diff --git a/book/2_potential_fields/gravity_field/density_anomalies.md b/book/2_potential_fields/gravity_field/density_anomalies.md new file mode 100644 index 0000000..b6437d5 --- /dev/null +++ b/book/2_potential_fields/gravity_field/density_anomalies.md @@ -0,0 +1,265 @@ +# Density Anomalies and Gravity Observations + +Poisson's equation connects the gravitational potential to the local mass density, + +$$ +\nabla^2\Phi=4\pi G\rho. +$$ + +Together with superposition, it also gives us a convenient way to construct potentials for bodies that are less symmetric than a complete sphere. We will use this to move from idealized density distributions towards a simple gravity-observation problem. + +## An off-centre spherical cavity + +Consider a uniform sphere of radius $R$ and density $\rho_0$, centred at the origin. A spherical cavity of radius $a$ is centred at the position $\vec d$, with + +$$ +|\vec d|+aa$, its potential and field are + +$$ +\boxed{ +\delta\Phi(\vec r)=-\frac{G\Delta M}{|\vec r-\vec d|} +} +$$ (eq:spherical-density-anomaly-exterior-potential) + +and + +$$ +\boxed{ +\delta\vec g(\vec r) +=-G\Delta M +\frac{\vec r-\vec d}{|\vec r-\vec d|^3} +}. +$$ (eq:spherical-density-anomaly-exterior-field) + +```{admonition} Exercise: indistinguishable spherical anomalies +:class: exercise + +Consider two spherical anomalies with the same centre $\vec d$ but different radii $a_1$ and $a_2$. + +1. Find the relation between $\Delta\rho_1$, $a_1$, $\Delta\rho_2$, and $a_2$ that gives both anomalies the same $\Delta M$. +2. Show that their potentials and gravity fields are identical at every point outside both anomalies. +3. Explain why measurements made outside the body cannot distinguish between these two density-radius combinations. +4. Could a measurement made inside one of the anomalous spheres distinguish them? Use Poisson's equation to support your answer. +``` + +This is a first example of **non-uniqueness** in gravity inversion: distinct density models can produce identical observations. The conclusion here is exact because the anomalies are spherical, have the same centre, and are observed from outside. For arbitrary disturbances, equal total mass alone is not sufficient to guarantee identical exterior fields; their geometry and distribution can contribute additional spatial structure. + +## Coding exercise: observing the anomaly from different heights + +We will now examine a spherical density anomaly buried at depth $D$ below Earth's surface. Since every observation point is outside the anomaly, Equation {eq}`eq:spherical-density-anomaly-exterior-field` tells us that only its anomalous mass $\Delta M$ and centre matter. + +Place the anomaly beneath the point $\theta=0$ at + +$$ +\vec d=(0,R_{\mathrm E}-D), +$$ + +and place observations along a circular track at height $h$, + +$$ +\vec r(\theta,h) +=(R_{\mathrm E}+h) +\left(\sin\theta,\cos\theta\right). +$$ + +The radial unit vector is $\hat r=\vec r/|\vec r|$. We will compare the downward radial anomaly + +$$ +\delta g_{\mathrm{down}} +=-\delta\vec g\cdot\hat r +$$ (eq:downward-radial-gravity-anomaly) + +at the surface, at a GOCE-like low-Earth orbit of $254\ \mathrm{km}$, and at a higher orbit. + +````{admonition} Write and investigate the model +:class: exercise + +Complete the function below and use it to calculate $\delta g_{\mathrm{down}}$ along the observation track. One microgal is $10^{-8}\ \mathrm{m\,s^{-2}}$. + +```python +import numpy as np +import matplotlib.pyplot as plt + +G = 6.67430e-11 # m^3 kg^-1 s^-2 +R_E = 6.371e6 # m +depth = 100e3 # m below the surface +delta_mass = 1.0e15 # kg + +theta = np.deg2rad(np.linspace(-10, 10, 501)) +heights = [0.0, 254e3, 500e3] + + +def downward_radial_anomaly(theta, height, delta_mass, depth): + """Return the downward radial gravity anomaly in m/s^2.""" + orbit_radius = R_E + height + + # Construct the observation positions r(theta, height). + # Construct the anomaly centre d. + # Evaluate delta_g using Eq. (spherical-density-anomaly-exterior-field). + # Project delta_g onto the local downward direction. + raise NotImplementedError + + +distance_along_surface_km = R_E * theta / 1e3 + +for height in heights: + anomaly = downward_radial_anomaly( + theta, height, delta_mass, depth + ) + plt.plot( + distance_along_surface_km, + anomaly / 1e-8, + label=f"h = {height / 1e3:.0f} km", + ) + +plt.xlabel("Distance from point above anomaly (km)") +plt.ylabel(r"Downward radial anomaly ($\mu$Gal)") +plt.legend() +plt.grid(True) +plt.show() +``` + +Use your results to answer the following questions: + +1. How do the peak amplitude and horizontal width of the