From 8ac46f0d644135372424b600c2bf608fc02c5acf Mon Sep 17 00:00:00 2001 From: Paco Lopez-Dekker Date: Thu, 27 Aug 2026 17:12:11 +0200 Subject: [PATCH 1/6] Developing material on gravity --- .../figures/point_mass_gravity_geometry.png | Bin 0 -> 1518534 bytes .../figures/point_mass_gravity_geometry.svg | 69 ++++++++ .../2_potential_fields/gravity_field/intro.md | 12 +- .../spherically_symmetric_bodies.md | 157 ++++++++++++++++++ .../gravity_field/the_gravity_field.md | 155 +++++++++++++++++ book/_toc.yml | 3 + book/references.bib | 22 +++ 7 files changed, 417 insertions(+), 1 deletion(-) create mode 100644 book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.png create mode 100644 book/2_potential_fields/gravity_field/figures/point_mass_gravity_geometry.svg create mode 100644 book/2_potential_fields/gravity_field/spherically_symmetric_bodies.md create mode 100644 book/2_potential_fields/gravity_field/the_gravity_field.md diff --git 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zkp!diL7?%(DJL>xiSD~M;6uch-aY-b`kbMJag##uJQ5(SvHxuWY@Z~|VLr_!*^sX_ zz#DM_8cev~M@OrdV1^H)2X&{p!qWgU6|m}? zV#T2Kyy^qKxX_>%!fM8)qcC|3NO?qhR$E0+}7?a-UT-DzICnW-mIgphf{-ovyaIlR@z)2Y~1!d2$4i(b6>MyHfW_50a-r{B}TL#FHdnTi(fQSwYI z@pVXVX$Dc@`oTlAZieL{RDn}U?$^BALu%%;+BvIdXJn$(@x8Q8Ct&sF6lbA8j|ajy z;8!{&OJI^~?h)gQNBCP}QzMhUyzuc#n}dWcS*7JfVdUw3Ctu*1lZK{pj1b3DShnye zED3cC-r(r*X=1d+DlEg-c;ip5w_T_1(7X%1Bw z^i-NoFp2_d&tYOwjyMuy0$B@_YnBJ_g{0*-cnJoyF&r4RGfirH8~B+30QGPpkg9oD zX!Zem1R|z!t#|p_#`VKS^<^hr!mzx?!gG;&gz(H4fP2sl9TaJwfQPcmgnPBdkn`r5 z0nZUe1wOhQZ1XDs3cl&>jcYeoR!bvaj`}{+grdl>!7z22*Fdhh;z-ApMoM>&^G@u^(L31*yc$_O4=MTBVUNgU{eHs)Mz)N+d$( zO#(9xtp$~APIb#oY<`0DL|`2v5%lHu*djCWY^)^{x_zfsqpHMhcOxMkr$IR*iAhyT-H{B zyFjU-_5PCyba3%o6~}NB@AhorqH#1#zWVY)i(NePHC#jGnv)CSLf*d zI^n{}*)s!9ly>Xt0 + 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 + + + + + + + + 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/spherically_symmetric_bodies.md b/book/2_potential_fields/gravity_field/spherically_symmetric_bodies.md new file mode 100644 index 0000000..552b969 --- /dev/null +++ b/book/2_potential_fields/gravity_field/spherically_symmetric_bodies.md @@ -0,0 +1,157 @@ +# 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. 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We will not reproduce the proof here, but its main idea gives useful physical intuition. + +Imagine an object at some point $P$ inside a thin spherical shell. If $P$ is not at the centre, the nearby part of the shell pulls more strongly than the more distant part, because gravitational attraction decreases as $1/r^2$. It might therefore seem that the object should be pulled toward the nearest wall of the shell. + +Now consider a very narrow cone passing through $P$ and extending in both directions until it intersects the shell. The cone selects a small patch of the shell on each side of $P$. The more distant patch is farther away and therefore pulls more weakly per unit mass. However, the cone also intercepts a larger area of the shell on that side. The area grows with the square of the distance in just the way needed to compensate for the $1/r^2$ decrease of the gravitational attraction. + +```{figure} figures/principia_book1_plate21_figure4.png +:name: principia-book1-plate21-figure4 +:width: 48% + +Newton's construction for Proposition 70. Lines passing through the interior point $P$ select opposing elements of the spherical shell near $H,I$ and $K,L$. The geometrical construction is used to compare their areas, distances, and attractions. Book I, Plate 21, Figure 4 from Andrew Motte's 1729 English translation of the *Principia*. [Source: Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Principia_-_1729_-_Book_1,_Plate_21,_Figure_4.png), public domain. +``` + +The two patches therefore exert equal forces in opposite directions. Newton showed geometrically that the entire spherical shell can be considered as such pairs of opposing patches, so all their contributions cancel: + +$$ +\vec{g}=0. +$$ + +The exact cancellation is closely connected to the inverse-square nature of gravity: the same factor $r^2$ that makes a patch selected by a given angular extent grow with distance appears inversely in the gravitational force. + +This argument gives the intuition, but not a complete proof. Newton's geometric proof can be found in Book I, Section XII, Proposition 70 of the *Principia*. +:::: + ## The exterior field Outside the body, every shell lies inside the observation radius. 