KEPLER’S EQUATIONS
a kinematical deduction
Raul Nunes
mailto:[email protected]
P.O. Box 45350 - S. Paulo - SP - Brazil - CEP 04010-970
Kepler’s photo was reproduced from Stadtmuseum Graz
( http://www.graz.at/grazinfo/personen/da kepler.htm )
Abstract: Kepler’s equations are considered as central to Celestial Mechanics since their
solutions permit ’to find the position of a body for a given time’. To obtain a kinematical
deduction of them, only one new concept is introduced with the name of ”maccel”. By
assuming the simultaneous conservation of the maccel and of the angular momentum, it is
shown that the position vector always describes a conic section. Then, the proof includes
the three main types of Kepler’s equations for elliptical, parabolic, and hyperbolic trajectories. The theoretical innovations here introduced concern mathematicians, physicists,
astronomers, astronauts, cosmologists, historians, philosophers, and everyone minimally
interested in the development and improvement of Science.
Keywords: Lagrange’s identity, inertia, maccel, angular momentum, conic sections.
2000 Mathematics Subject Classification: 53A17, 70B05, 70F15, 70M20.
Acknowledgments: To Dr. Thomas E. Phipps Jr. for his stimulant correspondence,
to the editors of Who’s Who in the World for their encouraging recognition, to the
staff of The Mathematics Preprint Server for their web support.
Dedication: To Eng. Hideki Ishitani, an author’s schoolmate at Escola Politécnica of
the Universidade de São Paulo, SP, Brazil, where both were graduated as civil engineers.
Solicitation: The author makes a generic solicitation directed to everybody who is
able to contribute with some significant assistance in improving, supporting or spreading
(i.e., succeeding) his researches. May we live in a world full of wiser, healthier and richer
(i.e., happier) people.
2
Kepler’s Equations
Contents
Abstract, Keywords and Classification
1
Acknowledgments, Dedication and Solicitation
1
1. Prologue
3
2. The Starting Constructs
4
2.1. Kinematical vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2. Combination of the kinematical vectors . . . . . . . . . . . . . . . . . . . .
5
2.3. Lagrange’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.4. Definition of ’maccel’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3. Solving t = t(r)
9
3.1. The fundamental identity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3.2. The principle of inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2
3.3. Solving r̈ =
M J
+ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
r2
r
4. Solving r = r[r(θ(t)), θ(t)]
13
4.1. The equation of conics r = r(θ) . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2. The equations of the angle θ = θ(t) . . . . . . . . . . . . . . . . . . . . . . . 15
4.3. Solving Kepler’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5. Epilogue
19
Appendixes
21
A Solving the integral
R
(Qr2 − 2Mr − J2 )−1/2 r dr
21
B Solving the integral
R
(1 + e cos θ)−2 dθ
24
References
28
Raul Nunes
1.
3
PROLOGUE
If you want to learn about Nature, to appreciate Nature,
it is necessary to understand the language she speaks in.
Richard P. Feynman
A basic problem in Celestial Mechanics is ’to find the position of a body for a given time’. A Basic
The so-called Kepler’s equations has been traditionally used to solve it as a dynamical Problem
problem (that is, using the concepts of mass and force), in a relatively satisfactory manner.
Currently, however, this scientific paradigm needs some theoretical adaptations in order
to be compatible with recent astronomical discoveries (e.g., rotation curves of galaxies).
Researching about, Nunes found a very simple conceptual modification which seems to
fulfill a number of those demands. It consists in the introduction of only one kinematical
concept that he named ’maccel’. It can be qualified as kinematical because it can be defined
only as function of the concepts of length and time. In fact, it is a scalar quantity having
the physical dimensions of length3 /time2 , the same ones of the known Keplerian harmonic
constants existing among the lengths and the periods of the trajectories of planets and
satellites of our solar system.
As emphasized by Richard P. Feynmann [18]: ”It is surprising that people do not believe The New
that there is imagination in science. It is a very interesting kind of imagination, unlike Concept
that of the artist. The great difficulty is in trying to imagine something that you have
never seen, that is consistent in every detail with what has already been seen, and that is
different from what has been thought of; furthermore, it must be definite and not a vague
proposition. That is indeed difficult.” Thus, maccel has been imagined to be a well-defined
concept which is completely consistent with the traditional and actual observational data.
Moreover, it seems sufficiently powerful to answer old questions (e.g., what is gravity?)
and to solve new enigmas (e.g., the nature of the called ’dark matter’), provided a new
interpretation of the adapted theory is considered as plausible by open-minded people.
The scope of this paper, however, is limited to show only a new mathematical deduction
of Kepler’s equations based on purely kinematical concepts. Much of the supra cited
theoretical consequences and of the mathematical reasoning presented in this paper are
contained in a more comprehensive article [35].
The main reasoning of this paper is developed along four sections. Section 2 remembers Paper’s
the starting constructs: the triplet of kinematical vectors r, v, a (namely, position, velocity, Structure
and acceleration vectors), the scalar product and the vectorial product, the combinations
of the kinematical vectors, the definitions of norm and unit vector, the Lagrange’s identity,
jointly with the definition of ’maccel’ M and the proof that a = (M/r2 )r̂, whenever r 6= 0
and the angular momentum J is constant. Section 3 shows how to find the time t as
function of the norm r of the position vector r (that is, t = t(r)) considering the Nunesian
fundamental identity r3 r̈ − Mr − J2 = 0 as a second-order nonlinear differential equation
for r(t), when the maccel M and the norm J have constant values. Section 4 shows initially
how to find the norm of the position vector r as a function of the angle θ implicitly defined
by the differential dθ ≡ (J/r2 ) dt (that is, r = r(θ)). After, it shows the solutions of
θ = θ(t) which serve as ’bridges’ to determine the position vector r as a function of the
time t (that is, r = r[r(θ(t)), θ(t)]). In other words, in this last section, the kinematical
deduction of the Kepler’s equations is finished, under the assumptions of the simultaneous
conservation of the maccel M and of the angular momentum J. Their numerical solutions
permit one, at least approximately, ’to find the position of a body for a given time’.
4
2.
2.1.
