where u and y measure distance along the two perpendicular horizontal directions, z measures distance in the vertical direction, and α (alpha) is a positive number that determines the shape of the cone. (Our reason for using u instead of x for one of the horizontal coordinates will become clear soon.) The apex of this cone, where u = y = 0, is at z = 0. A plane that cuts the cone at an oblique angle can be defined as the set of points satisfying the condition that
where β (beta) and γ (gamma) are two more numbers, which respectively specify the tilt and height of the plane. (We are defining the coordinates so that the plane is parallel to the y-axis.) Combining Eq. (9) with the square of Eq. (8) gives
u2 + y2 = α2(βu + γ)2
or equivalently
This is the same as the defining Eq. (1) if we identify
Note that this gives e = αβ, so the eccentricity depends on the shape of the cone and on the tilt of the plane cutting the cone, but not on the plane’s height.
19. Elongations and Orbits of the Inner Planets
One of the great achievements of Copernicus was to work out definite values for the relative sizes of planetary orbits. A particularly simple example is the calculation of the radii of the orbits of the inner planets from the maximum apparent distance of these planets from the Sun.
Consider the orbit of one of the inner planets, Mercury or Venus, in the approximation that it and the Earth’s orbit are both circles with the Sun at the center. At what is called “maximum elongation,” the planet is seen at the greatest angular distance θmax (thetamax) from the Sun. At this time, the line from the Earth to the planet is tangent to the planet’s orbit, so the angle between this line and the line between the Sun and the planet is a right angle. These two lines and the line from the Sun to the Earth thus form a right triangle. (See Figure 13.) The hypotenuse of this triangle is the line between the Earth and the Sun, so the ratio of the distance rP between the planet and the Sun to the distance rE of the Earth from the Sun is the sine of θmax. Here is a table of the angles of maximum elongation, their sines, and the actual orbital radii rP of Mercury and Venus, in units of the radius rE of the Earth’s orbit:
Figure 13. The positions of the Earth and an inner planet (Mercury or Venus) when the planet is at its maximum apparent distance from the Sun. The circles are the orbits of the Earth and planet.
The small discrepancies between the sine of θmax and the observed ratios rP/rE of the orbital radii of the inner planets and the Earth are due to the departure of these orbits from perfect circles with the Sun at the center, and to the fact that the orbits are not in precisely the same plane.
20. Diurnal Parallax
Consider a “new star” or another object that either is at rest with respect to the fixed stars or moves very little relative to the stars in the course of a day. Suppose that it is much closer to Earth than the stars. One can assume that the Earth rotates once a day on its axis from east to west, or that this object and the stars revolve around the Earth once a day from west to east; in either case, because we see the object in different directions at different times of night, its position will seem to shift during every evening relative to the stars. This is called the “diurnal parallax” of the object. A measurement of the diurnal parallax allows the determination of the distance of the object, or if it is found that the diurnal parallax is too small to be measured, this gives a lower limit to the distance.
To calculate the amount of this angular shift, consider the object’s apparent position relative to the stars seen from a fixed observatory on Earth, when the object just rises above the horizon, and when it is highest in the sky. To facilitate the calculation, we will consider the case that is simplest geometrically: the observatory is on the equator, and the object is in the same plane as the equator. Of course, this does not accurately give the diurnal parallax of the new star observed by Tycho, but it will indicate the order of magnitude of that parallax.
The line to the object from this observatory when the object just rises above the horizon is tangent to the Earth’s surface, so the angle between this line and the line from the observatory to the center of the Earth is a right angle. These two lines, together with the line from the object to the center of the Earth, thus form a right triangle. (See Figure 14.) Angle θ (theta) of this triangle at the object has a sine equal to the ratio of the opposite side, the radius rE of the Earth, to the hypotenuse, the distance d of the object from the center of the Earth. As shown in the figure, this angle is also the apparent shift of the position of the object relative to the stars during the time between when it rises above the horizon and when it is highest in the sky. The total shift in its position from when it rises above the horizon to when it sets below the horizon is 2θ.