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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/poisson_laplace_equations.md b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md index 5ecec0e..2dc8e1a 100644 --- a/book/2_potential_fields/gravity_field/poisson_laplace_equations.md +++ b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md @@ -7,11 +7,11 @@ $$ =-G\int_V\frac{\rho(\vec{r'})}{|\vec{r}-\vec{r'}|}\,\mathrm{d}V' $$ -relates the gravitational potential at one point to the mass distribution throughout space. There is also a **local** relation between the potential and the density. This relation is Poisson's equation. +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. 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 the divergence of its gradient: +The Laplacian of a scalar field is, by definition, the divergence of its gradient: $$ \nabla^2\Phi @@ -27,6 +27,51 @@ $$ +\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). +$$ + +The symbol $s$ is used here for cylindrical radius to avoid confusing it with the mass density $\rho$. +``` + The gradient measures how the potential changes in space. Taking its divergence asks whether those changes produce a net outward or inward flux around a point. The Laplacian therefore measures the local curvature of the potential field. ## A physical two-dimensional example @@ -37,7 +82,7 @@ $$ \lambda=\pi R^2\rho_0. $$ -Because the source does not change along $z$, neither the potential nor the gravity field depends on $z$. The three-dimensional Laplacian therefore reduces exactly to +Because the source does not change along $z$, neither the potential nor the gravity field depends on $z$, which implies that al partial derivatives with respect to $z$ are zero. The three-dimensional Laplacian therefore reduces to $$ \nabla^2\Phi @@ -84,14 +129,119 @@ whereas outside, where $\rho=0$, the positive and negative curvatures in differe ## From gravitational flux to Poisson's equation -For any closed surface $S$, the gravitational flux depends on the mass enclosed by that surface: +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) -The area vector $\mathrm{d}\vec{A}$ points outwards. Since gravity points towards positive mass, the flux is negative. Writing the enclosed mass as a volume integral and applying the divergence theorem gives +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 $$ \int_V \left(\vec{\nabla}\cdot\vec{g}\right)\mathrm{d}V @@ -143,13 +293,13 @@ $$ (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 spherical body is +For example, the potential outside a spherically symmetric body is $$ \Phi(r)=-\frac{GM}{r}. $$ -For a spherically symmetric scalar field, the Laplacian is +Because of the spherical symmetry it is natural to evaluate its Laplacian in spherical coordinates, which reduces to $$ \nabla^2\Phi From 5cc17209e5823834a27a48e863f71d3b16c0e13c Mon Sep 17 00:00:00 2001 From: Paco Lopez-Dekker Date: Tue, 1 Sep 2026 17:45:17 +0200 Subject: [PATCH 5/6] developed further Laplace example --- .../gravity_field/making_of.md | 3 + .../poisson_laplace_equations.md | 2 +- .../postulating_laplace_solutions.md | 337 ++++++++++++++++++ book/_toc.yml | 7 +- book/references.bib | 7 + 5 files changed, 353 insertions(+), 3 deletions(-) create mode 100644 book/2_potential_fields/gravity_field/making_of.md create mode 100644 book/2_potential_fields/gravity_field/postulating_laplace_solutions.md diff --git a/book/2_potential_fields/gravity_field/making_of.md b/book/2_potential_fields/gravity_field/making_of.md new file mode 100644 index 0000000..2810b74 --- /dev/null +++ b/book/2_potential_fields/gravity_field/making_of.md @@ -0,0 +1,3 @@ +# Making Of + +The notebooks collected