Pedal Equation and Kepler Kinematics

Canadian Journal of Physics

Pedal equation and Kepler kinematics

Journal: Canadian Journal of Physics

Manuscript ID cjp-2019-0347.R2

Manuscript Type: Tutorial

Date Submitted by the For11-Aug-2020 Review Only Author:

Complete List of Authors: Nathan, Joseph Amal; Bhabha Atomic Research Centre, Reactor Physics Design Division

Pin and string construction of conics, Pedal equation, Central force field, Keyword: Conservation laws, Trajectories and Kepler laws

Is the invited manuscript for consideration in a Special Not applicable (regular submission) Issue? :

Pedal equation and Kepler kinematics

Joseph Amal Nathan Reactor Physics Design Division Bhabha Atomic Research Center, Mumbai-400 085, India

August 12, 2020 For Review Only Abstract: Kepler’s laws is an appropriate topic which brings out the signif- icance of pedal equation in Physics. There are several articles which obtain the Kepler’s laws as a consequence of the conservation and gravitation laws. This can be shown more easily and ingeniously if one uses the pedal equation of an Ellipse. In fact the complete kinematics of a particle in a attractive central force ﬁeld can be derived from one single pedal form. Though many articles use the pedal equation, only in few the classical procedure (without proof) for obtaining the pedal equation is mentioned. The reason being the classical derivations can sometimes be lengthier and also not simple. In this paper using elementary physics we derive the pedal equation for all conic sections in an unique, short and pedagogical way. Later from the dynamics of a particle in the attractive central force ﬁeld we deduce the single pedal form, which elegantly describes all the possible trajectories. Also for the purpose of completion we derive the Kepler’s laws.

Keywords: Pin & string construction of conics, Pedal equation, Central force ﬁeld, Conservation laws, Trajectories & Kepler laws.

1.Introduction: The accurate observations of Tycho Brahe and the dis- covery of the three laws by Johannes Kepler using it, can be considered an epitome of Observational Science. Kepler after presenting the first two laws in 1609, required 10 years of tediousness, tenacity and pertinence to formulate the third law. This showed what a herculean task it must have been to make such fine and precise sense from the mammoth data. For this extra-ordinary phenomenal work Kepler is considered a central figure of Sci- entific Revolution. Even today he is seen as a magician who pulled tricks out of his hat and his laws like magic which reveals the geometrical plan of the universe. But the secrets of this magic could be understood if the pillars of physics-the laws of conservation, are combined with the Newton’s law of gravitation. Because this combination describes the dynamics of the system, the laws of Kepler which are the kinematics of the system become a consequence. We will also see that this combination brings out much more.

Among several papers on Kepler’s laws, we have listed the relevant ones from 1941 to 2017. A innovative derivation of the elliptical orbits for the Kepler problem is presented in Feynman’s lost lecture [1]. All the three laws as a consequence of the conservation & gravitation laws are discussed in [2],[3] and in [4] based on geometry & gravitational law. Though papers [5],[6],[7],[8],[9],[10],[11],[12],[13] mention to have a simpliﬁed derivation, they discuss only about one or two laws and few among them about their relation to the gravitational law. But none derive the pedal equation and also do not show all the possible trajectories. The classical derivation of the pedal equation for ellipse and hyperbola is shown in [2], along with obtaining the RutherfordFor scattering Review law from the Only hyperbolic trajectory. Though in [14] all trajectories are deduced from the pedal form, the pedal equations and Kepler’s laws are not derived. So to sum up, no article has covered all aspects of this problem. Hence the motivation to present the comprehensive picture using elementary methods to make it easier to follow and appreciate.