For instance, if we take the object to be at the distance of the Moon, then d 250,000 miles, while rE 4,000 miles, so sin θ 4/250, and therefore θ 0.9°, and the diurnal parallax is 1.8°. From a typical spot on Earth, such as Hven, to an object at a typical location in the sky like the new star of 1572, the diurnal parallax is smaller, but still of the same order of magnitude, in the neighborhood of 1°. This is more than large enough for it to have been detected by a naked-eye astronomer as expert as Tycho Brahe, but Tycho could not detect any diurnal parallax, so he was able to conclude that the new star of 1572 is farther than the Moon. On the other hand, there was no difficulty in measuring the diurnal parallax of the Moon itself, and in this way finding the distance of the Moon from the Earth.
Figure 14. Use of diurnal parallax to measure the distance d from the Earth to some object. Here the view is from a point above the Earth’s north pole. For simplicity, the observer is supposed to be on the equator, and the object is in the same plane as the equator. The two lines separated by an angle θ are the lines of sight to the object when it just arises above the horizon and six hours later, when it is directly above the observer.
21. The Equal-Area Rule, and the Equant
According to Kepler’s first law, the planets, including the Earth, each go around the Sun on an elliptical orbit, but the Sun is not at the center of the ellipse; it is at an off-center point on the major axis, one of the two foci of the ellipse. (See Technical Note 18.) The eccentricity e of the ellipse is defined so that the distance of each focus from the center of the ellipse is ea, where a is half the length of the major axis of the ellipse. Also, according to Kepler’s second law, the speed of each planet in its orbit is not constant, but varies in such a way that the line from the Sun to the planet sweeps out equal areas in equal times.
There is a different approximate way of stating the second law, closely related to the old idea of the equant used in Ptolemaic astronomy. Instead of considering the line from the Sun to the planet, consider the line to the planet from the other focus of the ellipse, the empty focus. The eccentricity e of some planetary orbits is not negligible, but e2 is very small for all planets. (The most eccentric orbit is that of Mercury, for which e = 0.206 and e2 = 0.042; for the Earth, e2 = 0.00028.) So it is a good approximation in calculating the motions of the planets to keep only terms that are independent of the eccentricity e or proportional to e, neglecting all terms proportional to e2 or higher powers of e. In this approximation, Kepler’s second law is equivalent to the statement that the line from the empty focus to the planet sweeps out equal angles in equal times. That is, the line between the empty focus of the ellipse and the planet rotates around that focus at a constant rate.
Specifically, we show below that if is the rate at which area is swept out by the line from the Sun to the planet, and (dotted phi) is the rate of change of the angle ϕ between the major axis of the ellipse and the line from the empty focus to the planet, then
where O(e2) denotes terms proportional to e2 or higher powers of e, and R is a number whose value depends on the units we use to measure angles. If we measure angles in degrees, then R = 360° /2π = 57.293 . . . °, an angle known as a “radian.” Or we can measure angles in radians, in which case we take R = 1. Kepler’s second law t
ells us that in a given time interval the area swept out by the line from the Sun to the planet is always the same; this means that is constant, so ϕ is constant, up to terms proportional to e2. So to a good approximation in a given time interval the angle swept out from the empty focus of the ellipse to the planet is also always the same.
Now, in the theory described by Ptolemy, the center of each planet’s epicycle goes around the Earth on a circular orbit, the deferent, but the Earth is not at the center of the deferent. Instead, the orbit is eccentric—the Earth is at a point a small distance from the center. Furthermore, the velocity at which the center of the epicycle goes around the Earth is not constant, and the rate at which the line from the Earth to this center swivels around is not constant. In order to account correctly for the apparent motion of the planets, the device of the equant was introduced. This is a point on the other side of the center of the deferent from the Earth, and at an equal distance from the center. The line from the equant (rather than from the Earth) to the center of the epicycle was supposed to sweep out equal angles in equal times.