here generate figures used in the preceding sections and provide interactive versions of several examples. They are included as part of the course for students who want to inspect, reproduce, or modify the calculations, while keeping the main conceptual flow focused on the physics and mathematics. diff --git a/book/2_potential_fields/gravity_field/poisson_laplace_equations.md b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md index 2dc8e1a..493a5ff 100644 --- a/book/2_potential_fields/gravity_field/poisson_laplace_equations.md +++ b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md @@ -7,7 +7,7 @@ $$ =-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. 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. +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 diff --git a/book/2_potential_fields/gravity_field/postulating_laplace_solutions.md b/book/2_potential_fields/gravity_field/postulating_laplace_solutions.md new file mode 100644 index 0000000..6ee6c81 --- /dev/null +++ b/book/2_potential_fields/gravity_field/postulating_laplace_solutions.md @@ -0,0 +1,337 @@ +# Postulating and Verifying Solutions to Laplace's Equation + + +Laplace's equation, + +$$ +\nabla^2 \Phi = 0, +$$ + +is a **partial differential equation (PDE)**: an equation that a function and its partial derivatives must satisfy. Unlike an algebraic equation, it does not determine a single value, and unlike an ordinary differential equation, the unknown is a function of several variables. + +There are many functions that satisfy Laplace's equation. Finding the particular solution that describes a physical situation requires additional information about the geometry of the problem and the conditions at its boundaries. + +Before developing systematic methods for solving PDEs, we will take a simpler approach: **postulate a physically plausible form for the potential and check whether it satisfies the equation**. + +## A uniform gravitational field + +Close to the surface of the Earth, and over sufficiently small distances, we often approximate the gravitational field as uniform, + +$$ +\vec{g} = -g\,\hat{z}. +$$ + +A potential corresponding to this field is + +$$ +\Phi(z)=gz, +$$ + +since + +$$ +\vec{g}=-\vec{\nabla}\Phi=-g\,\hat{z}. +$$ + +There is no mass in the region in which we are describing the field, so the potential should satisfy Laplace's equation. In Cartesian coordinates, + +$$ +\nabla^2\Phi += +\frac{\partial^2\Phi}{\partial x^2} ++ +\frac{\partial^2\Phi}{\partial y^2} ++ +\frac{\partial^2\Phi}{\partial z^2}. +$$ + +For $\Phi=gz$, all three second derivatives vanish, and therefore + +$$ +\nabla^2\Phi=0. +$$ + +The familiar approximation of constant gravity near the Earth's surface is therefore consistent with Laplace's equation. + +```{admonition} A solution belongs to a region +:class: important + +The statement $\nabla^2\Phi=0$ does **not** mean that there are no masses producing the gravitational field. It means that there is no mass **in the region where we are solving the equation**. + +The Earth's mass lies below us, while the potential in the space above the surface satisfies Laplace's equation. +``` + +## A gravity field that varies in space + +A uniform gravitational field is a particularly simple case. Suppose instead that we postulate the potential + +$$ +\Phi(x,z)=A(x^2-z^2), +$$ + +where $A$ is a constant. + +Does this represent a possible gravitational potential in a region containing no mass? + +We can test the proposed solution directly: + +$$ +\frac{\partial^2\Phi}{\partial x^2}=2A, +$$ + +and + +$$ +\frac{\partial^2\Phi}{\partial z^2}=-2A. +$$ + +There is no dependence on $y$, so + +$$ +\nabla^2\Phi += +2A-2A += +0. +$$ + +The proposed potential is therefore a possible solution of Laplace's equation in a source-free region. + +The corresponding gravitational field is + +$$ +\vec{g} += +-\vec{\nabla}\Phi += +-2Ax\,\hat{x} ++ +2Az\,\hat{z}. +$$ + +Unlike the previous example, the gravitational field now changes with position even though there is no mass locally. + +Such spatial variations of the gravitational field occur