gravitational field strength is generated by masses, and it gives rise to an attractive force that decays with distance. Gravity is the weakest force among the four fundamental forces.
• Since gravitational field strength depends on mass and distance, planets feature different values of gravitational field strength on their surfaces.

The gravitational field strength is the intensity of the gravitational field sourced by a mass. If multiplied by a mass subject to it, one obtains the gravitational force.

To calculate the gravitational field strength, we apply Newton’s formula with the universal constant of gravitation, the mass of the source, and the radial distance from the object to the point where we want to calculate the field.

The gravitational field strength is measured in m/sor N/kg.

The gravitational field strength on the Moon is approximately 1.62m/sor N/kg.

The gravitational field strength on Earth is 9.81m/sor N/kg.

## Final Gravitational Field Strength Quiz

Question

The gravitational field strength is sourced by mass.

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The gravitational field strength is sourced by mass and affects masses.

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Newton’s description of the gravitational field strength is compatible with more modern descriptions.

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Radial dependence is a sign of isotropy, i.e. equivalence of all spatial directions.

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The gravitational field strength on the Sun is greater than on Jupiter.

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Who stated the first rigorous formulation of the gravitational field strength?

Newton stated the first rigorous formulation of the gravitational field strength.

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Question

What are two characteristics of a theory to describe the gravitational field strength?

The two characteristics of a theory to describe the gravitational field strength are

• sourced by masses
• attractive (decaying with distance from the source)

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Question

Do masses play an equivalent role when computing the forces caused by the gravitational field strength?

Yes, masses play an equivalent role when computing the forces caused by the gravitational field strength because one can describe the attractive force from the point of view of one mass or the other.

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Question

Can you name another theory about gravitational field strength that is more fundamental than Newton’s?

Einstein’s theory of general relativity.

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Question

Is the gravitational field strength constant on the surface of planets and stars?

Yes, the gravitational field strength is constant on the surface of planets and stars because they have an approximately spherical shape and the surface is at a constant radial distance.

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Is Earth’s gravitational field strength constant when an object is inside it and travelling towards its core?

No, Earth’s gravitational field strength is not constant when an object is inside it and travelling towards its core because the radial distance is decreasing (as well as the amount of mass attracting towards the core).

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Question

Is Earth’s gravitational field strength constant while getting away from the surface?

No, Earth’s gravitational field strength is not constant while getting away from the surface because the radial distance is increasing.

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Which astronomical object in the Solar System has the biggest surface gravitational field strength?

The Sun has the biggest surface gravitational field strength (it is the biggest astronomical object in the Solar System, but this is not the only variable because its mass is important too).

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Which planet in the Solar System has the biggest surface gravitational field strength?

Jupiter has the biggest surface gravitational field strength (it is the biggest planet in the Solar System, (but this is not the only variable because its mass is important too).

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Does the gravitational field strength have an infinite range?

Yes, the gravitational field strength has infinite range. It can reach arbitrarily long distances with a non-zero value of intensity.

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