Notation
Kepler’s Equations
THE STARTING CONSTRUCTS
Kinematical vectors
Let t be an independent parameter representing the variable ’time’ whose values varying
inside an interval I which is a subset of the real numbers (i.e.,∀t ∈ I ⊂ < ). Let f or
f (t) represent any continuous function of the parameter t. Let the nth derivative of f
with respect to t be indicated by f (n) = dn f (t)/dtn , for all integer n ≥ 0, or, in a shorter
...
form (invented by own Newton) by f , f˙, f¨, and f when n is equal to 0, 1, 2, and 3,
respectively (i.e., f = f (0) = f (t), f˙ = f (1) = df (t)/dt, f¨ = f (2) = d 2 f (t)/dt2 , and
...
f = f (3) = d 3 f (t)/dt3 ). Let Dn denote the class of functions which are differentiable with
respect to t, at least up to the nth order, all along the interval I (that is, if f ∈ Dn , then
there exist the functions f , f˙, f¨, . . . , f (n) ). Thus, f ∈ D∞ means that the function f is
infinitely differentiable. The symbol D is being here used as a mnemonic for the words
‘differentiable’ or ’derivable’. As habitual, the vectorial functions are typographically
represented by means of single boldface letters. Thus,
Let r be a vectorial function of the parameter t, of class D∞ ,
with an image = in the Euclidean three-dimensional space
(i.e., r = r(t) : = −→ <3 , ∀t ∈ I ⊂ <, r ∈ D∞ ).
(0)
r, v, and a
Thinking of such vector r as an arrow-shaped segment, if its tail is attached to the
origin of a coordinate system, its head corresponds to a determined point of the space;
thereupon, it is called the position vector. As t assumes sequentially the values of
the interval I, the free extremity of the position vector r describes, in general, an arc of
curve through the space. For the purposes of this paper, its first derivative ṙ is called
the velocity vector and its second derivative r̈ is called the acceleration vector.
Thence the mnemonic use of their initial letters ‘v’ and ‘a’ to denote them, respectively
(i.e., v ≡ ṙ and a ≡ r̈). For the position vector, however, instead of the initial ‘p’, the
letter ‘r’ is very often used as a mnemonic of the name ‘radius’ attributed to the cited
arrow-shaped segment (i.e., r ≡ r(t)).
Norms
Now, let ‘•’ and ‘×’ indicate the scalar and vectorial products, respectively. Usually
they are also called ‘dot product’ and ‘cross product’, respectively. From the vectorial
analysis, it is so known that the squared norm of any vector can be calculated as the
scalar product of it by itself (for example, p2 ≡ p • p). Thus, let r2 ≡ r • r , v 2 ≡ v • v
and a2 ≡ a • a . Usually, the scalars r, v, and a are calculated, by convention, as the
nonnegative values of the square roots of the squared norms of the vectors r, v, and a,
√
√
√
respectively (i.e., r ≡ r • r , v ≡ v • v , and a ≡ a • a ).
Unit
Vectors
It is worthy to recall that it is possible, for any vector p, to write p = p p̂, where the
scalar p is the nonnegative value of the square root of the squared norm of the vector p
√
√
(i.e., p = p • p) and p̂ is a unit vector (i.e., p̂ = p̂ • p̂ = 1). Thus, the magnitude of
the vector p is determined by its norm p while the unit vector p̂ furnishes its complete
spatial orientation (e.g., r = r r̂, v = v v̂, a = a â).
Raul Nunes
2.2.
5
Combination of the kinematical vectors
Three useful expressions can be obtained from the squared norms of the vectors r and v Useful
by deriving them with respect to t:
Expressions
d 2
d
(r ) =
(r • r)
dt
dt
d 2
d
(v ) =
(v • v)
dt
dt
⇒ rṙ = r • v
(1)
⇒ v v̇ = v • a
(2)
⇒ ṙ 2 + rr̈ = v • v + r • a
(3)
and also
d
d
(rṙ) =
(r • v)
dt
dt
To avoid misinterpretations, one must keep in mind that, although the next vectorial
expressions: ṙ = v and r̈ = v̇ = a are always valid, the next scalars inequalities: ṙ 6= v,
r̈ 6= v̇, r̈ 6= a, and v̇ 6= a occur in the great majority of the cases.
The following stairslike diagrams of the Figure 1 show all possible combinations of the Combinations
vectors r, v, and a when paired off through the scalar and vectorial products.
scalar products
r ≡ r (t)
r
r • r ≡ r2
v
r • v = rṙ
a
r • a ≡ W v • a = v v̇
r
v
a
r×r=0
r×v ≡J
r×a=I
r
v×v =0
v×a≡K
v
a×a=0
a
v ≡ ṙ
v • v ≡ v2
v
r
a • a ≡ a2
a
a ≡ r̈
vectorial products
Figure 1: The combinations of the vectors r, v, and a.
Here ’W’ is a mnemonic abbreviation of the word ’Work’, but the other three consecutive
symbols: I, J, and K, were arbitrarily adopted. As supra, the squared norms of these
vectors are defined as I2 ≡ I • I , J2 ≡ J • J , K2 ≡ K • K . It is important to notice that
the vector J ≡ r × v is called angular momentum and the vector I is its derivative with
respect to t (since J̇ = v × v + r × a = I).
6
Kepler’s Equations
2.3.
Proofs
Lagrange’s identity
From the Vector Calculus, it is known that, for any pair of vectors p and q, are valid the
following pair of expressions:
p • q = p q cos βp,q
and
p × q = p q sin βp,q ô ,
(4)
where βp,q is the angle existing between the vectors p and q which is computable as
cos βp,q = p̂ • q̂ and ô is a unit vector which is computable as ô = p̂ × q̂. Since, from
the Trigonometry, cos2 β + sin2 β = 1 holds for any angle β, it is easy to verify that the
expressions (4) justify the called Lagrange’s identity:
(p × q) • (p × q) = (p • p)(q • q) − (p • q)(p • q)
.
