Physics · Ch 5 — Gravitation

Kepler's Laws

5.2

Kepler's Laws

Kepler's laws of planetary motion describe the actual orbits followed by the planets around the Sun. Kepler published the first two laws in 1609 and the third in 1619, all derived purely from painstaking analysis of Tycho Brahe's decades of naked-eye observational data -- Kepler did not know, and could not derive, WHY the planets should obey these laws; that explanation came only later, from Newton.

Law of orbits (First law): Every planet moves in an elliptical orbit around the Sun, with the Sun located at ONE of the two foci of the ellipse, not at its geometric centre. In the orbit, the point closest to the Sun is called the perihelion and the point farthest from the Sun is called the aphelion; the major axis (through both foci) has length $2a$ where $a$ is the semi-major axis, and the minor axis has length $2b$ where $b$ is the semi-minor axis (Fig. 5.1).

Law of areas (Second law): The line joining a planet to the Sun sweeps out equal areas in equal intervals of time. Because planets do not move with uniform speed -- they move faster when nearer the Sun and slower when farther away -- this law captures exactly how that speed must vary to keep the swept area constant (Fig. 5.2). The law of areas is really a special case of the conservation of angular momentum for any CENTRAL force (a force always directed along the line joining the object to a fixed point, here the Sun). If the Sun is at the origin, a planet of mass m at position $\vec{r}$ with perpendicular momentum component $\vec{p}_\perp$ sweeps area $dA=\frac{1}{2}|\vec{r}\times\vec{v}|\,dt$ in time dt, i.e. $\frac{dA}{dt}=\frac{1}{2}|\vec{r}\times\vec{v}|$ . Writing $\vec{v}=\vec{p}/m$ and using the definition of angular momentum $\vec{L}=\vec{r}\times\vec{p}$ , this becomes $\frac{dA}{dt}=\frac{L}{2m}$ . Since gravity is a central force, angular momentum $\vec{L}$ is conserved (there is no torque about the Sun, as the force always passes through it), so $\frac{dA}{dt}$ is constant -- exactly the law of areas. …

Figure 5.1Fig. 5.1: An ellipse traced by a planet with the Sun at the focus

What this figure shows. A diagram of an elliptical planetary orbit drawn around the Sun. Two foci S and S' are marked inside the ellipse, with the Sun located AT focus S (not at the centre of the ellipse). Point P, the point on the orbit closest to S, is labelled 'Perihelion'; point A, the point farthest from S (on the opposite end of the major axis), is labelled 'Aphelion'. The major axis runs from P to A through both foci and the geometric centre O of the ellipse, with PA = 2a (a = semi-major axis, so PO = OA = a). A minor axis MN is drawn perpendicular to the major axis through O, with MN = 2b (b = semi-minor axis, so MO = ON = b). The figure establishes the geometric vocabulary (foci, perihelion, aphelion, semi-major/semi-min …

Figure 5.2Fig. 5.2: The orbit of a planet P moving around the Sun (law of areas)

What this figure shows. An elliptical orbit with the Sun S at one focus and a planet at a position P on the ellipse, with the connecting radius vector SP drawn. Two or more thin, roughly triangular/wedge-shaped shaded regions are marked at different points along the orbit -- each bounded by two radius-vector positions (a short time interval apart) and the arc of the orbit between them -- representing the AREA swept out by the line SP in a fixed interval of time $\Delta t$ . The shaded wedges are drawn with visibly different arc-lengths (a longer, thinner wedge where the planet is farther from S and moving slowly, and a shorter, fatter wedge where the planet is closer to S and moving fast), while their AREAS are equal, illustrating that a planet sweeps equal areas in equal times -- moving faster near the Sun (perihelion) and slower far from it (aphelion). The perpendicular component of m …

Table T5.1Table 5.1: Kepler's third law -- planetary data confirming $T^2/r^3$ is constant

Planet | Semi-major axis (units of $10^{10}$ m) | Period (years) | $T^2/r^3$ (units of $10^{-34}$ y^2 m^-3)

Mercury | 5.79 | 0.24 | 2.95

Venus | 10.8 | 0.615 | 3.00

Earth | 15.0 | 1 | 2.96

Mars | 22.8 | 1.88 | 2.98

Jupiter | 77.8 | 11.9 | 3.01

Saturn | 143 | 29.5 | 2.98 …

Misc Ex.1Example 5.1: Change in the length of a year if the Earth-Sun distance is tripled / doubled

Worked out. Using Kepler's law of periods in ratio form, $T_2^2/T_1^2=(r_2/r_1)^3$ , with $T_1=365$ days as the present year length: (A) if the distance becomes 3 times the present distance ( $r_2=3r_1$ ), $T_2^2/T_1^2=3^3=27$ , so $T_2=T_1\sqrt{27}\approx365\times5.196\approx1897$ days; (B) if the distance is doubled ( $r_2=2r_1$ ), $T_2^2/T_1^2=2^3=8$ , so $T_2=T_1\sqrt{8}\approx365\times2.828\approx1032$ days. The example demonstrates how strongly period scales with orbital radius via the $r^{3/2}$ dependence. …

Misc Activity.1Constructing an ellipse with two pins and a thread loop (Do you know? activity box)

Worked out. A hands-on construction procedure for drawing an ellipse, defined as the locus of points whose sum of distances from two fixed points (the foci) is constant: (1) insert two tacks/drawing pins A and B a distance 'd' apart into a sheet of drawing paper; (2) tie the two ends of a thread whose length is greater than 2d into a loop and place the loop around A and B; (3) place a pencil inside the loop, pull the thread taut, and move the pencil sideways all the way around while keeping the thread taut -- the pencil traces a complete ellipse with A and B as its two foci. This activity makes concrete the definition of the ellipse used in Kep …

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