anomaly change with observation height? +2. Directly above the anomaly, show that the magnitude of the field is + + $$ + |\delta\vec g|=\frac{G|\Delta M|}{(D+h)^2}. + $$ + + Use this result to check the central value returned by your code. +3. By what factor is the central anomaly reduced between the surface and the GOCE-like orbit? +4. Choose two radius-density combinations with the same $\Delta M$. Confirm numerically that their exterior gravity curves overlap. +5. Repeat the calculation for several depths $D$. Which anomalies are most strongly affected by increasing the observation height? +```` + +````{dropdown} Hints for the implementation + +Store the observation positions as an array with one row for each value of $\theta$. If `r` contains those positions and `d` is the anomaly centre, then + +```python +separation = r - d +distance = np.linalg.norm(separation, axis=1) +``` + +The factor $|\vec r-\vec d|^3$ must be applied once for every observation point. `distance[:, None]` can be used to make a one-dimensional array broadcast across the two vector components. + +To calculate the radial projection, construct `r_hat` and take a row-by-row dot product. For example, `np.sum(delta_g * r_hat, axis=1)` evaluates the dot product for every row. +```` + +The loss of amplitude and broadening of the anomaly with increasing height are the spatial-domain manifestation of the upward-continuation transfer function introduced in the previous section. Flying lower preserves more of the short-scale gravity signal, but, as GOCE demonstrated, it also makes atmospheric drag a much more serious engineering constraint {cite}`esa_goce_operations`. diff --git a/book/2_potential_fields/gravity_field/figures/gravitational_flux_surface_patch.png b/book/2_potential_fields/gravity_field/figures/gravitational_flux_surface_patch.png new file mode 100644 index 0000000..66ab67c Binary files /dev/null and b/book/2_potential_fields/gravity_field/figures/gravitational_flux_surface_patch.png differ diff --git a/book/2_potential_fields/gravity_field/figures/gravitational_flux_surface_patch.svg b/book/2_potential_fields/gravity_field/figures/gravitational_flux_surface_patch.svg new file mode 100644 index 0000000..40aac9e --- /dev/null +++ b/book/2_potential_fields/gravity_field/figures/gravitational_flux_surface_patch.svg @@ -0,0 +1,260 @@ + + + + Gravity field and area vector on a spherical surface + + A two-dimensional slice through a spherical surface surrounds a central mass. At a highlighted surface patch, + the area vector points outward and the gravitational field points inward, so their dot product is negative. + + + + + + + + + + + + + + + + + + + + Sr + + + M + + + + + surface patch dA + + + + + dA⃗ + outward normal + + g⃗ + towards M + + + The vectors are antiparallel: + g⃗ · dA⃗ = |g⃗| dA cos θ + θ = π + g⃗ · dA⃗ = -|g⃗| dA + negative flux + diff --git a/book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.png b/book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.png new file mode 100644 index 0000000..3d9bc19 Binary files /dev/null and b/book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.png differ diff --git a/book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.svg b/book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.svg new file mode 100644 index 0000000..bf2000a --- /dev/null +++ b/book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.svg @@ -0,0 +1,69 @@ + + Geometry of the gravitational force between two point masses + + Position vectors locate masses M and m from an arbitrary origin. The separation vector points from M to m, + while the gravitational force on m points back toward M. Equal unit vectors are shown at m and at the origin. + + + + + + + + + + + + + + + + + + + + + + + + x + y + O + + + + + r⃗′ + r⃗ + + + + + M + m + + + + R⃗ = r⃗ - r⃗′ + + + + F⃗ + + + + R̂ + + + R̂ + diff --git a/book/2_potential_fields/gravity_field/figures/principia_book1_plate21_figure4.png b/book/2_potential_fields/gravity_field/figures/principia_book1_plate21_figure4.png new file mode 100644 index 0000000..85f2168 Binary files /dev/null and b/book/2_potential_fields/gravity_field/figures/principia_book1_plate21_figure4.png differ diff --git a/book/2_potential_fields/gravity_field/figures/uniform_cylinder_laplacian.png b/book/2_potential_fields/gravity_field/figures/uniform_cylinder_laplacian.png new file mode 100644 index 0000000..7b14aa5 Binary files /dev/null and b/book/2_potential_fields/gravity_field/figures/uniform_cylinder_laplacian.png differ diff --git a/book/2_potential_fields/gravity_field/intro.md b/book/2_potential_fields/gravity_field/intro.md