(5)
In order to offer a supplementary proof of this fundamental identity, let r = r[r1 , r2 , r3 ] be
another notation for the vector r in terms of three orthogonal coordinates [r1 , r2 , r3 ], which
are single-valued real functions of the time t (i.e., ri = ri (t), ∀i ∈ {1, 2, 3}). Besides, let
hı̂, ̂, k̂i be a triplet of unit vectors that is the vectorial basis of an orthogonal coordinate
system such that r = r1 ı̂ + r2 ̂ + r3 k̂. A common arbitrary unit of length is associated
to these unit vectors. Thus, for arbitrary vectors p = [p1 , p2 , p3 ] and q = [q1 , q2 , q3 ], it is
known, from the Vector Calculus, that
and
p • q = (p1 ı̂ + p2 ̂ + p3 k̂) • (q1 ı̂ + q2 ̂ + q3 k̂) = p1 q1 + p2 q2 + p3 q3
ı̂
̂ k̂ p × q = p1 p2 p3 = (p2 q3 − p3 q2 )ı̂ + (p3 q1 − p1 q3 )̂ + (p1 q2 − p2 q1 )k̂ .
q q q 1
2
3
Thereupon, the above Lagrange’s identity can be rewritten as
(p2 q3 − p3 q2 )2 + (p3 q1 − p1 q3 )2 + (p1 q2 − p2 q1 )2 =
2
(p21 + p22 + p23 )(q12 + q22 + q32 ) − (p1 q1 + p2 q2 + p3 q3 ) ,
the validity of which can be easily confirmed by means of elementary operations of Algebra.
Applications
In consequence, three very important expressions can be promptly obtained by means
of the Lagrange’s identity as given by (5):
(a) by replacing p by r and q by a, it results
(r × a) • (r × a) = (r • r)(a • a) − (r • a)(r • a)
|
{z
}
|
{z
}
|
{z
}
h
i
2
I2
=
(ra)2
−
W2
= r2 a2 − (W/r)
|
{z
}
only if r6=0
(b) by replacing p by r and q by v, it results
(r × v) • (r × v) = (r • r)(v • v) − (r • v)(r • v)
|
{z
}
|
{z
}
|
{z
}
J2
=
(rv)2
−
(rṙ)2
= r2 v 2 − ṙ 2 ;
(6)
(7)
(c) by replacing p by v and q by a, it results
(v × a) • (v × a) = (v • v)(a • a) − (v • a)(v • a)
|
{z
}
|
{z
}
|
{z
}
K2
=
(va)2
−
(v v̇)2
= v 2 a2 − v̇ 2 .
(8)
Raul Nunes
7
Because, by definition, an identity holds good for all admissible values of its variables,
the employment of the remarkable Lagrange’s identity assures the generality of this
vectorial approach. In fact, it is the cornerstone of the mathematical argumentation shown
henceforth.
2.4.
Definition of ’maccel’
At this point, let the concept of ’maccel’ be introduced as a scalar quantity M whose Maccel
mathematical definition is given through the following formula:
M ≡ r (r • a)
(9)
where r is the norm of the position vector r and (r • a) is the scalar product between
the position vector r and the acceleration vector a. It is easy to verify that its physical
dimensions are length3 /time2 . Then it can be considered as a kinematical concept because
it depends only of measurements of length and time which are the essential variables of
Kinematics. Despite the simplicity of the above formula, the importance of the concept of
’maccel’ cannot be overestimated. The name ‘maccel’ (whose pronunciation is ’mak-sel’)
was attributed to it in virtue of its close connection with the acceleration vector a and
by its phonetic similarity with the word ‘mass’ (when pronounced alternatively as ’mas’).
In fact, there exists a profound relationship between these two essential concepts whose
analysis, although interesting it may be, is beyond of the scope of this paper.
From Vectorial Calculus, it is known that a vector may be decomposed in two or more Projections
vectors according to given directions. So, let us decompose the velocity vector v into two
velocities vk and v⊥ such that v = vk + v⊥ , vk is always parallel to r, v⊥ is always
perpendicular to r, and both are situated in the same plane determined by the nonzero
vectors r and v, as illustrated in the left side of the Figure 2. If v = 0, then vk = v⊥ = 0.
Similarly, let us decompose the acceleration vector a into two accelerations ak and a⊥ such
that a = ak + a⊥ , ak is always parallel to r, a⊥ is always perpendicular to r, and both are
situated in the same plane determined by the nonzero vectors r and a, as illustrated in
the right side of the Figure 2. If a = 0, then ak = a⊥ = 0. Since the symbols k and ⊥
denote the called radial and transverse projections, they can be replaced by the subscripts
‘r’ and ‘t’, respectively. Thus, vr ≡ vk , vt ≡ v⊥ , ar ≡ ak , and at ≡ a⊥ .
J ≡ r × v = r × v⊥ = r × vt
pv
v⊥ p p p p p p ppp
ppp
BM
p
B pppp
1v
1B
k
r
O r
M ≡ r(r • a) = r(r • ak ) = r2 ar
a
ppppppp ⊥
aX
pppp p pX
y
MBB
ppp X
p
1r
)
a
k
O r
Figure 2: The radial and transverse decompositions of v and a.
8
Kepler’s Equations
Now, let vr2 ≡ vr • vr , vt2 ≡ vt • vt , whence v 2 ≡ v • v = vr2 + vt2 . On the other hand,
J ≡ r × v = r × vr + r × vt = r × vt ; then J = ± r vt sin(π/2) = ± r vt , whence J2 = (rvt )2 .
But, because J2 = r2 (v 2 − ṙ 2 ), according to (7), it implies that vt2 = v 2 − ṙ 2 and vr2 = ṙ 2 .
Thence, (7) can be rewritten as
J2 = (rvt )2
where
vt2 = v 2 − ṙ 2 .
(10)
Similarly, let a2r ≡ ar • ar , a2t ≡ at • at , whence a2 ≡ a • a = a2r + a2t . On the other
hand, M ≡ r (r • a) = r (r • ar + r • at ) = r (r • ar ); then, according to (4), M = r2 ar .
So, (9) can be rewritten as
M = r2 ar
a=
M
r̂
r2
where
a2r = a2 − a2t .
