5: Gravitational Field and Potential
- Page ID
- 6818
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This chapter deals with the calculation of gravitational fields and potentials in the vicinity of various shapes and sizes of massive bodies. The reader who has studied electrostatics will recognize that this is all just a repeat of what he or she already knows.
- 5.1: Introduction
- This page discusses calculating gravitational fields and potentials, comparing them to electrostatics. It emphasizes similarities in force equations while noting key differences: the lack of negative masses and the uniform acceleration of all masses in a gravitational field, in contrast to charged particles. The text also mentions a rationalized gravitational constant but does not elaborate on it.
- 5.2: Gravitational Field
- The region around a gravitating body (by which I merely mean a mass, which will attract other masses in its vicinity) is a gravitational field. Although I have used the words “around” and “in its vicinity”, the field in fact extents to infinity. All massive bodies (and by “massive” I mean any body having the property of mass, however little) are surrounded by a gravitational field, and all of us are immersed in a gravitational field.
- 5.3: Newton's Law of Gravitation
- Newton noted that the ratio of the centripetal acceleration of the Moon in its orbit around the Earth to the acceleration of an apple falling to the surface of the Earth was inversely as the squares of the distances of Moon and apple from the centre of the Earth. Together with other lines of evidence, this led Newton to propose his universal law of gravitation:
- 5.4: The Gravitational Fields of Various Bodies
- This page discusses calculating electric fields around different objects, similar to introductory electricity courses. While some concepts may not relate directly to celestial mechanics, they offer useful practice in field and potential calculations, establishing a foundational understanding for future studies.
- 5.5: Gauss's Theorem
- The total normal outward gravitational flux through a closed surface is equal to \(−4 \pi G\) times the total mass enclosed by the surface.
- 5.6: Calculating Surface Integrals
- While the concept of a surface integral sounds easy enough, how do we actually calculate one in practice?
- 5.7: Potential
- We have defined only the potential difference between two points. If we wish to define the potential at a point, it is necessary arbitrarily to define the potential at a particular point to be zero. We might, for example define the potential at floor level to be zero, in which case the potential at a height h above the floor is gh ; equally we may elect to define the potential at the level of the laboratory bench top to be zero, where the potential at a height z above the bench top is gz.
- 5.8: The Gravitational Potentials Near Various Bodies
- This page highlights the benefits of using scalar potentials for gravitational calculations over vector fields. It notes that determining gravitational potential simplifies calculations and enables easier derivation of the gravitational field through the potential's gradient, thereby enhancing problem-solving efficiency in gravitational physics.
- 5.9: Work Required to Assemble a Uniform Sphere
- This page describes the assembly of a uniform solid sphere, addressing the work done against gravitational forces as atoms are brought together from infinite separation. It derives the potential energy at the surface and calculates the work required to add layers to the sphere, ultimately concluding with the formula for total work: \(-\frac{3GM^2}{5a}\).
- 5.10: Nabla, Gradient and Divergence
- We are going to meet, in this section, the symbol ∇ . In North America it is generally pronounced “del”, although in the United Kingdom and elsewhere one sometimes hears the alternative pronunciation “nabla”, called after an ancient Assyrian harp-like instrument of approximately that shape.
- 5.11: Legendre Polynomials
- In this section we cover just enough about Legendre polynomials to be useful in the following section.
- 5.12: Gravitational Potential of any Massive Body
- This page covers gravitational potential at an external point of a massive body, emphasizing moments of inertia. It details the calculation of potential \(ψ\) through mass elements and their positions relative to the center of mass. Key equations for symmetric shapes, particularly oblate spheroids, are derived, illustrating the influence of moments of inertia on gravitational potential.
- 5.13: Pressure at the Centre of a Uniform Sphere
- What is the pressure at the centre of a sphere of radius a and of uniform density ρ?
Thumbnail: Gravitational field lines around the Earth. (Public Domain; Sjlegg).