index 75a039a..02425ac 100644 --- a/book/2_potential_fields/gravity_field/intro.md +++ b/book/2_potential_fields/gravity_field/intro.md @@ -1 +1,11 @@ -# Gravity Field +# Earth's Gravity + +Gravity is the most familiar of the fundamental interactions, yet in important ways it remains one of the least understood. It governs falling objects and planetary orbits, but also the formation of stars and galaxies and the evolution of the Universe on its largest scales. + +In Newtonian physics, gravity is described as a force field produced by mass. Einstein's general theory of relativity gives a deeper interpretation: gravity is not an ordinary force, but a manifestation of the curvature of spacetime produced by matter and energy. General relativity has been remarkably successful, but it does not yet fit into a complete quantum description of nature. Understanding gravity at the smallest scales therefore remains one of the major open problems in physics. + +In this chapter we will work mainly with the Newtonian gravity field. This description is accurate for many problems involving the Earth and has the great advantage of making the connection between sources, fields, potentials, and measurements particularly clear. Its mathematical simplicity should not hide the depth of the physics behind it. + +While gravity is an interesting topic of study by itself, at least assuming an interest in physics, we are interested in its practical applications. Aside from keeping us on the ground and our Earth-observation satellites in orbit, direct and indirect measurements of the gravity field give us otherwise inaccessible information about the world we live in, very broadly speaking. For example, by studying the gravity-controlled motion of celestial bodies, astronomers can infer the masses of other bodies. Newton's generalized form of Kepler's third law makes this possible when the orbital period and separation are known, and remains the basis for many measurements of the masses of objects in and beyond the Solar System {cite}`nasa_orbits_kepler_laws`. + +In the context of this course, we are particularly interested in what measurements of the gravity field at Earth's surface and from satellites can tell us about the distribution of mass within the Earth and how that distribution changes with time. Missions such as the **Gravity Recovery and Climate Experiment (GRACE)** and the **Gravity field and steady-state Ocean Circulation Explorer (GOCE)** have shown how satellite observations can reveal both spatial and temporal variations in Earth's gravity field. At a more local scale, volcanologists use precise gravity measurements to infer subsurface mass changes associated with moving magma and to help monitor volcanic unrest. At a global scale, satellite gravimetry provides unique information about changes in groundwater storage and the loss of ice from polar regions. diff --git a/book/2_potential_fields/gravity_field/making_of.md b/book/2_potential_fields/gravity_field/making_of.md new file mode 100644 index 0000000..2810b74 --- /dev/null +++ b/book/2_potential_fields/gravity_field/making_of.md @@ -0,0 +1,3 @@ +# Making Of + +The notebooks collected here generate figures used in the preceding sections and provide interactive versions of several examples. They are included as part of the course for students who want to inspect, reproduce, or modify the calculations, while keeping the main conceptual flow focused on the physics and mathematics. diff --git a/book/2_potential_fields/gravity_field/poisson_laplace_equations.md b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md new file mode 100644 index 0000000..241b1b8 --- /dev/null +++ b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md @@ -0,0 +1,422 @@ +# Poisson's and Laplace's Equations + +The continuous-density potential introduced in Equation {eq}`eq:continuous-mass-gravitational-potential`, + +$$ +\Phi(\vec{r}) +=-G\int_V\frac{\rho(\vec{r'})}{|\vec{r}-\vec{r'}|}\,\mathrm{d}V' +$$ + +relates the gravitational potential at one point to the mass distribution throughout space. If we were merely interested in Earth's gravity field, we could now make the problem harder by considering a non-spherical Earth. We could start with an ellipsoid, use some trigonometric relations and use series expansions and patiently develop a fairly complex and accurate model of the gravity field. Instead of that, what we will do now is do introduce mathematical tools that allow us to set up the equations from which we can solve a general gravity (or another conservative potential fiedl): the Poisson's and Laplace's equations. The