The geometry of isochrone orbits: from Archimedes’ parabolae to Kepler’s third law

Abstract

The Kepler potential $\propto {-1/r}$ and the harmonic potential $\propto r^2$ share the following remarkable property: In either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and harmonic potentials are called isochrone. In this paper, we solve the following general problem: Are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Hénon (Annales d’Astrophysique 22:126–139, 1959a, 22:491–498, 1959b), in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometrical generalization of Kepler’s third law, and give a similar law for the apsidal angle, of any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations and use this to derive an analytic parameterization of any isochrone orbit. Along the way, we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem and argue that our geometrical methods can be exported to more generic orbits in potential theory.

Appendices

Dynamical system

In order to draw the orbit, we write the equations of motion as a three-dimensional dynamical system. Although, in general, a generic three-dimensional motion in classical mechanics involves 6 degrees of freedom, namely the three coordinates and their associated momenta, the spherical symmetry here at play reduces this number to three. Moreover, the radial motion is decoupled from the polar one. To see this, differentiate Eq. (2) with respect to r to obtain a second-order ODE for r(t), or equivalently, a two-dimensional dynamical system for the radial motion in $(r,\dot{r})$. To get the polar motion, and thus the full orbit $(r(t),\theta (t))$, one may simply use the definition of the angular momentum $\varLambda =r^2\dot{\theta }$, which gives $\theta (t)$ directly from r(t). These three pieces together give the following three-dimensional dynamical system in $(r,\dot{r},\theta )$

$$\begin{aligned} \frac{{\mathrm {d}}r}{{\mathrm {d}}t} = \dot{r} \,, \quad \frac{{\mathrm {d}}\dot{r}}{{\mathrm {d}}t} = \frac{\varLambda ^2}{r^3} - \psi ^{\prime }(r) \quad \text {and} \quad \frac{{\mathrm {d}}\theta }{{\mathrm {d}}t} = \frac{\varLambda }{r^2} \, . \end{aligned}$$

(90)

The system (90) is sufficient to compute the trajectory of any particle in any central potential $\psi (r)$. In particular, once $\psi (r)$ is plugged into Eqs. (90) and some initial conditions $(r(0),\dot{r}(0),\theta (0))$ are provided, the motion can be solved using, e.g., a classical Runge—Kutta numerical method. Since we are interested in periodic, bounded orbits, we must, however, choose the initial conditions carefully. In order to find these orbits more easily, we choose to express $(r(0),\dot{r}(0))$ in terms of the two constants of motion $\xi$ and $\varLambda$, and take $\theta (0)=0$, as the latter does not change the periodic nature of an orbit. Since the set of $(\xi ,\varLambda )$ that produces periodic orbits is precisely the one we found in Sect. 4.3 depicted in Figs. 9, 10 and 11, this procedure allows for an easy picking of initial conditions and allows us to draw any periodic orbit in any isochrone potential. This has been used to draw the orbits in Figs. 15 through 17 and to check the validity of all our analytic isochrone formulae.

Alternative form of the third laws

We have seen that the Hénon’s formula (21) gives the period $T(\xi )$ of an orbit $(\xi ,\varLambda )$ in any isochrone potential. For any value of $\xi$, there exists a unique value $\varLambda _C$ such that the orbit is circular, corresponding to the line ${\mathscr {L}}_C:y=\xi x-\varLambda ^2_C$ being tangent to the isochrone parabola. Using the notations h and L(h) introduced in Sect. 3.2, this circular limit corresponds to $h\rightarrow 0$. Taking this well-defined limit in Eq. (25) gives

$$\begin{aligned} T^2=\frac{\pi ^2}{16(1+\xi ^2)^{3/2}} \lim _{h\rightarrow 0} \frac{L(h)^2}{h}\, . \end{aligned}$$

(91)

As it can be intuited from the discussion of Sect. 3.2.4, it turns out that the limit on the right-hand side of Eq. (91) is independent of the global aspect of the curve. In fact, this limit is simply eight times the radius of curvature $R_C$ at the point C corresponding to the circular orbit.²¹ In other words, we have

$$\begin{aligned} T^2=\frac{\pi ^2}{2}\frac{R_C}{(1+\xi ^2)^{3/2}} \, . \end{aligned}$$