2.Classical derivation of pedal Y-axis equation: Refer to Fig-1. For a curve (Y-y1)=m1(X-x1) g(x, y)=0 and a fixed point Pf (u, v), the pedal equation is the relation between r (x1, y1) and β, where r is the distance from Pf to a point P on g and β is the perpendic- g(x,y)=0=G(r, β) r ular distance from Pf to the tangent at 1 β1 (x , y ) P . The fixed point Pf is called the pedal 2 2 β2 point. The quantities r and β called the r 2 (Y-y )=m (X-x ) pedal coordinates are associated with ev- 2 2 2 P (u,v) ery point on g, similar to the Polar co- f ordinates or any other Curvilinear coor- (0,0) X-axis dinate pair. Then the curve g(x, y) = 0 Fig-1 Pedal Coordinates can be represented by a pedal equation G(r, β) = 0. The locus of the foot of the perpendicular from Pf to the tangent at g(x, y) = 0 is called the ‘Pedal Curve’. The pedal curve is usually different from g(x, y) = 0 unless for all points on g, β = r (for example: Circle). See [15] for an introduction to pedal equations and [16] for pedal curves. For this article pedal curve is of no relevance, we require only pedal equation. To find the pedal equation for any curve we use the standard result from coordinate geometry that the perpendicular distance√ from a point (k, l) to a line AX +BY +C =0 is given by |Ak +Bl +C|/ k2 + l2. For the function g(x, y)=0 the slope of the tangent at (x, y) is dy/dx=−gx/gy =m and its equation will be (Y − y) = m(X − x), where gx, gy are partial derivatives. Using the above result from coordinate geometry, the perpendicular distance from Pf (u, v) to the tangent (Y − y) − m(X − x) = 0, 2 2 1/2 β =[(x − u)gx + (y − v)gy]/[gx + gy] . Then the pedal equation G(r, β)=0

is found by eliminating x, y between equations r =[(x−u)2 +(y −v)2]1/2 and β. Here we do not use the above procedure to derive the pedal equations, instead derive them using simple concepts from physics.

3.Description of the problem: The exercise is to find the possible trajectories of a mass m moving under the influence of the gravitational field due to a mass M m. There is no loss of generality in assuming that all trajectories of m are in XY -plane and their points of closest approach to M(impact parameter) lie on X-axis. We first derive the pedal equations of ellipse, parabola and hyperbola with the focus as the pedal(fixed) point. Though for an individual plot of a trajectory in the cartesian coordinates the origin could beFor arbitrarily Review anywhere, conventionally Only the origin for the ellipse and hyperbola is at the midpoint of their respective foci and for parabola it is at the vertex. But for the purpose of plotting all trajectories together in one axis (explained in Section-8 ), except the ellipse the cartesian forms of the parabola and hyperbola will be written with the origin shifted along X-axis. Next while describing the dynamics of m and deriving the Kepler’s laws, we show the conic sections are the only possible trajectories and that M is at the focus of each trajectory, which is also the pedal point. Finally we plot the trajectories together for a simple case.

4.Ellipse: Refer to Fig-2. Y-axis T Let ae and be be the semi-

major and semi-minor axis S(0,be) P β=rcosθ of the ellipse respectively, P’

β be the perpendicular dis- b r e 2θ tance from the focus F to e 2ae-r the tangent at the point P θ X-axis R 푭′ (-c ,0) Q(a ,0) on the ellipse, r the radial 풆 e Oe(0,0) 푭풆(ce,0) e distance from F to P and e 풄 = 풂ퟐ − 풃ퟐ 0 풆 풆 풆 ∠FePFe =2θ. When a string 0 with constant length=FeP + 0 PFe > FeFe and ends tied to ae 0 Fe and Fe is stretched by the Fig-2 Ellipse pencil tip and moved the ellipse is traced. Let P be a point anywhere on the ellipse. To keep the string stretched, the force applied should be along the normal at P . Let τ be the tension in the string and the normal divide the angle 2θ into θ1 and θ2. To mark the point P the tension in the string resolved perpendicular to the normal should balance, then τ sin θ1 = τ sin θ2 ⇒ θ1 = θ2 = θ, hence the normal 0 at P bisects ∠FePFe. Since the normal and FeT are parallel ∠PFeT = θ. 0 0 0 Now consider point Q, since RFe = FeQ, FeP + PFe = FeFe + 2FeQ = 0 0 0 RFe+FeFe+FeQ=2ae. Then for any point P on the ellipse, FeP +PFe =2ae. 0 0 p 2 2 Consider point S, FeS = SFe = ae, OS = be, so FeFe = 2 ae − be = 2ce. Us-

0 2 2 2 2 ing law of cosines in 4FePFe (2ae − r) + r − 2(2ae − r)r cos 2θ =4(ae − be) 2 2 2 gives 4ae − 2r(2ae − r)(1 + cos 2θ) = 4ae − 4be. Since β = r cos θ, we get 2 2 2 2 4 [(2ae − r)/r] β = 4be ⇒ be/β − 2ae/r = −1, with Fe(ce, 0) as the pedal point. Then the cartesian and pedal form of the ellipse are 2 2 2 x y be 2ae 2 + 2 = 1 and 2 − = −1. (1) ae be β r