Poisson equation gives us the set of Partial Differential Equations (PDEs) describing the **local** relation between the potential and the mass-density. The Laplace equation describes the special case at points where the mass density is zero. + +## The Laplacian + +The Laplacian of a scalar field is, by definition, the divergence of its gradient: + +$$ +\nabla^2\Phi +=\vec{\nabla}\cdot\left(\vec{\nabla}\Phi\right). +$$ (eq:scalar-laplacian-definition) + +In Cartesian coordinates, + +$$ +\nabla^2\Phi +=\frac{\partial^2\Phi}{\partial x^2} ++\frac{\partial^2\Phi}{\partial y^2} ++\frac{\partial^2\Phi}{\partial z^2}. +$$ (eq:scalar-laplacian-cartesian) + +```{admonition} Laplacian in spherical and cylindrical coordinates +:class: note + +In spherical coordinates $(r,\theta,\varphi)$, where $\theta$ is the polar angle measured from the positive $z$-axis and $\varphi$ is the azimuthal angle, + +$$ +\begin{aligned} +\nabla^2\Phi +={}&\frac{1}{r^2}\frac{\partial}{\partial r} +\left(r^2\frac{\partial\Phi}{\partial r}\right)\\ +&+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta} +\left(\sin\theta\frac{\partial\Phi}{\partial\theta}\right) ++\frac{1}{r^2\sin^2\theta}\frac{\partial^2\Phi}{\partial\varphi^2}. +\end{aligned} +$$ + +For a spherically symmetric potential, $\Phi=\Phi(r)$, the angular derivatives vanish and only the radial term remains: + +$$ +\nabla^2\Phi +=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r} +\left(r^2\frac{\mathrm{d}\Phi}{\mathrm{d}r}\right). +$$ + +In cylindrical coordinates $(s,\varphi,z)$, where $s$ is the perpendicular distance from the $z$-axis, + +$$ +\nabla^2\Phi +=\frac{1}{s}\frac{\partial}{\partial s} +\left(s\frac{\partial\Phi}{\partial s}\right) ++\frac{1}{s^2}\frac{\partial^2\Phi}{\partial\varphi^2} ++\frac{\partial^2\Phi}{\partial z^2}. +$$ + +If the potential is rotationally symmetric and does not vary along the axis, as for the infinite-cylinder example, $\Phi=\Phi(s)$ and + +$$ +\nabla^2\Phi +=\frac{1}{s}\frac{\mathrm{d}}{\mathrm{d}s} +\left(s\frac{\mathrm{d}\Phi}{\mathrm{d}s}\right). +$$ + +The symbol $s$ is used here for cylindrical radius to avoid confusing it with the mass density $\rho$. +``` + +The gradient measures how the potential changes in space. Taking its divergence asks whether those changes produce a net outward or inward flux around a point. The Laplacian therefore measures the local curvature of the potential field. + +## A physical two-dimensional example + +Consider an infinitely long cylinder of radius $R$ and uniform density $\rho_0$, aligned with the $z$-axis. The mass per unit length is + +$$ +\lambda=\pi R^2\rho_0. +$$ + +Because the source does not change along $z$, neither the potential nor the gravity field depends on $z$, which implies that al partial derivatives with respect to $z$ are zero. The three-dimensional Laplacian therefore reduces to + +$$ +\nabla^2\Phi +=\frac{\partial^2\Phi}{\partial x^2} ++\frac{\partial^2\Phi}{\partial y^2}. +$$ (eq:two-dimensional-laplacian) + +This makes a cross-section through the cylinder a genuinely two-dimensional physical problem. Choosing the surface value $\Phi(R)=0$, the potential is + +$$ +\Phi(r)= +\begin{cases} +G\lambda\left(\dfrac{r^2}{R^2}-1\right), & 0\leq r\leq R,\\[6pt] +2G\lambda\ln\left(\dfrac{r}{R}\right), & r\geq R. +\end{cases} +$$ (eq:uniform-cylinder-potential) + +Unlike the potential of a bounded mass, this potential cannot be chosen to vanish at infinity: an infinite cylinder has infinite total mass, and its exterior potential grows logarithmically. Only potential differences have physical significance. + +The corresponding gravity field is directed towards the axis, + +$$ +\vec{g}(\vec{r})= +\begin{cases} +-\dfrac{2G\lambda}{R^2}\left(x\hat{x}+y\hat{y}\right), & r\leq R,\\[8pt] +-\dfrac{2G\lambda}{r^2}\left(x\hat{x}+y\hat{y}\right), & r\geq R. +\end{cases} +$$ (eq:uniform-cylinder-gravity) + +```{figure} figures/uniform_cylinder_laplacian.png +:name: uniform-cylinder-laplacian +:width: 100% + +Cross-section of a uniform infinite cylinder. Left: gravitational potential $\Phi$ and gravity field $\vec{g}=-\vec{\nabla}\Phi$. Arrow direction shows the inward attraction, while arrow length indicates field strength. Right: the two-dimensional Laplacian calculated numerically from the potential. It is $4\pi G\rho_0$ inside the matter and zero outside, apart from finite-grid smoothing near the boundary. The calculation is available in the accompanying {doc}`uniform_cylinder_laplacian` notebook. +``` + +The potential varies both inside and outside the cylinder, but its Laplacian distinguishes the two regions. Inside, where mass is present, + +$$ +\nabla^2\Phi=4\pi G\rho_0, +$$ + +whereas outside, where $\rho=0$, the positive and negative curvatures in different directions cancel and $\nabla^2\Phi=0$. The Laplacian is therefore not simply a measure of whether a surface is curved: it measures the **net** curvature obtained by adding the second derivatives in all coordinate directions. + +## From gravitational flux to Poisson's equation + +Let us first return to the exterior of a spherically symmetric body. On a spherical surface $S_r$ of radius $r>R$, the gravity field is + +$$ +\vec{g}=-\frac{GM}{r^2}\hat{r}. +$$ + +To picture the flux, imagine covering the spherical surface with a mosaic of very small patches. Attach an arrow to each patch that points perpendicular to the surface and out of the sphere. This is the area vector + +$$ +\mathrm{d}\vec{A}=\hat{r}\,\mathrm{d}A. +$$ + +Its direction describes the orientation of the patch, while its length represents the patch area. The dot product $\vec{g}\cdot\mathrm{d}\vec{A}$ measures how much of the gravity field passes through that patch. A field pointing straight out gives positive flux, a field pointing straight in gives negative flux, and a field tangent to the surface gives no flux because it does not cross the surface. + +```{figure} figures/gravitational_flux_surface_patch.svg +:name: gravitational-flux-surface-patch +:width: 92% + +A two-dimensional slice through the spherical surface. The highlighted arc represents a small surface patch. Its area vector $\mathrm{d}\vec{A}$ is perpendicular to the surface and points outwards, while the gravity field $\vec{g}$ points inwards towards the mass. The vectors are antiparallel, so $\vec{g}\cdot\mathrm{d}\vec{A}<0$: gravity enters rather than leaves the enclosed volume. +``` + +For the spherical surface, every area arrow points outwards and every gravity arrow points directly inwards. The two vectors are antiparallel at every patch, so + +$$ +\vec{g}\cdot\mathrm{d}\vec{A} +=-\frac{GM}{r^2}\,\mathrm{d}A. +$$ + +The total flux is obtained by adding the contributions from all the small patches: + +$$ +\begin{aligned} +\oint_{S_r}\vec{g}\cdot\mathrm{d}\vec{A} +&=-\frac{GM}{r^2}\oint_{S_r}\mathrm{d}A\\ +&=-\frac{GM}{r^2}\left(4\pi r^2\right)\\ +&=-4\pi GM. +\end{aligned} +$$ (eq:spherical-gravitational-flux) + +```{admonition} Visualizing the cancellation +:class: tip + +Imagine inflating the spherical surface like a transparent balloon while leaving the mass at its centre. As the radius increases, each gravity arrow becomes shorter as $1/r^2$. At the same time, a fixed cone drawn from the centre intercepts a patch whose area grows as $r^2$. + +If the cone subtends a small solid angle $\mathrm{d}\Omega$, the patch area is + +$$ +\mathrm{d}A=r^2\,\mathrm{d}\Omega. +$$ + +The flux through that patch is therefore + +$$ +\mathrm{d}\mathcal{F} +=-\frac{GM}{r^2}\mathrm{d}A +=-GM\,\mathrm{d}\Omega. +$$ + +The field has become weaker, but it acts across a proportionally larger patch. Each cone carries the same flux through every concentric sphere. The complete sphere contains a total solid angle of $4\pi$, giving $\mathcal{F}=-4\pi GM$. +``` + +The flux is consequently independent of the radius of the spherical surface. For a concentric surface inside a spherically symmetric body, the same calculation applies with $M$ replaced by the enclosed mass $M_{\mathrm{enc}}(r)$. + +This result is more general than the spherical calculation suggests. For any closed surface $S$, of any shape, the net gravitational flux depends only on the total mass enclosed by that surface: + +$$ +\oint_S \vec{g}\cdot\mathrm{d}\vec{A} +=-4\pi G M_{\mathrm{enc}}. +$$ (eq:gauss-law-gravity-integral) + +This is **Gauss's law for gravity**. One way to visualize the generalization is to imagine deforming the transparent spherical balloon into an irregular closed shape without moving it across the mass. The gravity arrows are no longer perpendicular to every patch, and their strengths vary across the surface. However, the dot product automatically counts only the component crossing each patch. The same cones from the mass now meet tilted patches at different distances, but their total solid angle remains $4\pi$, so the total flux does not change. + +Mass inside the surface therefore contributes a net inward flux. A mass outside the surface produces no net flux: its field enters the volume through some patches and leaves through others, and those contributions cancel. Because gravitational fields obey superposition, the result extends from individual point masses to arbitrary mass distributions. The area vector $\mathrm{d}\vec{A}$ always