(92)

Equation (92) provides a geometrical way to find the period of any given orbit in an isochrone potential, without any algebraic reference to the parabola itself. First take a line ${\mathscr {L}}$ intersecting an isochrone parabola ${\mathscr {P}}$ at P and A, and then, perform a translation of this line to construct ${\mathscr {L}}_C$, tangent to ${\mathscr {P}}$ at C. The curvature radius $r_C$ of the parabola at the tangency point C gives the period, via Eq. (92). This is the local version of the result given in Eq. (21).

In a similar fashion, the law for the apsidal angle can also be written in terms of curvature, albeit for the effective potential. If $\varPsi _e(u)=\psi _e(r)$ with $u=1/r$, we have

$$\begin{aligned} \varTheta ^2 = \frac{4\pi ^2\varLambda ^2}{\varPsi _e^{\prime \prime }{(u_C)}} \, . \end{aligned}$$

(93)

This law provides a way to compute the apsidal angle in the effective potential $\varPsi _e$ in the Binet variable $u=1/r$, or in the real effective potential $\psi _e$, using $\varPsi _e^{\prime \prime }(u_C)=r_C^4\psi _e^{\prime \prime }(r_C)$.

Hénon’s formula for $\varTheta$

We detail the computation of the integral (7) for $\varTheta$, with the method used to derive the Hénon formula (21) for T. According to the dictionary in Table. 1, this time we use the Binet variable $u:=1/r$ and define a potential $\varPsi _e(u)$ by $\psi _e(r)=\varPsi _e(u)$. Inserting these notations in (7) readily gives

$$\begin{aligned} \varTheta = \sqrt{2}\varLambda \int _{u_A}^{u_P} \frac{{\mathrm {d}}u}{\sqrt{D_\varTheta (u)}} \, , \quad \text {with} \quad D_\varTheta (u):=\xi - \varPsi _e(u) \, . \end{aligned}$$

(94)

This is the equivalent for $\varTheta$, of Eq. (14) for T. The bounds of the integral are $u_A:=1/r_A$ and $u_P=1/r_P \ge u_A$. In the (u, y) plane, the quantity $D_\varTheta (u):=\xi - \varPsi _e(u)$ appearing in Eq. (94) is the vertical distance between the curve ${\mathscr {C}}:y=\varPsi _e(u)$ and the line ${\mathscr {L}}:y=\xi$. Once again, the fact that $D_\varTheta (u)\ge 0$ follows from the requirement $\xi -\psi _e(r)\ge 0$, cf. (4).

Fig. 18

Illustration of the geometrical analysis involved in the computation of $\varTheta$, according to the dictionary in Table. 1. $D_\varTheta$ is the vertical distance between the straight line ${\mathscr {L}}:y=\xi$ and a generic curve ${\mathscr {C}}:y=\varPsi _e(u)$. The line ${\mathscr {L}}_C:y=\xi _C$ is the unique line both parallel to ${\mathscr {L}}$ and tangent to ${\mathscr {C}}$

Next we rewrite the distance $D_\varTheta$ as $D_\varTheta (u)=\varepsilon ^2 - z(u)^2$, with $\varepsilon ^2 := \xi - \xi _C$ the vertical distance between the two lines ${\mathscr {L}}$ and ${\mathscr {L}}_C$, and $z(u)^2:=\varPsi _e(u)-\xi _C$, as depicted in Fig. 18. As we did for T, we may conveniently choose z(u) to be negative on $[u_A,u_C]$ and positive on $[u_C,u_P]$. The formula for $\varTheta$ becomes

$$\begin{aligned} \varTheta = \sqrt{2} \varLambda \int _{u_A}^{u_P} \frac{{\mathrm {d}}u}{\sqrt{\varepsilon ^2 - z(u)^2}} \, . \end{aligned}$$

(95)