It will not escape the reader’s notice that this is very similar to what happens according to Kepler’s laws. Of course, the roles of the Sun and Earth are reversed in Ptolemaic and Copernican astronomy, but the empty focus of the ellipse in the theory of Kepler plays the same role as the equant in Ptolemaic astronomy, and Kepler’s second law explains why the introduction of the equant worked well in explaining the apparent motion of the planets.
For some reason, although Ptolemy introduced an eccentric to describe the motion of the Sun around the Earth, he did not use an equant in this case. With this final equant included (and with some additional epicycles introduced to account for the large departure of Mercury’s orbit from a circle), the Ptolemaic theory could account very well for the apparent motions of the planets.
Here is the proof of Eq. (1). Define θ as the angle between the major axis of the ellipse and the line from the Sun to the planet, and recall that ϕ is defined as the angle between the major axis and the line from the empty focus to the planet. As in Technical Note 18, define r+ and r– as the lengths of these lines—that is, the distances from the Sun to the planet and from the empty focus to the planet, respectively, given (according to that note) by
where x is the horizontal coordinate of the point on the ellipse—that is, it is the distance from this point to a line cutting through the ellipse along its minor axis. The cosine of an angle (symbolized cos) is defined in trigonometry by considering a right triangle with that angle as one of the vertices; the cosine of the angle is the ratio of the side adjacent to that angle to the hypotenuse of the triangle. Hence, referring to Figure 15,
Figure 15. Elliptical motion of planets. The orbit’s shape here is an ellipse, which (as in Figure 12) has ellipticity 0.8, much larger than the ellipticity of any planetary orbit in the solar system. The lines marked r+ and r– go respectively to the planet from the Sun and from the empty focus of the ellipse.
We can solve the equation at the left for x:
We then insert this result in the formula for cos ϕ, obtaining in this way a relation between θ and ϕ:
Since both sides of this equation are equal whatever the value of θ, the change in the left-hand side must equal the change in the right-hand side when we make any change in θ. Suppose we make an infinitesimal change δθ (delta theta) in θ. To calculate the change in ϕ, we make use of a principle in calculus, that when any angle α (such as θ or ϕ) changes by an amount δα (delta alpha), the change in cos α is –(δα/R) sin α. Also, when any quantity f, such as the denominator in Eq. (5), changes by an infinitesimal amount δf, the change in 1/f is –δf/f2. Equating the changes in the two sides of Eq. (5) thus gives
Now we need a formula for the ratio of sin ϕ and sin θ. For this purpose, note from Figure 15 that the vertical coordinate y of a point on the ellipse is given by y = r+ sin θ and also by y = r- sin ϕ, so by eliminating y,
Using this in Eq. (6), we have
Now, what is the area swept out by the line from the Sun to the planet when the angle θ is changed by δθ? If we measure angles in degrees, then it is the area of an isosceles triangle with two sides equal to r+, and the third side equal to the fraction 2πr+ × δθ/360° of the circumference 2πr+ of a circle of radius r+. This area is
(A minus sign has been inserted here because we want dA to be positive when ϕ increases; but as we have defined them, ϕ increases when θ decreases, so δϕ is positive when δθ is negative.) Thus Eq. (8) may be written
Taking δA and δϕ to be the area and angle swept out in an infinitesimal time interval δt, and dividing Eq. (10) by δt, we find a corresponding relation between the rates of sweeping out areas and angles:
So far, this is all exact. Now let’s consider how this looks when e is very small. The numerator of the second fraction in Eq. (11) is (1 – e cosθ)2 = 1 – 2e cosθ + e2 cos2 θ, so the terms of zeroth and first order in the numerator and denominator of this fraction are the same, the difference between numerator and denominator appearing only in terms proportional to e2. Equation (11) thus immediately yields the desired result, Eq. (1). To be a little more definite, we can keep the terms in Eq. (11) of order e2:
where O(e3) denotes terms proportional to e3 or higher powers of e.