points outwards, so the inward flux produced by positive mass is negative. + +### The divergence theorem + +Before writing another equation, let us start with the physical idea. Imagine surrounding a region with a closed, transparent boundary and observing a net flux passing out through it. That flux cannot simply appear at the boundary: it must emerge from somewhere inside the enclosed volume. Conversely, if more field enters than leaves, something inside behaves as a sink. + +Recall from the earlier chapter on [gradient, divergence, and curl](../../1_gradient_divergence_curl/intro.md) that the divergence $\vec{\nabla}\cdot\vec{u}$ measures this local balance for a vector field $\vec{u}$: + +- positive divergence means that, locally, more field leaves than enters, like a source; +- negative divergence means that more field enters than leaves, like a sink; +- zero divergence means that there is no net production or absorption at that point. + +Now imagine dividing the complete volume into many tiny cells. The divergence in each cell tells us its small net outward flux. When we add the fluxes from all cells, every shared internal face appears twice: flux leaving one cell enters its neighbour through the same face. These internal contributions cancel pair by pair. Only the flux through the outer boundary remains. + +This is the idea expressed mathematically by the **divergence theorem**. For any sufficiently smooth vector field $\vec{u}$ in a volume $V$ bounded by the closed surface $S=\partial V$, + +$$ +\boxed{ +\oint_{\partial V}\vec{u}\cdot\mathrm{d}\vec{A} +=\int_V\vec{\nabla}\cdot\vec{u}\,\mathrm{d}V. +} +$$ (eq:divergence-theorem) + +The left-hand side measures the net outward flux through the boundary. The right-hand side adds up all the local sources and sinks inside the volume. The equality follows from the cancellation of the internal faces: all net flux crossing the outer boundary must be accounted for by the divergence somewhere inside. + +For gravity, positive mass behaves as a **sink** of the gravity field rather than a source: the field arrows converge towards the mass. We should therefore expect $\vec{\nabla}\cdot\vec{g}$ to be negative wherever positive mass density is present. + +Applying the divergence theorem to the gravity field gives + +$$ +\oint_{\partial V}\vec{g}\cdot\mathrm{d}\vec{A} +=\int_V\vec{\nabla}\cdot\vec{g}\,\mathrm{d}V. +$$ + +Gauss's law supplies the value of the surface integral, while the enclosed mass can be written as + +$$ +M_{\mathrm{enc}}=\int_V\rho\,\mathrm{d}V. +$$ + +Combining these relations gives + +$$ +\int_V \left(\vec{\nabla}\cdot\vec{g}\right)\mathrm{d}V +=-4\pi G\int_V\rho\,\mathrm{d}V. +$$ + +Because this relation holds for any volume $V$, the integrands must be equal: + +$$ +\vec{\nabla}\cdot\vec{g} +=-4\pi G\rho. +$$ (eq:gauss-law-gravity-differential) + +Using $\vec{g}=-\vec{\nabla}\Phi$, we obtain + +$$ +-\vec{\nabla}\cdot\left(\vec{\nabla}\Phi\right) +=-4\pi G\rho, +$$ + +or + +$$ +\boxed{\nabla^2\Phi=4\pi G\rho.} +$$ (eq:gravitational-poisson-equation) + +This is **Poisson's equation for gravity**. It says that mass density is the source of the gravitational potential. The positive sign on the right-hand side follows from our conventions $\Phi=-GM/r$ and $\vec{g}=-\vec{\nabla}\Phi$. + +```{admonition} Reading the equation +:class: tip + +Poisson's equation relates two quantities evaluated at the same position: + +$$ +\rho(\vec{r}) +=\frac{1}{4\pi G}\nabla^2\Phi(\vec{r}). +$$ + +It can therefore be read in two directions. Given a density distribution and suitable boundary conditions, we can calculate the potential. Conversely, sufficiently detailed knowledge of the potential throughout a volume gives information about the density there. In practice, gravity is often measured only on or above Earth's surface, which makes the inverse problem much less direct. +``` + +## Laplace's equation in empty space + +In a region containing no mass, $\rho=0$, and Poisson's equation reduces to + +$$ +\boxed{\nabla^2\Phi=0.