As we argued for T, the function $u\mapsto z(u)$ is by construction monotonically increasing so that we can perform the change of variables $u\rightarrow z(u)$ and introduce f such that $u=f(z)$. Since $z(u_A)=-\varepsilon$ and $z(u_A)=\varepsilon$, the integral becomes

$$\begin{aligned} \varTheta = \sqrt{2} \varLambda \int _{u_A}^{u_P} \frac{ f^{\prime }(z) {\mathrm {d}}z}{\sqrt{\varepsilon ^2 - z^2}} = \sqrt{2} \varLambda \int _{-\pi /2}^{\pi /2} f^{\prime }(\varepsilon \sin \phi ) {\mathrm {d}}\phi \, , \end{aligned}$$

(96)

where the last equality follows from the change of variables $z\rightarrow \varepsilon \sin \phi$, with $\phi$ varying between $-\pi /2$ and $\pi /2$ when $z\in [-\varepsilon ,\varepsilon ]$. Now we assume for $f^{\prime }$ a Taylor expansion around 0 of the form $f(z)=a_0 + \sum _{n\ge 1} a_n z^n$, and integrate term by term to get

$$\begin{aligned} \varTheta = \sqrt{2} \pi \varLambda a_0 + 2\sqrt{2} \varLambda \sum _{n\ge 1} a_{2n} W_{2n} \varepsilon ^{2n} \, , \end{aligned}$$

(97)

with $W_n$ the Wallis integral as given in Eq. (18), and the odd terms vanishing by integration over the symmetric interval $[-\varepsilon ,\varepsilon ]$. Now if $\varTheta$ is to be independent of $\xi$, then it must also be independent of $\varepsilon$, since $\xi _C$ depends only on $\varLambda$. Therefore, we must have $a_{2n}=0$ for all $n\ge 1$. We thus obtain the formula $\varTheta = \sqrt{2} \pi \varLambda a_0$ and the Taylor expansion of $f'$ therefore writes

$$\begin{aligned} f'(z) = \frac{\varTheta }{\sqrt{2}\pi \varLambda } + \sum _{n \ge 1} a_{2n+1} z^{2n+1} \, . \end{aligned}$$

(98)

Integrating this equation over the interval $[z(u_A),z(u_P)]=[-\varepsilon ,\varepsilon ]$, we can make the same remarks as we did in the paragraph below Eq. (20), except that in this case $f(z_A)=u_A$ and $f(z_P)=u_P$. In the end, we obtain Eq. (22), which is the equivalent to Eq. (21) for T. The right-hand side of that equation is independent of $\xi$, even though the quantities $u_P,u_A$ depend explicitly on $\xi$.

Analysis of $\varTheta (\varLambda )$

Let $\psi _i$ be a potential in one of the four families $i\in \{1,2,3,4\}$ as defined in Sect. 3.2. In this appendix, we study the properties of the function $\varTheta _i(\varLambda )$ defined in Eq. (61). These formulae are used in Sect. 5.3 to classify the orbits in each isochrone potential $\psi _i$. The claims of Sect. 5.3 regarding each function $\varLambda \mapsto \varTheta _i(\varLambda )$ follow easily from the mathematical analysis detailed here, with the Latin parameters (a, b, c, d, e) replaced by the Greek ones $(\omega ,\varepsilon ,\lambda ,\mu ,\beta )$.

1.1 Harmonic and Kepler family, $\varTheta _1(\varLambda )$ and $\varTheta _4(\varLambda )$

For the harmonic potential $\psi _1$, the analysis of $\varTheta _1(\varLambda )=\pi \varLambda (\varLambda ^2+\lambda )^{-1/2}$ is straightforward. For $\lambda <0$, it is strictly decreasing and $\varTheta$ varies in $]\pi ,+\infty [$ when $\varLambda \in ]\sqrt{-\lambda },+\infty [$. For $\lambda =0$, $\varTheta =\pi$ for all $\varLambda \in \mathbb {R}$. For $\lambda >0$, it is strictly increasing and $\varTheta$ varies in $]0,\pi [$ when $\varLambda \in ]0,+\infty [$. For the Kepler family $\psi _4$, the analysis is also straightforward since we have for any $\varLambda$ and $\lambda$ the identity $\varTheta _4(\varLambda )=2\varTheta _1(\varLambda )$ by direct examination of Eq. (61) when $b=0$ (harmonic) and $d^2=4b^2e$ (Kepler).

