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Gravitational fields as vector fields

A gravitational field assigns a vector to every position in space. That vector tells us the acceleration a small freely falling test body would have at that position. Describing gravity as a field separates the source masses from the object used to probe their influence and makes direction as important as magnitude.

From gravitational force to field

Newton's law gives the magnitude of the force between two point masses:

\[F=G\frac{Mm}{r^2}\]

Here \(G\) is the universal gravitational constant, \(M\) and \(m\) are the masses, and \(r\) is the distance between their centers. The force is attractive and lies along the line joining the masses.

Divide the force on the test body by its mass. The resulting gravitational field, or free-fall acceleration, does not depend on the chosen test mass:

\[\mathbf g(\mathbf r)=-GM\frac{\mathbf r}{|\mathbf r|^3}\]

The vector \(\mathbf r\) points from the source to the probe. Its magnitude appears cubed in the denominator because one factor converts the position vector into a direction and the remaining two create the inverse-square magnitude. The minus sign points the acceleration back toward the source.

It is often convenient to define the gravitational parameter

\[\mu=GM\]

whose SI units are cubic meters per square second. Then the field needs only \(\mu\) and position. Astronomical orbit calculations often know this product more accurately than \(G\) and mass separately.

Reading the vector formula

The field magnitude is

\[|\mathbf g|=\frac{GM}{r^2}\]

so doubling distance reduces the magnitude to one quarter. At equal distances in different directions, the magnitudes are equal but the vectors point differently. At opposite positions, the vectors are opposites. These symmetry facts are excellent tests because they do not depend on a particular numerical value of \(G\).

import numpy as np

mu = 3.986004418e14
probe = np.array([6.371e6, 0.0, 0.0])
source = np.zeros(3)
offset = probe - source
radius = np.linalg.norm(offset)
acceleration = -mu * offset / radius**3
print("Acceleration (m/s^2):", acceleration)

This calculation should reject zero separation in the ideal point-mass model. Silently clamping the displayed arrow is not the same as defining the physical field there.

Many sources and superposition

Newtonian gravity obeys superposition. For point sources at positions \(\mathbf R_i\), the total field at probe position \(\mathbf r\) is

\[\mathbf g(\mathbf r)=-G\sum_i M_i\frac{\mathbf r-\mathbf R_i}{|\mathbf r-\mathbf R_i|^3}\]

Each source contribution must use its own displacement from source to probe. Summing scalar magnitudes loses direction and is generally wrong. For two identical masses, the field at their midpoint is zero because the two equal contributions point oppositely. That point is a force equilibrium, but it is not stable in every direction.

def point_mass_field(offset, mu):
    radius = np.linalg.norm(offset)
    if radius == 0:
        raise ValueError("The point-source field is singular at its source.")
    return -mu * np.asarray(offset) / radius**3


sources = [(np.array([-2.0, 0, 0]), 1.0), (np.array([2.0, 0, 0]), 1.0)]
probe = np.array([0.0, 1.0, 0.0])
total = sum((point_mass_field(probe - p, mu) for p, mu in sources), np.zeros(3))
print("Combined field in scaled units:", total)

When all gravitating bodies move, each body must receive the field from the other bodies using one common state snapshot. Updating a source position in the middle of force assembly introduces order dependence and breaks the intended simultaneous equations.

Extended spherical bodies

The point-mass equation applies exactly outside any spherically symmetric mass distribution if its full mass is treated as concentrated at the center. This shell-theorem result explains why the same equation works outside an ideal spherical planet or star. It does not mean that the field remains inverse-square inside the material.

For a spherically symmetric body, only mass enclosed within radius \(r\) contributes to the net field at that radius:

\[\mathbf g(\mathbf r)=-G\frac{M_{\mathrm{enc}}(r)}{r^2}\hat{\mathbf r}\]

Inside a uniform-density sphere, \(M_{\mathrm{enc}}(r)\propto r^3\), so field magnitude is proportional to radius and reaches zero at the center. A point-mass singularity would therefore be the wrong model for motion through an extended uniform planet.

For a general continuous density \(\rho(\mathbf R)\), contributions are integrated:

\[\mathbf g(\mathbf r)=-G\int\rho(\mathbf R)\frac{\mathbf r-\mathbf R}{|\mathbf r-\mathbf R|^3}\,d^3R\]

The point-source sum is a discrete approximation to this expression when each cell mass equals density times cell volume.