22. Focal Length
Consider a vertical glass lens, with a convex curved surface in front and a plane surface in back, like the lens that Galileo and Kepler used as the front end of their telescopes. The curved surfaces that are easiest to grind are segments of spheres, and we will assume that the convex front of the lens is a segment of a sphere of radius r. We will also assume throughout that the lens is thin, with a maximum thickness much less than r.
Suppose that a ray of light traveling in the horizontal direction, parallel to the axis of the lens, strikes the lens at point P, and that the line from the center C of curvature (behind the lens) to P makes an angle θ (theta) with the centerline of the lens. The lens will bend the ray of light so that when it emerges from the back of the lens it makes a different angle ϕ with the centerline of the lens. The ray will then strike the centerline of the lens at some point F. (See Figure 16a.) We are going to calculate the distance f of this point from the lens, and show that it is independent of θ, so that all horizontal rays of light striking the lens reach the centerline at the same point F. Thus we can say that the light striking the lens is focused at point F; the distance f of this point from the lens is known as the “focal length” of the lens.
First, note that the arc on the front of the lens from the centerline to P is a fraction θ/360° of the whole circumference 2πr of a circle of radius r. On the other hand, the same arc is a fraction ϕ/360° of the whole circumference 2πf of a circle of radius f. Since these arcs are the same, we have
and therefore, canceling factors of 360° and 2π,
So to calculate the focal length, we need to calculate the ratio of ϕ to θ.
For this purpose, we need to look more closely at what happens to the light ray inside the lens. (See Figure 16b.) The line from the center of curvature C to the point P where a horizontal light ray strikes the lens is perpendicular to the convex spherical surface of the lens at P, so the angle between this perpendicular and the light ray (that is, the angle of incidence) is just θ. As was known to Claudius Ptolemy, if θ is small (as it will be for a thin lens), then the angle α (alpha) between the ray of light inside the glass and the perpendicular (that is, the angle of refraction) will be proportional to the angle of incidence, so that
α = θ / n
Figure 16. Focal length. (a) Definition of focal length. The horizontal dashed line is the axis of the lens. Horizontal lines marked with arrows indicate rays of light that enter the lens parallel to this axis. One ray is shown entering the lens at point P, where the ray makes a small angle θ to a line from the center of curvature C that is perpendicular to the convex spherical surface at P; this ray is bent by the lens to make an
angle ϕ to the lens axis and strikes this axis at focal point F, at a distance f from the lens. This is the focal length. With ϕ proportional to θ, all horizontal rays are focused to this point. (b) Calculation of focal length. Shown here is a small part of the lens, with the slanted hatched solid line on the left indicating a short segment of the convex surface of the lens. The solid line marked with an arrow shows the path of a ray of light that enters the lens at P, where it makes a small angle θ to the normal to the convex surface. This normal is shown as a slanted dotted line, a segment of the line from P to the center of curvature of the lens, which is beyond the borders of this figure. Inside the lens this ray is refracted so that it makes an angle α with this normal, and then is refracted again when it leaves the lens so that it makes an angle ϕ with the normal to the planar back surface of the lens. This normal is shown as a dotted line parallel to the axis of the lens.
where n > 1 is a constant, known as the “index of refraction,” that depends on the properties of glass and the surrounding medium, typically air. (It was shown by Fermat that n is the speed of light in air divided by the speed of light in glass, but this information is not needed here.) The angle β (beta) between the light ray inside the glass and the centerline of the lens is then
β = θ − a = (1 − 1/n)θ
This is the angle between the light ray and the normal to the flat back surface of the lens when the light ray reaches this surface. On the other hand, when the light ray emerges from the back of the lens it makes a different angle ϕ (phi) to the normal to the surface. The relation between ϕ and β is the same as if the light were going in the opposite direction, in which case ϕ would be the angle of incidence and β the angle of refraction, so that β = ϕ/n, and therefore
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