(11)
A very interesting result can be established whenever r 6= 0 and J̇ = I = 0. In fact, in
this case, the expression (6) can be rewritten as:
"
2
I =r
2
2
a −
W
r
2 #
"
=r
2
2
a −
M
r2
2 #
,
because M ≡ r(r • a) = r W, according to (9). Then, it results a = ± M/r2 , whenever r 6= 0 and I = 0 (or equivalently I = J̇ = 0). But, it can be changed into
a = ± (r • a)/r = ± (r̂ • a) (because r = r r̂) or, according to (4), a = ±a cosβr,a or
a(1 ∓ cosβr,a ) = 0, where βr,a is the angle existing between the vectors r and a. Then,
either a = 0 or cosβr,a = ± 1, which means that βr,a must be a multiple of π radians
(that is, the vectors r and a have parallel directions). Recall that the null vector 0 is, by
definition, parallel to any other vector. In this case, r 6= 0 and I2 = (r a sinβr,a )2 = 0; then,
it implies either a = 0 or sinβr,a = 0 (whence, βr,a = n π, n ∈ {0, 1, 2, . . .}). Therefore,
a = (M/r2 ) r̂ where the ‘±’ signs were embedded, by convention, in the factor cosβr,a of the
scalar product r̂•a = a cosβr,a = ± a. In fact, a•a = a2 = (M/r2 )(r̂•a) = (M/r2 )a cosβr,a ,
or a[a − (M/r2 ) cosβr,a ] = 0, whence either a = 0 or a = (M/r2 ) cosβr,a = ± M/r2 ,
as above derived from (6). Inversely, whenever a = (M/r2 ) r̂ and r 6= 0, it results
I = J̇ = 0. In fact, the vectorial product of this expression by the nonzero vector r
produces I ≡ r × a = (M/r2 ) (r × r̂) = 0.
In sum, the above reasoning proves that
J̇ = 0 if and only if a =
M
r̂
r2
, whenever r 6= 0
.
(12)
It means that, whenever r 6= 0 and J̇ = I ≡ r × a = r × at = 0, it results at = 0, whence
a = ar and, consequently, ar = (M/r2 ) r̂. Therefore, in this case, the acceleration vector
a and the position vector r have parallel directions, but its orientation is given by the sign
of the maccel M. In other words, if M > 0 then a and r have equal orientations, but if
M < 0 then a and r have contrary orientations. It is worthy to observe that the maccel
M in the equation 12 is not necessarily constant but can vary with the time t.
Raul Nunes
3.
3.1.
9
SOLVING t = t(r)
The fundamental identity
By assuming that r 6= 0, it is mathematically permitted to multiply (3) by r2 in order
to obtain r2 (ṙ 2 ) + r2 (rr̈) = r2 (v • v) + r2 (r • a). Since r2 (v • v) = (rv)2 , according the
Figure 1, and r2 (r • a) = rM, according (9), it results that r2 (ṙ 2 ) + r2 (rr̈) = (rv)2 + rM,
which may be rearranged as r2 (rr̈) − rM − [(rv)2 − (rṙ)2 ] = 0, and rewritten as
r3 r̈ − rM − J2 = 0 ,
(13)
because J2 = (rv)2 − (rṙ)2 , according to (7). Finally, by noting that the expression (13)
is valid also for r = 0 because r is a factor of J and then a common factor of each term
of that expression, it can be regarded as an authentic mathematical identity. In fact, it is
the first of a quartet of Nunesian fundamental identities which is completely presented in
other more comprehensive article [35].
Apropos, it is worthwhile to notice that the fundamental identity (13) holds in arbitrary (theoretical) coordinate systems and, consequently, in the corresponding (empirical)
reference frames. Besides, it is so known that certain results can have simpler expressions
and clearer interpretations depending on the type of coordinate system adopted (e.g., the
use of polar coordinates to study conic sections and planetary movements is much more
elucidative and instructive than the use of rectangular coordinates).
In order to create an open-ended theory, Nunes has adopted an interesting tactical Whenever
expedient which assures extensible applicability and impressive fertility for his approach. r 6= 0
It consists in the generation of more specific consequences from his fundamental identities
by establishing a hierarchy of restrictions which are characterized by the word ‘whenever’
followed by a set of mathematical expressions that delimits precisely the context of validity
of the theoretical results obtained. Thus, the expression ‘whenever r 6= 0’ characterizes
the level of the ‘general properties’ which are valid everywhere in the Euclidean threedimensional space, excepting the origin of the coordinate system, which is recognizedly a
very singular point. In practice, the restriction r 6= 0 permits the norm r to appear in
denominators. So, the next general property can be immediately obtained from (13)
r̈ =
3.2.
M J2
+ 3
r2
r
whenever
r 6= 0
(14)
The principle of inertia
In the Nunesian approach, each of the so-called conservation principles corresponds, in Conservation
general, to the ‘constancy’ of a determined mathematical quantity. For instances, the Principles
old ‘principle of conservation of the angular momentum’ corresponds to the constancy of J
(that is, J̇ = 0) and the new ‘principle of conservation of the Maccel’ corresponds to the
constancy of M (that is, Ṁ = 0). Similarly, the historical ‘principle of conservation of linear
momentum’, that is more known as ‘principle of inertia’ or Newton’s ‘First Law of Motion’,
corresponds to the constancy of v (that is, v̇ = 0). Since, by definition, v 2 ≡ v • v and
then v v̇ = v • v̇, the constancy of v (that is, v̇ = a = 0) implies v v̇ = 0 whence v = 0
or v̇ = 0 (that is, the speed v is constant). Besides, it also implies J̇ = 0 because, by
10
Kepler’s Equations
definition, J ≡ r × v and then J̇ = r × a = 0, and too M = 0 because M ≡ r(r • a),
as defined in (9). Therefore, from (14), it results r̈ = J2 /r3 with J = constant. In sum,
the constancy of the vector velocity v implies that the position vector r describes either a
stationary point or a straight line satisfying the supra mentioned principles all together. It
also satisfies the ’principle of the conservation of the mechanical energy’ as explained in [35].
Supplementary philosophico-historical readings concerned the principle of inertia and the
foundations of classical dynamics are Whitrow [42], Hanson [26], Christensen [10], Mora
[33] (entries: inercia and ı́mpetu), Crombie [13], and some others from McMullin [32]. By
the way, the word inertia was first used in Physics by Kepler.
r̈ =
J2
r3
Subscripts i
It is easy to see that r̈ = J2 /r3 , whenever r 6= 0 and a = 0, from the general property
(14) and the definition of M (9). Thereafter, one is naturally conducted to work out the
solutions of the second-order differential equation r̈(t) = J2 (t)/r3 (t), where J(t) remains
constant for every value of t. At once, it is evident that there exist two complementary
cases to be examined: one with r̈ = 0 and another with r̈ 6= 0.