1.2 Bounded family, $\varTheta _2(\varLambda )$

For Bounded potentials $\psi _2$, the analysis is more involved. For any $\mu >0$ and $\beta >0$, we write $\alpha :=\lambda /(\lambda +4\mu \beta )$. We also define a function $f(\varLambda ,\lambda )$ of the real variables $\varLambda ,\lambda$ by the formula

$$\begin{aligned} f(\varLambda ,\lambda ):=\frac{\varLambda }{(\varLambda ^2+\lambda )^{1/2}} - \frac{\varLambda }{(\varLambda ^2+\lambda +4\mu \beta )^{1/2}} \, . \end{aligned}$$

(99)

With these notations, we have $\varTheta _2(\varLambda )=\pi f(\varLambda ,\lambda )$ (cf. Eq. (87)). We want to study the three cases $\lambda >0$, $\lambda =0$ and $\lambda <0$, used to classify the orbits in Sect. 5.3.

$\bullet$   Case $\lambda =0$. In this case, we simply plug $\lambda =0$ in Eq. (99) and we see that $f(\varLambda ,0)\in [0,1]$. Moreover, we have easily $\partial _\varLambda f<0$. Therefore, $\varTheta (\varLambda )$ is strictly decreasing and varies $[0,\pi ]$.

$\bullet$   Case $\lambda >0$. In this case, $0<\alpha <1$ and for a fixed $\lambda$, we have

$$\begin{aligned} \frac{\partial f}{\partial \varLambda } (\varLambda ,\lambda ) = \frac{\lambda }{(\varLambda ^2+\lambda )^{3/2}} - \frac{\lambda +4\mu \beta }{(\varLambda ^2+\lambda +4\mu \beta )^{3/2}} \, . \end{aligned}$$

(100)

Then, a few algebraic manipulation show that $\partial _\varLambda f(\varLambda ,\lambda )$ vanishes for a value $\varLambda _o$ given by

$$\begin{aligned} \varLambda _o^2 = \lambda \frac{\alpha ^{1/3}-1}{\alpha -\alpha ^{1/3}} \quad \Rightarrow \quad f(\varLambda _o,\lambda )=\frac{\varLambda _o(1-\alpha ^{1/3})}{(\varLambda _o^2+\lambda )^{1/2}} \, . \end{aligned}$$

(101)

Since $0<\alpha <1$ and $0<\varLambda _o<(\varLambda _o+4\mu \beta )^{1/2}$, we have readily $0<f(\varLambda _o,\lambda )<1$. Now, for any fixed $\lambda >0$, $\varLambda \mapsto f(\varLambda ,\lambda )$ is continuous, $\partial _\varLambda f$ vanishes only once at $\varLambda _o$ and furthermore $0<f(\varLambda _o,\lambda )<1$. Furthermore, it is clear that $f(\varLambda ,\lambda )$ goes to 0 as $\varLambda \rightarrow 0$ and $\varLambda \rightarrow +\infty$. With all these results, the general shape of the curve $\varLambda \mapsto f(\varLambda ,\lambda )$ can be easily inferred.

$\bullet$   Case $\lambda <0$. In this case, $\varLambda \mapsto f(\varLambda ;\lambda )$ is defined only when $\varLambda ^2>-\lambda$. First subcase: $\lambda <0$ and $\lambda +4\mu \beta <0$. Then we readily see in Eq. (100) that $\partial _\varLambda f(\varLambda ,\lambda )>0$. Second subcase: $\lambda <0$ but $\lambda +4\mu \beta \ge 0$, then setting $g(\varLambda ,\lambda ) := \lambda (\varLambda ^2+\lambda )^{-3/2}$, we have for any $\varLambda ^2>-\lambda$

$$\begin{aligned} \frac{\partial g}{\partial \lambda } (\varLambda ,\lambda ) = \frac{2\varLambda ^2-\lambda }{2(\varLambda ^2+\lambda )^{5/2}} \, . \end{aligned}$$

(102)