Near a planet's surface

At distance \(r\) from a spherical planet of mass \(M\), field magnitude is \(GM/r^2\). Near the surface, write \(r=R+h\), where \(R\) is planetary radius and \(h\) is height. If height is tiny compared with radius, the field is approximately constant:

\[g\approx\frac{GM}{R^2}\]

This is the basis of the constant- \(g\) model used in elementary projectile motion. It is a local approximation, not a universal field. Altitude, latitude, planetary rotation, nonspherical mass distribution, and local geology all affect measured effective gravity.

Potential and field

For a point mass, gravitational potential per unit test mass is

\[\Phi(r)=-\frac{GM}{r}\]

The field is the negative spatial gradient of potential:

\[\mathbf g=-\boldsymbol\nabla\Phi\]

The potential is scalar, so potentials add as scalars before taking a gradient. Field lines point toward decreasing potential. Equipotential surfaces are perpendicular to the field wherever the gradient is nonzero. Adding a constant to potential changes no field; only potential differences have dynamical meaning.

The differential source equation is Poisson's equation,

\[\nabla^2\Phi=4\pi G\rho\]

and in empty space it reduces to Laplace's equation. These forms are useful when a continuous density or boundary values are more natural than summing point sources.

The singularity and softening

The ideal point-mass field diverges at zero separation. A numerical model may stop at a finite surface, detect collision, adapt its step, regularize close encounters, or replace the point source with an extended density. A commonly used softened field is

\[\mathbf g_\varepsilon(\mathbf r)=-GM\frac{\mathbf r}{(r^2+\varepsilon^2)^{3/2}}\]

Softening length \(\varepsilon\) changes the force law; it is not merely protection against division by zero. Report its value and show that conclusions of interest are insensitive to smaller appropriate values.

Checks for a trustworthy field

Check units first: field output must have dimensions of acceleration. Test a source at the origin on each coordinate axis, opposite positions, and two radii whose ratio is simple. Verify that multiplying the test mass changes force but not field acceleration. For multiple sources, compare the result with separately computed contributions and test the symmetric midpoint case.

For a grid representation, refine the spacing while comparing field values at the same physical points. Never infer correctness from attractive arrows alone: normalized arrows can hide inverse-square magnitude, and screen coordinates can reverse an axis. Store the physical vectors and their units separately from all drawing data.

Newtonian gravity assumes instantaneous interaction in an inertial frame and is highly accurate for many weak-field, low-speed systems. Strong gravity, precision orbital effects, gravitational waves, and motion near compact bodies require general relativity.

A worked Python notebook

Compare the field outside an ideal spherical Earth at several distances from its centre. The question for this calculation is precise: how does gravitational field magnitude change when we change radial distance, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The exterior inverse-square expression does not describe points inside the Earth. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 6.371e+06 to 2e+07 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for radial distance, expressed in m. The output is gravitational field magnitude, expressed in m/s^2. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = \frac{398600441800000.0}{q^{2}}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1e+07 m, the expression gives 3.986 m/s^2. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("3.986004418e14/q**2", locals={"q": q})
reference_input = 10000000.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted gravitational field magnitude:", reference_output, "m/s^2")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 9.82025 m/s^2. The midpoint, at 1.31855e+07 m, gives 2.29269 m/s^2. The upper endpoint gives 0.996501 m/s^2. Read these together with the assumptions: they describe different choices of radial distance within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([6371000.0, 13185500.0, 20000000.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} m/s^2")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows radial distance; the vertical axis shows gravitational field magnitude. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(6371000.0, 20000000.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Radial distance (m)")
ax.set_ylabel("Gravitational field magnitude (m/s^2)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does gravitational field magnitude change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1e+07 m is -7.97201e-07 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = - \frac{797200883600000.0}{q^{3}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 10068145.0
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m/s^2")
print("Full expression:", full_prediction, "m/s^2")
print("Approximation difference:", full_prediction - linear_prediction, "m/s^2")

Here the changed input is 1.00681e+07 m. The tangent estimate is 3.93168 m/s^2, while the full expression gives 3.93223 m/s^2. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 13629000.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of radial distance inside the stated interval, predict whether gravitational field magnitude will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The exterior inverse-square expression does not describe points inside the Earth. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m/s^2: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(20000000.0) - evaluate(6371000.0))
for points in [101, 201, 401]:
    grid = np.linspace(6371000.0, 20000000.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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