naturally when observing the field produced by distant masses. Locally, the first-order spatial variation of gravity is called a **gravity gradient** or **tidal field**. + +```{exercise} +Verify that + +$$ +\Phi(x,z)=A(x^2+z^2) +$$ + +does *not* satisfy Laplace's equation. + +What would Poisson's equation imply about the mass density in a region described by this potential? +``` + +## Superposition and perturbations + +The examples above already give us more than two isolated solutions. Laplace's equation is **linear**: for any two functions $\Phi_1$ and $\Phi_2$ and any constants $a$ and $b$, + +$$ +\nabla^2\left(a\Phi_1+b\Phi_2\right) +=a\nabla^2\Phi_1+b\nabla^2\Phi_2. +$$ + +Therefore, if + +$$ +\nabla^2\Phi_1=0 +\qquad\text{and}\qquad +\nabla^2\Phi_2=0, +$$ + +then + +$$ +\nabla^2\left(a\Phi_1+b\Phi_2\right)=0. +$$ + +This is the **principle of superposition**. Any linear combination of solutions is itself a solution. Even a single non-zero solution can be multiplied by infinitely many different constants, and combining different solutions produces still more possibilities. Laplace's equation therefore does not, by itself, select one potential from the infinitely many functions that satisfy it. + +Superposition also gives us a useful physical interpretation of the previous example. In a small region near Earth's surface, we can neglect the curvature of Earth and take $z$ to be the local upward direction. The potential can then be written as a uniform background plus a smaller perturbation, + +$$ +\Phi_{\mathrm{total}}(x,z) +=gz+\delta\Phi(x,z), +$$ + +with gravity field + +$$ +\vec{g}_{\mathrm{total}} +=-g\,\hat{z}-\vec{\nabla}\delta\Phi. +$$ + +For example, $\delta\Phi=A(x^2-z^2)$ describes a local gravity gradient, or tidal perturbation, superimposed on the approximately uniform background field. The periodic gravity anomaly considered next will be interpreted in the same way: it is a perturbation caused by variations in the mass distribution, not the complete gravitational potential of Earth. + +## Boundary conditions: selecting a physical solution + +If Laplace's equation admits infinitely many solutions, what selects the one realized in a particular physical problem? We need to specify the **domain** in which we are solving the equation and provide information at its **boundaries**. + +For a gravitational potential, we might know the value of the potential on a boundary, the component of the gravity field normal to that boundary, or how the potential behaves far from the sources. Such information is called a **boundary condition**. Together, the equation, domain, and boundary conditions define the physical problem: + +$$ +\boxed{ +\text{differential equation} ++ +\text{domain} ++ +\text{boundary conditions} +\longrightarrow +\text{physical solution}. +} +$$ + +A proposed function that satisfies Laplace's equation is therefore only a possible solution. It becomes a solution to the physical problem only if it also satisfies the relevant boundary conditions. + +## A periodic gravity anomaly + +Now consider a more interesting problem. Imagine that the mass distribution below a locally flat reference surface at $z=0$ varies periodically in the $x$ direction. We might expect the resulting perturbation of the gravitational potential above it to have a similar horizontal variation. + +Suppose that the anomaly at the reference surface is + +$$ +\delta\Phi(x,0)=A\cos(kx), +$$ + +and that its influence must vanish far above the source, + +$$ +\delta\Phi(x,z)\longrightarrow 0 +\qquad\text{as}\qquad z\longrightarrow\infty. +$$ + +Let us therefore postulate + +$$ +\delta\Phi(x,z) += +A e^{-\alpha z}\cos(kx), +$$ + +where $\alpha>0$ describes how rapidly the potential perturbation decreases with height. This proposed form already satisfies both boundary conditions. We have not yet shown, however, that it satisfies Laplace's equation. + +```{admonition} Interpreting the wavenumber +:class: tip + +The **wavenumber** $k$ measures how rapidly a periodic pattern varies in space. It is related to the horizontal wavelength by + +$$ +\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. +$$ + +The function $H_z(k)$ is a **spatial transfer function**. It tells us how the amplitude of each horizontal spatial-frequency component changes between the reference surface and the observation height. Since $H_z(k)$ is close to one for small $k$ but rapidly approaches zero for large $k$, observing the field at altitude acts as a **spatial low-pass filter**: broad features pass more easily than fine details. + +This is directly analogous to the frequency response of a system, which you may encounter in another course. There, a transfer function describes how different temporal frequencies are amplified or attenuated. Here the independent frequency variable is the spatial wavenumber $k$, measured in inverse metres, rather than a temporal frequency measured in hertz. Components of the gravity field and its gradients acquire additional factors of $k$ when we differentiate the potential, but they retain the same exponential attenuation with height. +``` + +```{admonition} More than gravity +:class: note + +Laplace's equation is not specifically an equation of gravity. It appears whenever a potential-like quantity has no sources within the region under consideration. + +For example, it describes the electric potential in a region without electric charge and the steady-state temperature in a region without heat sources. + +The physical quantities are different, but the mathematical problem is the same: + +$$ +\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/_toc.yml b/book/_toc.yml index a5d984c..f7a7eb9 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -24,14 +24,17 @@ parts: - file: 2_potential_fields/introduction/work.md - file: 2_potential_fields/introduction/conservative_fields.md - file: 2_potential_fields/introduction/potential.md - - file: 2_potential_fields/introduction/earth_gravity_work.ipynb - file: 2_potential_fields/introduction/helmholtz_decomposition.md - file: 2_potential_fields/gravity_field/intro.md sections: - file: 2_potential_fields/gravity_field/the_gravity_field.md - file: 2_potential_fields/gravity_field/spherically_symmetric_bodies.md - file: 2_potential_fields/gravity_field/poisson_laplace_equations.md - - file: 2_potential_fields/gravity_field/uniform_cylinder_laplacian.ipynb + - file: 2_potential_fields/gravity_field/postulating_laplace_solutions.md + - file: 2_potential_fields/gravity_field/making_of.md + sections: + - file: 2_potential_fields/introduction/earth_gravity_work.ipynb + - file: 2_potential_fields/gravity_field/uniform_cylinder_laplacian.ipynb - file: 2_potential_fields/magnetic_field/intro.md # - file: 2_potential_fields/gradient_fields.md # - file: 2_potential_fields/divergence_free_fields.md diff --git a/book/references.bib b/book/references.bib index 4652c71..c071fb8 100644 --- a/book/references.bib +++ b/book/references.bib @@ -36,3 +36,10 @@ @misc{noaa_geoid_modeling url = {https://geodesy.noaa.gov/research/geopotential-datums/modeling-calculations.shtml}, note = {Accessed 27 August 2026} } + +@misc{esa_goce_operations, + author = {{European Space Agency}}, + title = {{GOCE} Operations}, + url = {https://www.esa.int/Enabling_Support/Operations/GOCE_operations}, + note = {Accessed 1 September 2026} +} From fea1b3b62a566e7342e328a147b65c617d8ffe73 Mon Sep 17 00:00:00 2001 From: Paco Lopez-Dekker Date: Tue, 1 Sep 2026 19:17:21 +0200 Subject: [PATCH 6/6] added a draft of a density anomaly section, needs work --- .../gravity_field/density_anomalies.md | 265 ++++++++++++++++++ .../poisson_laplace_equations.md | 55 +++- .../postulating_laplace_solutions.md | 2 +- book/_toc.yml | 1 + 4 files changed, 315 insertions(+), 8 deletions(-) create mode 100644 book/2_potential_fields/gravity_field/density_anomalies.md 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/poisson_laplace_equations.md b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md index 493a5ff..241b1b8 100644 --- a/book/2_potential_fields/gravity_field/poisson_laplace_equations.md +++ b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md @@ -320,16 +320,56 @@ $$ The potential is not constant and the gravity field is not zero, yet Laplace's equation is satisfied because there is no mass in the exterior region. For a point mass, the origin itself must be excluded from this calculation: that is precisely where the source is located and where $-GM/r$ is singular. -## Checking the uniform sphere +## Guided exercise: checking the uniform sphere -Inside the uniform sphere, Equation {eq}`eq:uniform-sphere-potential` gives +We have just verified that the potential outside a uniform sphere satisfies Laplace's equation. Inside the sphere the density is non-zero, so the potential should instead satisfy Poisson's equation. Let us check this directly. + +```{admonition} Verify the interior potential +:class: exercise + +For a uniform sphere of radius $R$, total mass $M$, and density $\rho_0$, Equation {eq}`eq:uniform-sphere-potential` gives the interior potential $$ \Phi(r) -=-\frac{GM}{2R}\left(3-\frac{r^2}{R^2}\right). +=-\frac{GM}{2R}\left(3-\frac{r^2}{R^2}\right), +\qquad 0\leq r0. -$$ +$$ (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. diff --git a/book/_toc.yml b/book/_toc.yml index f7a7eb9..e807a50 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -31,6 +31,7 @@ parts: - file: 2_potential_fields/gravity_field/spherically_symmetric_bodies.md - file: 2_potential_fields/gravity_field/poisson_laplace_equations.md - file: 2_potential_fields/gravity_field/postulating_laplace_solutions.md + - file: 2_potential_fields/gravity_field/density_anomalies.md - file: 2_potential_fields/gravity_field/making_of.md sections: - file: 2_potential_fields/introduction/earth_gravity_work.ipynb

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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..5ecec0e --- /dev/null +++ b/book/2_potential_fields/gravity_field/poisson_laplace_equations.md @@ -0,0 +1,231 @@ +# 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. There is also a **local** relation between the potential and the density. This relation is Poisson's equation. + +## The Laplacian + +The Laplacian of a scalar field is 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) + +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 $z$. The three-dimensional Laplacian therefore reduces exactly 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 + +For any closed surface $S$, the gravitational flux depends on the 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) + +The area vector $\mathrm{d}\vec{A}$ points outwards. Since gravity points towards positive mass, the flux is negative. Writing the enclosed mass as a volume integral and applying the divergence theorem 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 spherical body is + +$$ +\Phi(r)=-\frac{GM}{r}. +$$ + +For a spherically symmetric scalar field, the Laplacian is + +$$ +\nabla^2\Phi +=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r} +\left(r^2\frac{\mathrm{d}\Phi}{\mathrm{d}r}\right). +$$ (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. + +## Checking the uniform sphere + +Inside the uniform sphere, Equation {eq}`eq:uniform-sphere-potential` gives + +$$ +\Phi(r) +=-\frac{GM}{2R}\left(3-\frac{r^2}{R^2}\right). +$$ + +Applying the radial Laplacian gives + +$$ +\begin{aligned} +\nabla^2\Phi +&=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r} +\left(r^2\frac{GM}{R^3}r\right)\\ +&=\frac{1}{r^2}\frac{\mathrm{d}}{\mathrm{d}r} +\left(\frac{GM}{R^3}r^3\right) +=\frac{3GM}{R^3}. +\end{aligned} +$$ + +Using $M=4\pi\rho_0R^3/3$, this becomes + +$$ +\nabla^2\Phi=4\pi G\rho_0, +$$ + +as required by Poisson's equation. The same potential therefore satisfies Poisson's equation inside the matter and Laplace's equation outside it. + +At the surface of an ordinary body with a finite density, $\Phi$ and the normal component of $\vec{g}$ remain continuous, although the second derivatives of $\Phi$ need not be continuous. This is why the uniform-sphere solutions join continuously at $r=R$. An idealized infinitely thin surface layer requires more care because it concentrates mass directly on the boundary. + +## Why