1st Case: Supposing r̈ = 0, it implies that J = 0 too (whence vt = 0 and ṙ = ±v =
constant). Hence J ≡ r × v = 0 and then either v = 0 meaning that the (nonzero)
position vector r is constant (that is, it describes justly a stationary point which is at rest
in relation to the adopted coordinate system) or r k v meaning that the (nonzero) position
vector r is parallel to the (constant) vector velocity v (that is, it describes justly a straight
line with constant speed v). Therefore, in this
R r case, r̈ =R0t implies ṙ = ±v = constant, that
can be rewritten as dr = ±vdt and then r dr = ±v t dt, whence r − ri = ±v (t − ti ),
i
i
where ri and ti are arbitrary constants of integration. Every subscript ‘i’ qualifies the
respective symbol with the adjective ‘initial’ in order to associate its value with t = ti
(so, r = ri when t = ti ).
2nd Case: Supposing r̈ 6= 0, it implies that both ṙ 6= 0 and J 6= 0 too (whence
J2 = (rvt )2 = constant > 0 and v 2 = ṙ 2 + vt2 = constant > 0). For clarity’s sake, let
q ≡ ṙ, whence q̇ = r̈ = J2 /r3 . Now, multiplying it by the nonzero q it results q q̇ = (J2 /r3 )ṙ,
which can be rewritten as
2
Z q
Z r
q 2 − qi2
J
J2 1
dr
dr
1
dq
2
=
or
=
−
q
dq
=
J
whence
−
.
q
3
dt
r3 dt
2
2 r2
ri2
qi
ri r
Hence
2
2
J
J
2
= qi +
≡ vi2 = v 2 .
q +
r
ri
2
Thus, this result can be rewritten as
2
2
Z r
dr
J
2
=v −
or
dt
r
ri
"
whence
r dr
q
=
± (vr)2 − J2
Z
t
dt
ti
2 #1/2 "
2 #1/2
J
J
r2 −
− ri2 −
= ± v (t − ti ) .
v
v
Now, putting rJ2 ≡ (J/v)2 , R2 ≡ r2 − rj2 , and Ri2 ≡ ri2 − rj2 , it can be transformed into
R − Ri = ± v (u − ui ), which is analogous to that deduced justly for first case.
Law of
Areas
The Figure 3 illustrates the Pythagorean triangle formed by hr, rj , Ri and shows that
rJ is the smallest distance between the origin O of the coordinate system and the straight
line described by the position vector r. In fact, the above two cases can be distinguished
Raul Nunes
11
by the length of rJ : in the first case rJ = 0 and in the second case rJ > 0. Moreover,
the same figure evinces the called ‘law of areas’ for a motion with a = 0. For equal
intervals of length ∆l = AB = BC = . . . , the areas of the triangles AOB, BOC, . . . , are
exactly equals (because they have equal bases ∆l and a common height OV = rJ ). In
other words, the position vector r sweeps out equal areas in equal intervals of t given by
∆t ≡ ∆l/v, where v is a nonzero constant speed. From the Figure 3, a little remark can
still be made with respect to the constancy of J. It is easy to see that θ + βr,v = π/2
whence cos θ = sin βr,v and therefore rJ = r cos θ = r sin βr,v which multiplied by the
nonzero speed v gives (vrJ )2 = (vr sin βr,v )2 = J2 , in accordance with (4). Because both
v and rJ are constant, it results that J is also constant.
O
S
S
θ S
S
S
r
rS
J
S
S
S
S
r
v
SS
q
R
w
S
P S βr,v
A
B
C
V
S
Figure 3: Depicting the law of areas for a motion with a = 0.
3.3.
Solving r̈ =
M J2
+ 3
r2
r
(14)
For the purpose of this paper, let us interpret the general property (14) as a second-order Solving
nonlinear differential equation for r(t) in which the independent variable t is absent. The (14)
old papers of Jeans [28] and MacMillan [29], for instances, present interesting solutions of
(14) by supposing that M varies with the time t while J maintains a constant value. In
another recent paper of Shoucri [40], particular solutions of (14), expressed by his formula
(19), are discussed with both M and J depending on the time t. The Jeans’ formula (4)
and the MacMillan’s formula (2) are both equivalent to the general property (14), although
they have been obtained from the conjunction of the Newtonian law of gravitation with
the conservation of the angular momentum. In the Nunesian approach, however, it is
remarkable that (14) is a general property which is valid for any movement satisfying
only one very simple restriction: r 6= 0. Provided the mathematical expressions for the
functions M = M(t) and J = J(t), (14) can sometimes be solved using standard analytical
methods described in the specialized literature (e.g., lesson 35C of [41]). In particular,
when the movement satisfies both principles of conservation of the angular momentum J
(that is, J̇ = 0) and that of conservation of the Maccel M (that is, Ṁ = 0), the solutions
of (14) giving the time t as a function of r are relatively easy to obtain. However, ‘the
problem of solving the resulting equation for r as a function of t turns out to be exceeding
difficult.’ (comment 34.91 in [41]) In fact, it is immediately evident from (14) that there
exist two complementary cases to be examined: one with r̈ = 0 and another with r̈ 6= 0.
12
Kepler’s Equations
R r 1st Case:
R t Supposing r̈ = 0, it implies that ṙ = k, where k is a constant. Then,
dr = k t dt, whence r − ri = k (t − ti ), where ri and ti are two additional arbitrary
r
i
i
constants of integration. The constant speed k can be determined from Mi r + J2i = 0. If
Mi = 0, then J2i = (r vt )2 = 0 and v 2 = ṙ 2 , whence k = ±v; therefore, r = ri ± v (t − ti ) or
t = ti ∓ (r − ri )/v, when v 6= 0. That is, the position vector r describes either a rectilinear
segment with constant speed ṙ = ±v, or merely a stationary point when ṙ = v = 0. If
Mi 6= 0, then r = −J2i /Mi = ri , ṙ = k = 0, and J2i = (ri vt )2 = (ri v)2 = constant 6= 0,
for all values of t. That is, the position vector r describes a circle with constant angular
speed vt = ±v.