5.Parabola: If the inner side of the ellipse is re- Y-axis flecting, all light rays coming from a point source 0 0 at Fe will converge to Fe. By fixing Fe, when Fe R P(s,t) is moved to negative infinity the ellipse becomes 2θ a parabola in the limit, with F =F as focus and I For Reviewe p Only r line through Fp and vertex Q as axis of symme- θ β 0 Op(0,0) X-axis try. Then all light rays coming from Fe will be F (d -a ,0) Q parallel to X-axis and hence converge to Fp. The p p p pin and string construction of a parabola is based dp on this concept. Refer to Fig-3, the equation of 2 the parabola is y =4ap(−x + dp), where dp is the ap distance between Op and Q. Consider a parallel ray R getting reflected at P (s, t) on the parabola Fig-3 Parabola and passing through Fp. Then the normal at P will bisect ∠RPFp. Let I 2 be the intersection of the tangents at P and Q. From t =4ap(−s + dp) and the equation of tangent at P ,(y − t)=−(2ap/t)(x − s) co-ordinates of I are (dp, t/2). Since the slope of PI and FpI is −2ap/t and t/2ap respectively with their product as −1, PI ⊥ FpI. Then β =FpI and ∠PFpI =∠IFpQ=θ. Since in 4FpPI, cos θ = β/r and in 4FpIQ, cos θ = ap/β equating gives 2 β = apr, with Fp(dp − ap, 0) as the pedal point. Then the cartesian and pedal forms of the parabola are 2 2 y = 4ap(−x + dp) and β = apr. (2) 6.Hyperbola: We use the definition Y-axis that for any point P on the hyper- 0 bola, PFh − PFh = 2ah. If P is at R 0 the vertex Q then 2ah = QQ . Let P 0 2 2 2 2θ 2ch = FhF and b = c − a . Re- h h h h 2ah+r fer to Fig-4, the equation of the hyper- r θ 2 2 2 2 O (0,0) X-axis bola is (−x + dh) /ah − y /bh =1, where h 푭 (d -c ,0) Q β Q’ 푭′ dh is the distance between Oh and the 풉 h h 풉 0 d midpoint of FhFh. The line connect- h 0 ing P and Fh can be extended to be 2ah seen as a light ray R getting reflected 2ch at P to the focus Fh. This is the concept used in the pin and string construc- Fig-4 Hyperbola tion of a hyperbola. Since the normal

at P bisects ∠EPFh, we have PFh = r, β = r cos θ. Then the law of 0 2 2 2 cosines in 4FhPFh (2ah + r) + r − 2(2ah + r)r cos(π − 2θ)=(2ch) gives 2 2 2 2 2 4ah + 2r(2ah + r)(1 + cos 2θ) = 4ah + 4bh ⇒ 4 [(2ah + r)/r] β = 4bh, which 2 2 simpliﬁes to bh/β − 2ah/r = −1, with Fh(dh − ch, 0) as the pedal point. Then the cartesian and pedal form of the hyperbola are

2 2 2 (x − dh) y bh 2ah 2 − 2 = 1 and 2 − = 1. (3) ah bh β r 7.Trajectories: Let p be the linear momentum of m moving under the inﬂuence of the gravitational ﬁeld of M at a distance r from it. The angular momentum L about the position of M and the total energy E of m are constants of time.For Let r ⊥Review=β, then Only p2 GMm pβ = L and − = E, 2m r implying the position of M is the pedal point for all trajectories of m. Eliminating p from above expressions gives,

2 L GMm 2m − = E. (4) β2 r In an attractive potential the total energy E could be negative or zero or positive. Then these three are the only possible cases.

Case-1: When E = −Ee,Ee > 0, this case gives Kepler’s laws. For gen- eralisation let L = Le, m = me, then dividing the above equation by −E gives 2 Le GMme 2meEe − Ee = −1, β2 r

which from eq(1) is a pedal equation of an ellipse, with Fe(ce, 0) as the pedal point and by default the position of M. Hence the trajectory of me is a stable 2 2 2 2 elliptical orbit, with 2ae =GMme/Ee, be =Le/2meEe =Leae/GMme. Then s 2 2 2 2 2 GMme Le GM G M me Le E = −Ee = − , 2 2 = ⇒ ce = 2 − . (5) 2ae beme ae 4Ee 2meEe