Now, since $\varLambda ^2>-\lambda$, the right-hand side of Eq. (102) is strictly positive, and therefore, g is an increasing function of $\lambda$. In particular, we have $\lambda +4\mu \beta>\lambda \Rightarrow g(\varLambda ,\lambda +4\mu \beta ) > g(\varLambda ,\lambda )$ and by definition of g, the latter is exactly $\partial _\varLambda f(\varLambda ,\lambda )>0$. To conclude, in the $\lambda <0$ case, $\varLambda \mapsto f(\varLambda ,\lambda )$ is strictly decreasing. Furthermore, it is clear that $f(\varLambda ,\lambda )$ goes to $+\infty$ as $\varLambda \rightarrow (-\lambda )^{1/2}$ and to 0 as $\varLambda \rightarrow +\infty$. With all these results, the general shape of the curve $\varLambda \mapsto f(\varLambda ,\lambda )$ can be easily inferred.

1.3 Hénon family, $\varTheta _3(\varLambda )$

For Hénon potentials $\psi _3$, the analysis is similarly more involved. As for the Bounded potentials, we fix $\mu >0$ and $\beta >0$ and write $\alpha :=\lambda /(\lambda +4\mu \beta )$. This time we define a function $h(\varLambda ,\lambda )$ of the real variables $\varLambda ,\lambda$ by the formula

$$\begin{aligned} h(\varLambda ,\lambda ):=\frac{\varLambda }{(\varLambda ^2+\lambda )^{1/2}} + \frac{\varLambda }{(\varLambda ^2+\lambda +4\mu \beta )^{1/2}} \, . \end{aligned}$$

(103)

With these notations, we have $\varTheta _3(\varLambda )=\pi h(\varLambda ,\lambda -2\mu \beta )$ (cf. Eq. (89)). The analysis follows the same lines as what was done for $\varTheta _2(\varLambda )$. We want to study the three cases $\lambda >0$, $\lambda =0$ and $\lambda <0$, used to classify the orbits in Sect. 5.3.

$\bullet$   Case $\lambda >0$. Then we have $0<\alpha <1$ and there is no problem in showing that $\varLambda \mapsto h(\varLambda ,\lambda )$ is strictly increasing and that $0<h(\varLambda ,\lambda )<1$.

$\bullet$   Case $\lambda =0$. Once again, there is no problem in showing that $\varLambda \mapsto h(\varLambda ,\lambda )$ is strictly increasing and that $0<h(\varLambda ,\lambda )<2$.

$\bullet$   Case $\lambda <0$. In this case, h is only defined when $\varLambda ^2>-\lambda$. For any such $(\varLambda ,\lambda )$, we have

$$\begin{aligned} \frac{\partial h}{\partial \varLambda } (\varLambda ,\lambda ) = \frac{\lambda }{(\varLambda ^2+\lambda )^{3/2}} + \frac{\lambda +4\mu \beta }{(\varLambda ^2+\lambda +4\mu \beta )^{3/2}} \, . \end{aligned}$$

(104)

There are two subcases. First subcase: $\lambda <0$ and $\lambda +4\mu \beta <0$. Then from Eq. (104), $\partial _\varLambda h(\varLambda ,\lambda )<0$. Furthermore, $h(\varLambda ,\lambda )$ goes to $+\infty$ as $\varLambda \rightarrow (-\lambda )^{1/2}$, and to 2 as $\varLambda \rightarrow +\infty$. Second subcase: $\lambda <0$ and $\lambda +4\mu \beta <0$. If $|\alpha |<1$, then there is a value $\varLambda _o$ that makes $\partial _\varLambda h(\varLambda ,\lambda )$ vanish. It is given by

$$\begin{aligned} \varLambda _o^2 = \lambda \frac{|\alpha |^{1/3}+1}{|\alpha |-|\alpha |^{1/3}} \quad \Rightarrow \quad f(\varLambda _o,\lambda )=\frac{\varLambda _o(1+|\alpha |^{1/3})}{(\varLambda _o^2+\lambda )^{1/2}} \, . \end{aligned}$$

(105)

In this case, the function $\varLambda \mapsto h(\varLambda ,\lambda )$ decreases on $[(-\lambda )^{1/2},\varLambda _o]$ and increases on $[\varLambda _o,+\infty [$. The value $h(\varLambda _o,\lambda )$ is always strictly between 1 and 2. If $|\alpha |\ge 1$, then the function f is strictly decreasing. (It can be seen as the limit $\varLambda _o\rightarrow +\infty$.) The value $h(\varLambda _o,\lambda )$ is in this case always above 2.