} +$$ (eq:gravitational-laplace-equation) + +This is **Laplace's equation**. A field satisfying Laplace's equation is called **harmonic**. The equation applies in any source-free region, even if masses outside that region produce a non-zero potential and gravity field within it. + +For example, the potential outside a spherically symmetric body is + +$$ +\Phi(r)=-\frac{GM}{r}. +$$ + +Because of the spherical symmetry it is natural to evaluate its Laplacian in spherical coordinates, which reduces to + +$$ +\nabla^2\Phi +=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r} +\left(r^2\frac{\mathrm{d}\Phi}{\mathrm{d}r}\right). +$$ (eq:radial-scalar-laplacian) + +Consequently, for $r>R$, + +$$ +\nabla^2\left(-\frac{GM}{r}\right) +=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r} +\left[r^2\frac{\mathrm{d}}{\mathrm{d}r} +\left(-\frac{GM}{r}\right)\right] +=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r}(GM) +=0. +$$ + +The potential is not constant and the gravity field is not zero, yet Laplace's equation is satisfied because there is no mass in the exterior region. For a point mass, the origin itself must be excluded from this calculation: that is precisely where the source is located and where $-GM/r$ is singular. + +## Guided exercise: checking the uniform sphere + +We have just verified that the potential outside a uniform sphere satisfies Laplace's equation. Inside the sphere the density is non-zero, so the potential should instead satisfy Poisson's equation. Let us check this directly. + +```{admonition} Verify the interior potential +:class: exercise + +For a uniform sphere of radius $R$, total mass $M$, and density $\rho_0$, Equation {eq}`eq:uniform-sphere-potential` gives the interior potential + +$$ +\Phi(r) +=-\frac{GM}{2R}\left(3-\frac{r^2}{R^2}\right), +\qquad 0\leq r0$ describes how rapidly the potential perturbation decreases with height. This proposed form already satisfies both boundary conditions. We have not yet shown, however, that it satisfies Laplace's equation. + +```{admonition} Interpreting the wavenumber +:class: tip + +The **wavenumber** $k$ measures how rapidly a periodic pattern varies in space. It is related to the horizontal wavelength by + +$$ +\lambda=\frac{2\pi}{k}, +$$ + +so a large $k$ represents a short wavelength and a small $k$ a long wavelength. + +In this static example, the sign of $k$ does not change the pattern because + +$$ +\cos(-kx)=\cos(kx). +$$ + +We can therefore choose $k>0$ without losing any possible solution. Later, when we describe travelling waves or use complex exponentials, the sign of the wavenumber will matter: together with the time dependence, it indicates the direction of propagation. +``` + +As you work more with the PDEs discussed in this course, you should come to expect exponentials, sines, and cosines to appear as building blocks of solutions. Here we have postulated such a combination and will now test it. + +At this point we do **not** assume any relation between $k$ and $\alpha$. + +The region above the mass distribution contains no mass, so the potential must satisfy + +$$ +\nabla^2\delta\Phi=0. +$$ + +```{exercise} +Derive the expression for the Laplacian of $\delta\Phi(x,z)$ and use Laplace's equation to show that, for $A \ne 0$, + +$$\alpha^2=k^2.$$ + +``` + +Substitution into Laplace's equation gives + +$$ +\nabla^2\delta\Phi +=(\alpha^2-k^2)\delta\Phi +=0. +$$ + +For the non-trivial solution $A\ne0$, this requires + +$$ +\alpha^2=k^2. +$$ + +Because we chose both $\alpha>0$ and $k>0$, + +$$ +\boxed{\alpha=k}. +$$ + +Our proposed solution therefore becomes + +$$ +\boxed{ +\delta\Phi(x,z)=A e^{-kz}\cos(kx) +}. +$$ + +This result has an important physical consequence. The horizontal wavelength is + +$$ +\lambda=\frac{2\pi}{k}. +$$ + +A short-wavelength variation has a large $k$ and therefore decays rapidly with height. A long-wavelength variation has a small $k$ and persists to much greater heights. + +**Laplace's equation has linked the horizontal spatial scale of the gravity field to its vertical spatial scale.** + +This explains an important property of gravity observations: fine-scale variations in the mass distribution become increasingly difficult to observe as the distance from the sources increases. A satellite at high altitude is therefore much more sensitive to large-scale variations of the Earth's gravity field than to small-scale variations. + +This attenuation helped motivate the exceptionally low orbit of the **Gravity field and steady-state Ocean Circulation Explorer (GOCE)**. Its nominal science orbit was about $254\ \mathrm{km}$ above Earth, where short-wavelength variations remained stronger, but the residual atmosphere produced enough drag to threaten both the orbit and the very sensitive gravity measurements. GOCE combined a streamlined shape with a drag-free control system: an electric ion thruster continuously adjusted its thrust to compensate for the measured atmospheric drag {cite}`esa_goce_operations`. + +```{admonition} A spatial transfer function +:class: tip + +We can express the upward attenuation in the language of systems and signals. For one horizontal spatial frequency $k$, the potential anomaly at height $z$ is + +$$ +\delta\Phi(k,z) +=H_z(k)\,\delta\Phi(k,0), +$$ + +where + +$$ +\boxed{H_z(k)=e^{-kz}}, +\qquad k>0. +$$ (eq:upward-continuation-transfer-function) + +The function $H_z(k)$ is a **spatial transfer function**. It tells us how the amplitude of each horizontal spatial-frequency component changes between the reference surface and the observation height. Since $H_z(k)$ is close to one for small $k$ but rapidly approaches zero for large $k$, observing the field at altitude acts as a **spatial low-pass filter**: broad features pass more easily than fine details. + +This is directly analogous to the frequency response of a system, which you may encounter in another course. There, a transfer function describes how different temporal frequencies are amplified or attenuated. Here the independent frequency variable is the spatial wavenumber $k$, measured in inverse metres, rather than a temporal frequency measured in hertz. Components of the gravity field and its gradients acquire additional factors of $k$ when we differentiate the potential, but they retain the same exponential attenuation with height. +``` + +```{admonition} More than gravity +:class: note + +Laplace's equation is not specifically an equation of gravity. It appears whenever a potential-like quantity has no sources within the region under consideration. + +For example, it describes the electric potential in a region without electric charge and the steady-state temperature in a region without heat sources. + +The physical quantities are different, but the mathematical problem is the same: + +$$ +\nabla^2 u=0. +$$ + +This is one reason why learning how to reason about Laplace's equation is useful far beyond gravitational fields. +``` + +## What postulating has shown us + +The examples above illustrate an important way of thinking about partial differential equations. We can propose a potential and substitute it into the governing equation to determine whether it is mathematically possible. Superposition then allows us to combine such solutions into richer fields. + +The equation alone still permits infinitely many possibilities. The domain and boundary conditions supply the additional physical information needed to select among them. Later we will encounter systematic methods for constructing solutions from that information. For now, postulating and checking functions lets us begin to see how the equation constrains their possible form. diff --git a/book/2_potential_fields/gravity_field/spherically_symmetric_bodies.md b/book/2_potential_fields/gravity_field/spherically_symmetric_bodies.md new file mode 100644 index 0000000..46c3630 --- /dev/null +++ b/book/2_potential_fields/gravity_field/spherically_symmetric_bodies.md @@ -0,0 +1,184 @@ +# Spherically Symmetric Bodies + +We now return to the spherical mass distributions encountered in the discussion of Newton's *Principia*. Let a body of radius $R$ have a density + +$$ +\rho=\rho(r'), +$$ + +which may vary with distance $r'$ from the centre but not with direction. Such a body can be regarded as a collection of concentric thin spherical shells. Newton's shell theorem tells us that a shell produces no field at points inside it, while at exterior points it acts as if all its mass were concentrated at the centre. + +At a distance $r$ from the centre, only the mass at radii $r'R_o$. +3. Use Equation {eq}`eq:spherical-interior-gravity-general` to find $\vec{g}(\vec{r})$ in all three regions. +4. Verify that the field is continuous at $R_i$ and $R_o$. Where does its magnitude reach a maximum? +5. Taking $\Phi\rightarrow0$ at infinity, derive the potential in all three regions. Require $\Phi$ to be continuous at both boundaries. +6. Show that the limit $R_i\rightarrow0$ recovers the uniform solid sphere. +``` + +```{dropdown} Hints and physical checks +The enclosed mass within the material is the volume between $R_i$ and the observation radius $r$, multiplied by $\rho_0$. Matter at radii greater than $r$ contributes no field there, but it does contribute a constant to the potential. + +Your field should vanish everywhere in the cavity, vary within the material, and reduce to $-GM\hat{r}/r^2$ outside. The potential should be constant, but generally not zero, throughout the cavity. +``` + +The field and its radial profile can be explored in the existing {doc}`../introduction/spherical_shell_gravity` notebook. Try predicting each region before comparing your result with the figure. + +## From shells to a simple Earth model + +A spherically layered Earth can be constructed from a central sphere and a sequence of concentric thick shells. Let the boundaries be + +$$ +0=R_0