The total energy E = −GMme/2ae is negative, indicating bound system and is half the potential energy with “r = ae” (like in Bohr’s atom model). This is the First Law: All planets move in elliptic orbits with the Sun at p 2 2 one focus. Since M is at Fe, the impact parameter is ae − ae − be. Refer to Fig-2, let the particle move from P to P 0 in an inﬁnitesimal time δt, then

Area of sweep Area of(4FP 0P ) β P 0P βp L = = = = e (a constant), Time taken δt 2 δt 2me 2me

which is the Second Law: A line joining any planet to the Sun sweeps out equal areas in equal times. If T is the time period then for one complete rotation we get

2 2 2 Area of sweep Area of ellipse πaebe Le Le 4π ae = = = ⇒ 2 2 = 2 . Time taken T T 2me beme T

2 2 2 Equating the above expression with Le/beme in eq(5) gives a3 GM e = (a constant), T 2 4π2 the Third Law: The square of the period of any planet is proportional to the cube of theFor semi-major Review axis of the ellipse. Only

Case-2: When E = Ep = 0. Let L = Lp and m = mp. Putting E = 0 in 2 2 2 eq(4) gives β = (Lp/2GMmp)r and from eq(2) is the pedal equation of a parabola, with Fp(dp − ap, 0) as the pedal point, which again is the posi- 2 2 tion of M. Then ap = Lp/2GMmp, is the impact parameter with M at Fp(dp − ap, 0). Hence the trajectory is unbound and mp will escape to inﬁn- ity, where it will come to absolute rest (zero kinetic energy).

Case-3: When E = Eh > 0. Let L = Lh and m = mh. Dividing eq(4) by Eh gives 2 Lh GMmh 2mhEh − Eh = 1, β2 r which from eq(3) is the pedal equation of a hyperbola, with Fh(dh − ch, 0) as the pedal point and the position of M. Then ah = GMmh/2Eh, √ q 2 2 bh = Lh/ 2mhEh and ah + bh − ah is the impact parameter with M at Fh(dh − ch, 0). This trajectory will also be unbound and mh will escape to inﬁnity. The kinetic energy of mh at inﬁnity will be Eh.

This concludes that the above conic sections are the only possible trajectories of m, with M at the respective focus of each trajectory, which is also the pedal point.

8.Plot: Case-1 : varying Ee,Le, me independently such that ae ≥ be will give infinite elliptical trajectories. Case-2 : with Ep =0, varying Lp, mp independently gives infinite parabolic trajectories. Case-3 : varying Eh,Lh, mh independently will give infinite hyperbolic trajectories. Consider u elliptical, v parabolic and w hyperbolic trajectories. Plotting the trajectories together may seem a challenge. But the fact that M is at the focus of every trajectory makes it easier, because it implies that the focuses of all trajectories coincide. Choose the axis of the first ellipse e1 as the axis for plotting all trajectories.

This means the focuses of all trajectories should coincide with Fe1 (ce1 , 0), p 2 2 2 the position of M with respect to Oe1 , where ce1 = Ae1 /Ee1 − Be1 /Ee1 . Let 2 ≤ r ≤ u, 1 ≤ s ≤ v and 1 ≤ t ≤ w. Since the focuses of all trajec- 0 0 tories lie on the X-axis, shifting Oer by ce1 − cer = cer , shifts Fer by cer 0 0 0 0 shown as Fer (cer , 0) → Fer (cer , 0), making Fe1 = Fe2 = ··· = Feu . Then

to coincide the focuses of the parabola, hyperbola with Fe1 , we require

Fe1 (ce1 , 0)=Fps (dps − aps , 0)=Fht (dht − cht , 0) ⇒ dps − aps =ce1 =dht − cht ,

which gives dps =aps +ce1 and dht =cht +ce1 . This will imply Fe1 =Fp1 =···=

Fpv =Fh1 =···=Fhw , ﬁnally making the focuses of all trajectories coincide at Fe (ce , 0). Let Ae =GMme /2,Ap =GMmp /2,Ah =GMmh /2,Be = 1 √1 r p r s √ s t t r Ler / 2mer ,Bps =Lps / 2mps ,Bht =Lht / 2mht , then For Reviewr Only 2 2 2 Aer Ber Aer Ber Bps Aht Bht ae = , be = √ , ce = 2 − , ap = , ah = , bh = √ . r Ee r r E Ee s 2Ap t Eh t r Eer er r s t Eht Substituting the above expressions in the cartesian forms given in eqs(1), (2), (3) the equation of the ellipses, parabolas, and hyperbolas will be

