📚 Gravitational Fields | 引力场
Gravitational fields are fundamental to our understanding of the universe, from the orbital dance of planets to the trajectory of a thrown ball. In IB Physics, Topic D.1 explores the concepts of gravitational force, field strength, potential, and their applications to celestial mechanics. This article provides a comprehensive yet concise revision of these key ideas, pairing English explanations with Chinese translations to support bilingual learning.
引力场是理解宇宙的基础，从行星的轨道运行到抛出的球的轨迹。在IB物理中，主题D.1探讨了引力、场强、势等概念及其在天体力学中的应用。本文对这些核心概念进行了全面而简洁的复习，并配以中英文解释，支持双语学习。
1. Newton's Law of Gravitation | 牛顿万有引力定律
Every point mass attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The equation is: F = G M m / r², where G = 6.674 × 10⁻¹¹ N m² kg⁻² is the universal gravitational constant.
任何两个质点之间都存在引力，该力与两质点质量的乘积成正比，与它们中心之间距离的平方成反比。公式为：F = G M m / r²，其中 G = 6.674 × 10⁻¹¹ N m² kg⁻² 是万有引力常数。
The force acts along the line joining the centres of mass and is always attractive. In planetary contexts, we often treat spherical bodies as point masses located at their centres, which is justified by Newton's shell theorem.
力的方向沿两质心连线且始终为吸引力。在行星背景下，常将球体视为位于其中心的质点，这由牛顿壳层定理证明。
2. Gravitational Field and Field Strength | 引力场与场强
A gravitational field is a region of space where a mass experiences a force. The gravitational field strength g is defined as the force per unit mass on a small test mass placed at that point: g = F / m. It is a vector quantity pointing toward the source mass.
引力场是指空间中的一个区域，其中放置的质量会受到力的作用。引力场强度 g 定义为单位质量的质点在这一点所受到的引力，即 g = F / m。它是一个矢量，方向指向场源质量。
For a spherical mass M, the field strength at a distance r from its centre (outside the sphere) is given by g = G M / r². Near the Earth's surface, g is approximately 9.81 N kg⁻¹ (or m s⁻²) and is taken as uniform in simple problems.
对于一个球形质量 M，在距其中心 r 处（球体外）的场强为 g = G M / r²。在地球表面附近，g 约为 9.81 N kg⁻¹（或 m s⁻²），并在简单问题中视为均匀场。
Field lines are used to represent gravitational fields: for a point mass, they are radial and inward; for a uniform field, they are parallel and equally spaced. The strength is proportional to the density of field lines.
力线用于表示引力场：点质量的力线为径向向内；均匀场中力线平行且等距。场强与力线密度成正比。
3. Gravitational Potential | 引力势
Gravitational potential V at a point is defined as the work done per unit mass to bring a small test mass from infinity to that point. The potential is zero at infinity, and near a mass it is negative. For a spherical mass, V = −G M / r.
引力势 V 定义为将单位质量的检验质量从无穷远移至该点所做的功。无穷远处势为零，靠近质量处势为负值。对于球形质量，V = −G M / r。
Because the force is attractive, work is done by the field when a mass moves closer, decreasing potential energy; thus V becomes more negative as r decreases. The potential is a scalar field, and the field strength is the negative gradient of the potential: g = −dV/dr in the radial direction.
由于引力是吸引力，当质量靠近场源时场做正功，势能减小；因此随着 r 减小，V 变得更负。势是标量场，场强是势的负梯度：沿径向 g = −dV/dr。
4. Gravitational Potential Energy | 引力势能
The gravitational potential energy U of a system of two point masses M and m separated by distance r is given by U = −G M m / r. This is the work done by an external agent to assemble the system from infinite separation, assuming zero potential energy at infinity.
两个点质量 M 和 m 相距 r 时系统的引力势能为 U = −G M m / r。这是将两个质量从无穷远分离状态聚集到相距 r 时外力所做的功，无穷远处势能为零。
For a mass m near Earth's surface, the change in potential energy is often approximated as mgΔh. This approximation is valid only when height changes are small compared to Earth's radius. In orbital mechanics, the exact expression −GMm/r is used.
对于地球表面附近的质量 m，势能变化通常近似为 mgΔh。这个近似仅在高度变化远小于地球半径时成立。在轨道力学中，则需使用精确表达式 −GMm/r。
5. Equipotential Surfaces | 等势面
Equipotential surfaces are surfaces on which the gravitational potential is constant. No work is done when moving a mass along an equipotential surface because the component of the field strength along the surface is zero. These surfaces are always perpendicular to field lines.