2nd Case: Supposing r̈ 6= 0, it implies that ṙ 6= 0. For clarity’s sake, let q ≡ ṙ, whence
q̇ = r̈. Now, multiplying (14) by the nonzero q, it produces
!
Z r
Z q
Z r
J2i
dr
Mi
dr
2
+
ṙ
whence
q
dq
=
M
+
J
q q̇ =
i
i
2
3
2
3
r
r
ri r
qi
ri r
because Mi and J2i are constants. From these integrals, one obtains
q 2 − qi2
J2i
1
1
1
1
= −Mi
−
−
− 2
or
2
r
ri
2
r2
ri
2
q +
2Mi
r
+
Ji
r
2
2
= qi +
2Mi
ri
+
Ji
ri
2
≡ Qi
which multiplied by r2 and ri2 , respectively, and returning ṙ in the place of q, they can be
rearranged as
Qi r2 − 2 Mi r − J2i = (rṙ)2
and
Qi ri2 − 2 Mi ri − J2i = (ri ṙi )2 .
It is evident that Qi is a constant squared speed whose value can be computed from the
four constants Mi , Ji , ri , and ri0 , all determined for t = ti .
Solving
(15)
When solving
Qi r2 − 2 Mi r − J2i = (rṙ)2
with rṙ 6= 0
(15)
as a first-order differential equation, one finds the following integrals
Z t
Z r
r dr
q
dt =
,
ti
ri
Qi r2 − 2Mi r − J2i
whose solutions (excluded the possibilities involving complex numbers) require the separate
consideration of the following three cases (cf. Appendix A):
p
2
if Qi = 0
⇒ t − ti = K =
+
−2Mi r − Ji
3M2i
(−Mi r + J2i )
(16)
q
if Qi < 0
⇒ t − ti = K <
+
Qi r2 − 2Mi r − J2i
Qi
q
if Qi > 0
⇒ t − ti = K >
+
Qi r2 − 2Mi r − J2i
Qi


−Q r + Mi
Mi

+
cos−1  q i
(-Qi )3/2
M2i + Qi J2i


Q r − Mi
Mi

+ 3/2
cosh−1  q i
Qi
M2 + Q J 2
i
i
i
Raul Nunes
13
where K= , K< , and K> are the habitual constants of integration associated to the rightside integrals (substituting r by ri ). These expressions giving t as functions of r could be
used to compute the proper time of every movement, known the five constants Mi , Ji , ri ,
ṙi , and ti . As consequence of the complexity inherent to these results, the inverse problem
of finding from them the analytical solutions of (15) giving r as a function of t turns out
to be exceedingly difficult, even assuming the simultaneous constancies of J and M. In
truth, in the case when Qi = 0, solutions of (15) giving r as a function of t can be obtained
by solving the cubic equation (whose complete solution will not pursued here):
2M3i r3 − 3M2i J2i r2 + J6i ± 9M4i [(t − ti ) − K= ]2 = 0
(17)
which can be derived from (16) using a few algebraical operations.
4.
4.1.
SOLVING r = r[r(θ(t)), θ(t)]
The equation of conics r = r(θ)
Now, let us appeal to a known mathematical expedient in order to obtain the solutions Definitions
of the equation (14) (with r 6= 0, J = Ji = constant and M = Mi = constant) which of ρ and θ
give the inverse of the norm r, hereafter denoted by ρ (that is, ρ ≡ 1/r ) as a function of
an auxiliary variable θ = θ(t), implicitly defined by means of the expression θ̇ ≡ Ji /r2 .
The geometric meaning of θ will be explained just a few lines further on. Let us consider
separately two complementary cases: the first with θ̇ 6= 0 and the second with θ̇ = 0.
1st Case: Supposing θ̇ 6= 0, it implies that Ji 6= 0 too. Besides, two successive
derivations of r = 1/ρ , with respect to t, produce:
ṙ = −
ρ̇
1 dρ
1 dρ dθ
1 dρ Ji
dρ
=− 2
=− 2
=− 2
= −Ji
ρ2
ρ dt
ρ dθ dt
ρ dθ r2
dθ
r̈ = −Ji
2
d2 ρ
d2 ρ dθ
d2 ρ Ji
2 2 d ρ
θ̇
=
−J
=
−J
=
−J
ρ
i
i
i
dθ2
dθ2 dt
dθ2 r2
dθ2
Replacing these values of r̈ and r in (14), one obtains
−J2i ρ2
d2 ρ
= Mi ρ2 + J2i ρ3
dθ2
whence
J2i
d2 ρ
+ρ
dθ2
= −Mi
whose solution giving ρ as function of θ is J2i [ρ − ρi cos(θ − θi )] = −Mi , where ρi and θi
are arbitrary constants of integration, whose values are determined for the initial time ti .
There are two complementary subcases to be considered: one with Mi 6= 0 and another
with Mi = 0. If Mi 6= 0, then ρ − ρi cos(θ − θi ) = − Mi /J2i . By putting
1
M
≡ − 2i
p
Ji
and e ≡ ρi p , it results
1
1
=
1 + e cos(θ − θi )
r
p
.
(18)
It is very easy to show that (18) is the equation of a general conic section when expressed
in polar coordinates, the characteristic elements of which are illustrated in the Figure 4.
14
Kepler’s Equations
If Mi = 0, then ρ − ρi cos(θ − θi ) = 0 or ri = r cos(θ − θi ) which represents a straight line
forming an angle π2 ± θi with the polar axis.
By definition, a conic section is the geometrical locus of the points P of a plane such Graphic
that the ratio between the distance PF to a fixed focus and the distance PP to a fixed of Conics
directrix is always equal to its constant eccentricity e. Besides, every point P can be
perfectly determined as function of its distance r = FP to the focus F and the angle
d measured between the polar axis FV and the segment FP (that is, P = P(r, θ)).