s !2 r 2 2 A2 B2 2 Aer Ber e1 e1 Eer x+ 2 − E − 2 − E Ee er Ee e1 2 r 1 Eer y 2 + 2 = 1, 1 ≤ r ≤ u Aer Ber r 2B2 B2 A2 B2 y2 = ps −x + ps + e1 − e1 , 1 ≤ s ≤ v Ap 2Ap E2 Ee s s e1 1

s 2 2 s !2 A B A2 B2 2 ht ht e1 e1 Eh x− 2 + E − 2 − E t E ht Ee e1 2 ht 1 Eht y A2 − B2 = 1, 1 ≤ t ≤ w. ht ht The choice of the ellipse is arbitrary, instead one Y-axis can choose the axis of Parabola a parabola or a hyperbola to plot all trajectories. Then we have Ellipse-1 푭 (풄 , ퟎ) to shift the origin of the 풆ퟏ 풆ퟏ remaining trajectories so 푶 X-axis 풆ퟏ that their focuses coin- M cide with that of the cho- Ellipse-3 sen parabola or the hy- Ellipse-2 perbola. Plotting the Hyperbola trajectories for various Fig-5 Trajectories values of parameters will be an interesting exercise. We take 3 elliptical, 1 parabolic and 1 hyperbolic

trajectories with me1 = me2 = me3 = mp1 = mh1 = m and Le1 = Le2 = Le3 = L = L = L. This gives A = A = A = A = A = GMm/2 = A and p1 h1 e1 e√2 e3 p1 h1 Be1 = Be2 = Be3 = Bp1 = Bh1 = L/ 2m = B respectively. Fig-5 shows the

plot for A = 8.00,B = 2.50,Ee1 = 8.00,Ee2 = 5.00,Ee3 = 10.00,Eh1 = 20.00.

Notice the focuses of all trajectories coinciding at Fe1 (0.47, 0) the position of M. Since L, m are same, all trajectories also intersect on the axis Y = 0.47 at (0.47, ±0.78), which may diﬀer if L, m and other parameters are diﬀerent.

9.Pedal equation and kinematics: We will try to understand as to how the pedal equation could elegantly describe the kinematics. Consider a point particle moving in a plane. Let H[x, y] = 0 represent the path/trajectory traced by the particle, then x, y are functions of time, x ≡ x(t), y ≡ y(t). Let ~v(x, y) be the velocity of the particle at time t, then it will be along the direction of the tangent at H[x(t), y(t)]. In classical physics if the position and velocityFor of the Review particle is known Only at any time then the position and velocity at a later time can be predicted, i.e., H[x(t), y(t)] = 0 can be determined. When we transform from [x(t), y(t)] to pedal coordinates [r(x, y, t), β(x, y, t)], β~ will be along the normal to the curve H[x(t), y(t)] = 0. Letn ˆ be the unit normal to the plane containing β~ and ~v. Then at any time t coordinates (r, β) giving the position and β · v · nˆ = β~ × ~v giving the angular velocity makes it easier to describe the kinematics of the particle. Since the pedal system has the inbuilt advantage that the pedal point can be anywhere, judiciously the ‘reference point’ is taken as the pedal point. In our case since the reference point was at the position of M, we chose it as the pedal point for all the pedal equations. Hence from dynamics when we deduce the pedal equation with the reference point as the pedal point, describing the kinematics becomes easier.

10.Remarks: The pedal equation derived from the dynamics of the system easily help deduce all the possible trajectories and also the kinematics of the stable trajectory, the Kepler’s laws. It is also interesting to see that all the three possible trajectories lie in a plane. We also notice an important aspect that the three empirical laws of Kepler is replaced by three other empirical laws, two conservation laws and a gravitational law. This is because Kepler’s laws describes only the kinematics of a system whereas the three universal laws describes the dynamics and while Kepler’s laws are applicable only to a speciﬁc system the three universal laws are applicable to any system. So like the “axioms of a theory” the laws that replace Kepler’s are the axioms of Physics, which are more fundamental and universal. Kepler’s laws was an outcome of the observations by Tycho Brahe on a system which was ad- hering to these fundamental laws. So their work could be considered as an greatest evidence for the credibility, fundamentality and most importantly the universality of the conservation laws.

Acknowledgment: I sincerely thank Prof. Vijay Singh for reading the manuscript and suggesting improvements.

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