等势面是引力势恒定的曲面。沿着等势面移动质量时不做功，因为场强沿曲面切线分量为零。等势面总是与力线垂直。
For a single point mass, equipotential surfaces are concentric spheres centred on the mass. Near a planet's surface, over short distances, the surfaces are approximately horizontal planes, corresponding to the uniform field approximation.
对于单个点质量，等势面是以质量为中心的同心球面。在地球表面附近小范围内，这些面近似为水平面，对应于均匀场近似。
Spacing between equipotential surfaces for equal potential intervals indicates the field strength: closer spacing means a stronger field, as more work is done per unit distance.
相邻等势面（等势差相同）的间距反映场强大小：间距越小，场强越大，因为单位距离做功更多。
6. Graphing Fields and Potentials | 场与势的图像
Graphs of g versus r and V versus r provide insight into gravitational interactions. The field strength g follows an inverse-square law (g ∝ 1/r²) outside the mass, while potential V follows V ∝ −1/r. Inside a solid sphere of uniform density, g increases linearly with r (from centre to surface) and V forms a parabola opening downward.
g-r 和 V-r 图像揭示了引力相互作用的规律。在球外，场强 g 遵循平方反比规律（g ∝ 1/r²），而势 V 满足 V ∝ −1/r。在均匀密度的固体球内部，g 随 r 线性增加（从中心到表面），V 则为向下开口的抛物线。
res: g = 0 at the centre, maximum at the surface; V is negative everywhere outside, with V ∝ ro asymptotically. These graphs help visualize the relationship and are commonly examined.
关键特征：中心处 g = 0，表面处 g 最大；球外 V 处处为负，V ∝ −1/r 曲线渐近于零。这些图像有助于直观理解，也是常见考点。
7. Satellite Motion and Orbits | 卫星运动与轨道
A satellite in circular orbit around a planet of mass M has its centripetal force provided by gravity: G M m / r² = m v² / r, hence orbital speed v = √G M / r. The period T is T = 2π r / v = 2π √(r³/(G M)).
绕质量为 M 的行星做圆周运动的卫星所需的向心力由引力提供：G M m / r² = m v² / r，因此轨道速度 v = √G M / r。周期 T = 2π r / v = 2π √(r³/(G M))。
These equations show that outer satellites move slower and have longer periods. For geostationary satellites, the orbital period matches Earth's rotation (24 hours) and they orbit above the equator at a specific radius of about 42,200 km from Earth's centre.
这些公式表明，外层卫星速度较慢、周期较长。对于地球同步卫星，轨道周期等于地球自转周期（24小时），它们位于赤道上空，轨道半径约为地球中心起算 42,200 km。
In elliptical orbits, speed varies: faster at perihelion (closest approach) and slower at aphelion (farthest point), as derived from conservation of angular momentum and energy.
在椭圆轨道中，速度会变化：在近日点最快，在远日点最慢，这可以从角动量和能量守恒推导得出。
8. Kepler's Laws | 开普勒定律
Johannes Kepler formulated three empirical laws of planetary motion:
- First Law (Law of Ellipses): Planets move in elliptical orbits with the Sun at one focus.
- Second Law (Equal Areas): A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- Third Law (Harmonic Law): The square of the orbital period T is proportional to the cube of the semi-major axis r: T² ∝ r³. For circular orbits, T² = (4π²/GM) r³.
开普勒提出了行星运动的三个经验定律：
- 第一定律（椭圆定律）： 行星沿椭圆轨道运动，太阳位于椭圆的一个焦点上。
- 第二定律（面积定律）： 连接行星和太阳的线段在相等时间内扫过相等的面积。
- 第三定律（调和定律）： 轨道周期 T 的平方与半长轴 r 的立方成正比：T² ∝ r³。对于圆轨道，T² = (4π²/GM) r³。
Newton later derived Kepler's laws from his law of gravitation, demonstrating the universal nature of the inverse-square force.
牛顿随后从万有引力定律推导出了开普勒定律，证明了平方反比力的普适性。
9. Escape Speed | 逃逸速度
The escape speed from a planet is the minimum speed an object needs at its surface to completely escape the gravitational field, assuming no further propulsion. By equating kinetic energy to the magnitude of gravitational potential energy, we get: ½ m vₑ² = G M m / r, so vₑ = √2 G M / r.
逃逸速度是一个物体在星球表面为完全脱离引力场所需的最小初速度，假设没有后续推力。令动能等于引力势能的绝对值，得到：½ m vₑ² = G M m / r，因此 vₑ = √2 G M / r。
For Earth, vₑ ≈ 11.2 km s⁻¹. Escape speed does not
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