θ = VFP
By putting p ≡ FQ, one can write for the point Q = Q(p, π2 ) that QQ = p/e and for the
point P = P(r, θ) that PP = r/e. Finally, noting from the figure that FR = r cos θ and
QQ = PP + FR, it results
p
r
1
1
=
+ r cos θ whence
=
1 + e cos θ
e
e
r
p
which is the equation (18) without the θi corresponding to an initial rotation of the polar
axis. Additionally, it is so known that for e = 0, e < 1, e = 1, and e > 1, the conic section
is, respectively, a circle, an ellipse, a parabola, and a hyperbola.
Q
qQ
e=
FQ
= FP = FV
QQ
PP
VV
P
semifocal
width
p
qP
r
directrix
θ
O≡F
origin
focus
polar axis
r cos θ
R
V
vertex
q
V
Figure 4: The characteristic elements of the conic sections.
Degenerate
Conics
2nd Case: Supposing θ̇ = 0, it implies that Ji = 0 too. Hence J ≡ r × v = 0 and
then either v = 0 meaning that the (nonzero) position vector r is constant (that is, it
describes justly a stationary point P which is at rest in relation to the adopted coordinate
system) or r k v meaning that the (nonzero) position vector r is parallel to the vector
velocity v (that is, it describes justly a straight line containing the rectilinear segment FP
and forming an angle θ = θi = constant with the polar axis).
CONCLUSION: From the reasoning shown for both cases above, one concludes that,
whenever r 6= 0, J̇ = 0 and Ṁ = 0, the position vector r describes a conic section.
Raul Nunes
4.2.
15
The equations of the angle θ = θ(t)
In spite of this, the mathematical problem of finding analytical solutions of (14), giving r
as a function of t, still remains unsolved, even in the simpler case when J̇ = 0 and Ṁ = 0,
excepting the supra cited case of the cubic equation (17). By this reason, we are now
going to determine functions θ = θ(t) in order to use them as ‘bridges’ among the values
of the time t and the vector position r (that is r = r[r(θ(t)), θ(t)]).
From θ̇ ≡ Ji /r2 , r = p (1 + e cos θ)−1 , and p ≡ −J2i /Mi , one obtains
Ji
p2
Z
t
dt =
ti
M2i
Z
t
Z
θ
dt =
3
Ji
θi
ti
dθ
,
(1 + e cos θ)2
the solutions of which (excluded the points at which the second integrand is indefinite)
require the separate consideration of the following three cases (cf. Appendix B):
tan3
M2i
(t − ti ) = K= +
3
6
Ji
θ
2
tan
2
θ
2
for e = 1
⇒
for e < 1
M2
e sin θ
2
1
tan−1
⇒ 3i (t − ti ) = K< −
−√
1 − e2 1 + e cos θ
Ji
1 − e2
for e > 1
M2
1
e sin θ
2
−√
⇒ 3i (t − ti ) = K> + 2
tanh−1
2
e
−
1
1
+
e
cos
θ
Ji
e −1
+
"
r
1−e
θ
tan
1+e
2
"
r
e−1
θ
tan
e+1
2
!#
!#
where K= , K< , and K> are the habitual constants of integration associated to the rightside integrals (substituting θ by θi ).
When e = 1, which corresponds to the case of parabolas, the function θ(t) can be
obtained by solving the cubic equation for tan(θ/2) (known as Barker’s equation):
M2i
J3i
(t − ti ) − K= =
tan3
6
θ
2
+
tan θ2
.
2
e=1
(19)
It can be solved analytically by means of
the next expedient: (a) Let X ≡ tan3 (θ/2) + 3 tan(θ/2) = 6 (M2i /J3i )(t − ti ) − K= .
(b) Let tan(θ/2) ≡ 2 cot(2α) ; then X = [2 cot(2α)]3 + 3[2 cot(2α)].
Replacing
cot(2α) =
cot2 α − 1
2 cot α
and simplifying X, it results X =
cot6 α − 1
.
cot3 α
(c) Let cot3 α ≡ cot(β/2)
; therefore, X = 2cot(β/2).
(d) Let cot(β/2) ≡ 3 (M2i /J3i )(t − ti ) − K= .
In order to obtain θ(t), it is enough to proceed in the inverse order calculating cot(β/2),
cot(α), cot(2α), tan(θ/2), and θ.
When 0 ≤ e < 1, which corresponds to the case of ellipses, the function θ(t) can be 0 ≤ e < 1
obtained through a more elaborate procedure involving an auxiliary angle η defined by
the expression:
r
η
1−e
θ
1 − cos η
1 − e 1 − cos θ
tan ≡
tan
whence
=
,
2
1+e
2
1 + cos η
1 + e 1 + cos θ
16
Kepler’s Equations
according to the formula for half angles. In consequence, after some algebraical operations,
it results the following set of expressions:
cos θ + e
1 + e cos θ
√
1 − e2 sin θ
sin η =
1 + e cos θ
cos η =
cos η − e
1 − e cos η
√
1 − e2 sin η
sin θ =
1 − e cos η
cos θ =
Thus, the above solutions for 0 ≤ e < 1 can be rewritten as
η M2i
e sin η
2
1
√
−√
(t − ti ) = K< −
1 − e2
J3i
1 − e2
1 − e2 2
"
2 3/2
(1 − e )
M2i
J3i
or
#
(t − ti ) − K< = −e sin η + η
,
(20)
which is equivalent to the traditionally known as Kepler’s equation. The special case of
circles corresponds to ideal timepieces because η(t) = θ(t) varies linearly with t:
M2i
J3i
(t − ti ) − K< = θ
(when e = 0) .
Moreover, putting η(ti ) = 0 (whence K< = 0) and η(t − ti = P) = 2π, (20) gives
!