Proof that $c+d\xi <0$ for isochrone orbits around finite central mass

In Sect. 5.2, we used the fact that $a+b\xi <0$ and $c+d\xi <0$ for isochrone orbits in order to prove that our formula (82) covers all isochrone orbits. The former identity follows from the generalized Kepler’s third law, and here, we prove the latter identity. By assumption, we have a particle $(\xi ,\varLambda )$ on an isochrone orbit in a potential with finite central mass whose parabola ${\mathscr {P}}:(ax+by)^2+cx+dy=0$ verifies all hypotheses $(H_i)$ of Sect. 4.2. First we can check easily that $cx+dy=0$ is an equation for the tangent to ${\mathscr {P}}$ at the origin. Geometrically, since two intersections exist between ${\mathscr {L}}$ and ${\mathscr {P}}$, the slope of ${\mathscr {L}}$ must be bigger than that of this tangent, i.e., we must have $\xi >-c/d$. We just have to show that $d\le 0$ and the result will follow. First, if $b=0$ (harmonic case), then we necessarily have $d<0$ (top-oriented parabola). Second, if $b\ne 0$, then since $\lambda =0$ (${\mathscr {P}}$ crosses the origin) we have by Eq. (36) the equality $-d=\sqrt{d^2}$, which implies $d\le 0$. Therefore, we always have $d\le 0$ and thus $c+d\xi <0$.

Peaks of orbits in Bounded potentials

Let an arbitrary orbit be given by a polar equation $r(\theta )$, and compute the value of $|{\mathrm {d}}r/{\mathrm {d}}\theta |$. The latter is a measure of the change of ${\mathrm {d}}r$ when moving from $\theta$ to $\theta +{\mathrm {d}}\theta$. It vanishes for circles $r=\text {cst}$ and is infinite for straight lines $\theta =\text {cst}$. With the help of Eq. (2) and $\varLambda =r^2\dot{\theta }$, we obtain easily $|{\mathrm {d}}r/ {\mathrm {d}}\theta |^2 = 2r^4(\xi -\psi _e(r))/ \varLambda ^2$. Using a Taylor expansion of $\psi (r)$ and Eq. (3), we can linearize this equation around the apoapsis $r_A$. We then obtain

$$\begin{aligned} \biggl |\frac{{\mathrm {d}}r}{{\mathrm {d}}\theta }\biggr |^2 = \frac{2r_A^4}{\varLambda ^2} \biggl ( \psi ^\prime (r_A)-\frac{\varLambda ^2}{r_A^3} \biggr ) (r_A-r) + o(r_A-r). \end{aligned}$$

(106)

Examining Eq. (106), we see that as $r\rightarrow r_A$ the right-hand side goes to zero as every term is finite in front of $(r_A-r)$. The orbit is therefore smooth and differentiable around the apoapsis. However, the quantity $\psi ^\prime (r_A)$ turns out to be very large for the Bounded family, in general. This is because the slope of a Bounded potential increases to infinity as r grows toward $\beta$ from below, as can be seen readily on Eq. (86). Therefore, a line ${\mathscr {L}}$ can intersect ${\mathscr {C}}$ such that $r_A$ is very close to $\beta$, and it is clear from Eq. (86) that $\psi _2^\prime (r) \rightarrow \infty$ as $r \rightarrow \beta$. As a conclusion, before the apoapsis, the term $(r_A-r)$ does not yet compensate the $\psi ^\prime (r_A)$ which is large for the Bounded potential, making $\frac{{\mathrm {d}}r}{{\mathrm {d}}\theta }$ large and the curve resembles a $\theta =\text {cst}$ line. This is why we see such abrupt and pointy turns on Fig. 16.