these equations are useful + +The integral formulation adds the contribution from every mass element directly. Poisson's and Laplace's equations offer a different strategy: solve a differential equation in each region and use physical boundary conditions to connect the solutions. Typically, we specify that + +- $\Phi$ approaches zero far from a bounded mass distribution; +- $\Phi$ remains finite wherever there is no point-like source; +- $\Phi$ is continuous across material boundaries; +- the resulting field $\vec{g}=-\vec{\nabla}\Phi$ has the required symmetry and boundary behaviour. + +This approach becomes especially powerful for layered bodies and for regions where the density vanishes. Much of the difficulty shifts from carrying out a volume integral to finding the solution of a differential equation that satisfies the geometry and boundary conditions. + +```{admonition} Exercise: identifying the governing equation +:class: exercise + +Consider the uniform thick shell from the previous section. + +1. State whether $\Phi$ satisfies Poisson's equation or Laplace's equation in the cavity, in the shell material, and outside the shell. +2. The gravity field vanishes in the cavity. What does this imply about the potential there? +3. Explain why the potential in the cavity need not be zero. +4. Use Equation {eq}`eq:radial-scalar-laplacian` to show that the most general spherically symmetric solution of Laplace's equation is + + $$ + \Phi(r)=A+\frac{B}{r}. + $$ + +5. Which value of $B$ is allowed in a source-free region containing the centre? Which value of $A$ is allowed in an exterior region where $\Phi\rightarrow0$ at infinity? +``` diff --git a/book/2_potential_fields/gravity_field/uniform_cylinder_laplacian.ipynb b/book/2_potential_fields/gravity_field/uniform_cylinder_laplacian.ipynb new file mode 100644 index 0000000..fc839e6 --- /dev/null +++ b/book/2_potential_fields/gravity_field/uniform_cylinder_laplacian.ipynb @@ -0,0 +1,177 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "cylinder-introduction", + "metadata": {}, + "source": [ + "# Potential and Laplacian of a Uniform Cylinder\n", + "\n", + "This notebook visualizes the gravitational potential of an infinitely long cylinder with uniform density. The cylinder is parallel to the $z$-axis, so the potential depends only on $x$ and $y$. We use dimensionless units with $G=\\lambda=R=1$, where $\\lambda$ is the mass per unit length and $R$ is the cylinder radius." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "cylinder-model", + "metadata": { + "execution": { + "iopub.execute_input": "2026-08-27T16:32:06.531476Z", + "iopub.status.busy": "2026-08-27T16:32:06.531252Z", + "iopub.status.idle": "2026-08-27T16:32:07.028961Z", + "shell.execute_reply": "2026-08-27T16:32:07.028512Z" + } + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "from pathlib import Path\n", + "\n", + "G = 1.0\n", + "line_mass = 1.0\n", + "R = 1.0\n", + "rho_0 = line_mass / (np.pi * R**2)\n", + "\n", + "def cylinder_potential(x, y):\n", + " r = np.hypot(x, y)\n", + " phi = np.empty_like(r, dtype=float)\n", + " inside = r <= R\n", + " phi[inside] = G * line_mass * (r[inside]**2 / R**2 - 1)\n", + " phi[~inside] = 2 * G * line_mass * np.log(r[~inside] / R)\n", + " return phi\n", + "\n", + "def cylinder_gravity(x, y):\n", + " x, y = np.broadcast_arrays(np.asarray(x, dtype=float), np.asarray(y, dtype=float))\n", + " r = np.hypot(x, y)\n", + " factor = np.empty_like(r)\n", + " inside = r <= R\n", + " factor[inside] = -2 * G * line_mass / R**2\n", + " factor[~inside] = -2 * G * line_mass / r[~inside]**2\n", + " return factor * x, factor * y\n", + "\n", + "def figure_directory():\n", + " cwd = Path.cwd()\n", + " if cwd.name == 'gravity_field':\n", + " result = cwd / 'figures'\n", + " elif cwd.name == '2_potential_fields':\n", + " result = cwd / 'gravity_field' / 'figures'\n", + " elif (cwd / '2_potential_fields').exists():\n", + " result = cwd / '2_potential_fields' / 'gravity_field' / 'figures'\n", + " else:\n", + " result = cwd / 'book' / '2_potential_fields' / 'gravity_field' / 'figures'\n", + " result.mkdir(parents=True, exist_ok=True)\n", + " return result\n", + "\n", + "figure_dir = figure_directory()" + ] + }, + { + "cell_type": "markdown", + "id": "numerical-laplacian-heading", + "metadata": {}, + "source": [ + "## Calculating the two-dimensional Laplacian\n", + "\n", + "The potential is sampled on a regular grid. We then approximate $\\partial^2\\Phi/\\partial x^2+\\partial^2\\Phi/\\partial y^2$ using finite differences. Poisson's equation predicts $\\nabla^2\\Phi/(4\\pi G\\rho_0)=1$ inside the cylinder and $0$ outside it." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "cylinder-laplacian-figure", + "metadata": { + "execution": { + "iopub.execute_input": "2026-08-27T16:32:07.030589Z", + "iopub.status.busy": "2026-08-27T16:32:07.030464Z", + "iopub.status.idle": "2026-08-27T16:32:07.603257Z", + "shell.execute_reply": "2026-08-27T16:32:07.602682Z" + } + }, + "outputs": [], + "source": [ + "coordinates = np.linspace(-2.2 * R, 2.2 * R, 401)\n", + "X, Y = np.meshgrid(coordinates, coordinates)\n", + "Phi = cylinder_potential(X, Y)\n", + "spacing = coordinates[1] - coordinates[0]\n", + "dPhi_dx = np.gradient(Phi, spacing, axis=1)\n", + "dPhi_dy = np.gradient(Phi, spacing, axis=0)\n", + "laplacian = (\n", + " np.gradient(dPhi_dx, spacing, axis=1)\n", + " + np.gradient(dPhi_dy, spacing, axis=0)\n", + ")\n", + "normalized_laplacian = laplacian / (4 * np.pi * G * rho_0)\n", + "\n", + "fig, (ax_phi, ax_lap) = plt.subplots(1, 2, figsize=(11.5, 4.9), constrained_layout=True)\n", + "\n", + "potential_levels = np.linspace(Phi.min(), Phi.max(), 28)\n", + "potential_map = ax_phi.contourf(\n", + " X / R, Y / R, Phi / (G * line_mass),\n", + " levels=potential_levels / (G * line_mass), cmap='coolwarm'\n", + ")\n", + "ax_phi.contour(X / R, Y / R, Phi / (G * line_mass), levels=12, colors='white',\n", + " linewidths=0.45, alpha=0.6)\n", + "arrow_coordinates = np.linspace(-2.0 * R, 2.0 * R, 17)\n", + "Xq, Yq = np.meshgrid(arrow_coordinates, arrow_coordinates)\n", + "gx, gy = cylinder_gravity(Xq, Yq)\n", + "ax_phi.quiver(\n", + " Xq / R, Yq / R, gx / (G * line_mass / R), gy / (G * line_mass / R),\n", + " color='#202020', scale=18, width=0.006, pivot='mid'\n", + ")\n", + "fig.colorbar(potential_map, ax=ax_phi, label=r'$\\Phi/(G\\lambda)$', shrink=0.84)\n", + "\n", + "laplacian_map = ax_lap.contourf(\n", + " X / R, Y / R, np.clip(normalized_laplacian, 0, 1),\n", + " levels=np.linspace(0, 1, 21), cmap='viridis'\n", + ")\n", + "fig.colorbar(laplacian_map, ax=ax_lap,\n", + " label=r'$\\nabla^2\\Phi/(4\\pi G\\rho_0)$', shrink=0.84)\n", + "\n", + "theta = np.linspace(0, 2 * np.pi, 500)\n", + "for ax in (ax_phi, ax_lap):\n", + " ax.plot(np.cos(theta), np.sin(theta), color='white', lw=1.6)\n", + " ax.plot(np.cos(theta), np.sin(theta), color='0.15', lw=0.7, alpha=0.8)\n", + " ax.set(xlim=(-2.2, 2.2), ylim=(-2.2, 2.2), xlabel='$x/R$', ylabel='$y/R$')\n", + " ax.set_aspect('equal')\n", + "\n", + "ax_phi.set_title('Potential and gravity field')\n", + "ax_lap.set_title('Laplacian reveals the source')\n", + "fig.savefig(figure_dir / 'uniform_cylinder_laplacian.png', dpi=200, bbox_inches='tight')\n", + "plt.close(fig)" + ] + }, + { + "cell_type": "markdown", + "id": "cylinder-static-figure", + "metadata": {}, + "source": [ + "```{figure} figures/uniform_cylinder_laplacian.png\n", + ":width: 100%\n", + "\n", + "Potential, gravity field, and numerically calculated Laplacian of a uniform infinite cylinder.\n", + "```" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.13" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/book/_toc.yml b/book/_toc.yml index fd3b074..a5d984c 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -30,6 +30,8 @@ parts: sections: - file: 2_potential_fields/gravity_field/the_gravity_field.md - file: 2_potential_fields/gravity_field/spherically_symmetric_bodies.md + - file: 2_potential_fields/gravity_field/poisson_laplace_equations.md + - file: 2_potential_fields/gravity_field/uniform_cylinder_laplacian.ipynb - file: 2_potential_fields/magnetic_field/intro.md # - file: 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