M2i
2 3/2
P = 2π ,
(1 − e )
J3i
where P denotes the period that the position vector r spends sweeping once the whole
ellipse. By symmetry, P is the double time necessary for it to move from the perihelion
where θp = 0 and rp ≡ p(1+e)−1 until the aphelion where θa = π and ra ≡ p(1−e)−1 . The
terms perihelion and aphelion were coined by Kepler and first appeared in his Mysterium
cosmographicum (1596) (cf. Applebaum [4], p.454). The arithmetic mean of these values
defines rm ≡ (rp + ra )/2, which corresponds to the half major axis of the ellipse; whence,
rm (1 − e2 ) = p. Now, recalling that p ≡ −J2i /Mi , a very significant result can be derived
for ellipses:
3
rm
−Mi = 4π2
(21)
P2
e>1
When e > 1, which corresponds to the case of hyperbolas, the function θ(t) can be
obtained through a very similar procedure now involving an auxiliary angle ς defined by
the expression:
r
ς
θ
cosh ς − 1
e − 1 1 − cos θ
e−1
tanh ≡
tan
whence
=
,
2
e+1
2
cosh ς + 1
e + 1 1 + cos θ
according to the formulas for half angles. In consequence, after some algebraical operations, it results the following set of expressions:
e + cos θ
1 + e cos θ
√
e2 − 1 sin θ
sinh ς =
1 + e cos θ
cosh ς =
e − cosh ς
e cosh ς − 1
√
e2 − 1 sinh ς
sin θ =
e cosh ς − 1
cos θ =
Raul Nunes
17
Thus, the above solutions for e > 1 can be rewritten as
ς M2i
2
1
e sinh ς
√
√
−
(t
−
t
)
=
K
+
i
>
e2 − 1
J3i
e2 − 1
e2 − 1 2
"
2
3/2
(e − 1)
M2i
J3i
or
#
(t − ti ) − K> = +e sinh ς − ς
,
(22)
which is analogous to the Kepler’s equation given in (20).
Although historically the denomination ’Kepler’s equation’ had been attributed only to
the equation (20), which is associated with elliptic motions, more recently that name has
been also applied to the equations (19) and (22), which are associated with parabolic and
hyperbolic motions, respectively.
"
2 3/2
(1 − e )
#
M2i
(t − ti ) − K<
J3i
= −e sin η + η
η
where tan ≡
2
M2i
J3i
"
2
3/2
(e − 1)
M2i
J3i
(t − ti ) − K= =
r
1−e
θ
tan
1+e
2
tan3
6
θ
2
+
tan θ2
2
and when 0 ≤ e < 1
(for ellipses)
when e = 1
(for parabolas)
and when e > 1
(for hyperbolas)
#
(t − ti ) − K> = +e sinh ς − ς
ς
where tanh ≡
2
r
θ
e−1
tan
e+1
2
Figure 5: The so-called Kepler’s Equations.
4.3.
Solving Kepler’s equations
The consequent problem of solving the Kepler’s equation (20) (that is, of finding the angle
η and the corresponding angle θ, for a determined time t) was first posed in Kepler’s
Astronomia Nova (1609). Due to the transcendental nature of the equation, Kepler was
forced to solve it by means of methods of approximation so tedious that he implored
the assistance of geometers in finding more suitable alternatives. In fact, during the last
four centuries, hundreds of methods have been devised for it, in virtue of its central role
played in Celestial Mechanics, by the foremost mathematicians and astronomers (such
as Newton, Euler, Lagrange, Bessel, Gauss, Cauchy, among many others). An extensive
bibliography is presented in Colwell’s book [12] which includes geometric constructions,
analytic approximations, iterative methods, expansions and power series, graphical and
analog devices, tables, etc. Kepler himself produced some very creditable procedures for
use in his Rudolphine Tables [34]. With the advent of computers, there is almost no
18
Kepler’s Equations
difficulty to compute quickly very accurate numeric solutions for any eccentricity, even
using ‘not-so-good’ methods; but the improving researches go on continuously, motivated
specially by the pragmatical requirements of the space sciences associated with artificial
satellites and interplanetary probes.
Additional readings about Kepler’s equations are the Fukushima’s papers [19] thru [24]
and publications (http://chiron.mtk.nao.ac.jp/∼toshio/), Danby [15] (chapter 6),
Danby-Burkardt [14], Burkardt-Danby [9], Odell-Gooding [36], Maeyama [30], Bottazzini
[8], Fernandes [17], Markley [31], Serafin [38] and [39], and Dutka [16].
Kepler’s
Laws
According to Plummer [37], the principal astronomical discoveries of Kepler may be
summarized in the following four points:
(a) The heliocentric motions of the planets (i.e., their motions relative to
the Sun) take place in fixed planes passing through the actual position of the
Sun.
(b) The area of the sector traced by the radius vector from the Sun, between
any two positions of a planet in its orbit, is proportional to the time occupied
in passing from one position to the other.
(c) The form of a planetary orbit is an ellipse, of which the Sun occupies
one focus.
(d) The square of the periodic time is proportional to the cube of the mean
distance (i.e., the semi-axis major).
These deductions from observation are given here in the order in which
they were discovered. The third (c) is generally known as Kepler’s first law,
the second (b) as his second law, and the fourth (d) as his third law. But the
first statement is of equal importance.
From the above exposition and assuming that the Sun occupies the origin of the adopted
reference frame, (a) results from the fixed spatial orientation which is embedded in the
constancy of the angular momentum J (that is, J̇ = 0); (b) results from the fact that, for
an elementary area dA, 2 dA = r2 dθ = Jdt whence 2(A − Ai ) = J(t − ti ) or ∆A = (J/2)∆t,
where J/2 is the constant of proportionality; (c) results from both conditions J̇ = 0 and
Ṁ = 0, when applied to the case of ellipses (that is, for 0 ≤ e < 1); and (d) results promptly
from (21), where the constant of proportionality is 4π 2 /(-M). However, for truth’s sake,
Kepler’s results assert additionally that 4π 2 /(-M) is approximately constant for all planets,
asteroids, comets, and the like, when Sun is placed at the origin of a common reference
frame. This is a very important empirical information that happens similarly for all natural
satellites of a same planet, when it is placed at the origin of a common reference frame.
Besides, in both cases, the observed trajectories are not exactly elliptical. Undeniably
complementary justifications to these local discrepancies and those global harmonies are
still required — probably deduced from a better gravitational approach and derived from
a plausible conjecture explaining the formation of our solar system.
Supplementary historical readings about the life and astronomical discoveries of Kepler
are Gingerich [25], Applebaum [4], Wilson [43] and [44], Aiton [1], [2], and [3], Baigrie [5]
and [6], Boccaletti [7], Hallyn [27], and Cohen [11]. Also ’The MacTutor History of
Mathematics archive’ (http://www-groups.dcs.st-and.ac.uk/∼history/) gives a
Kepler’s biography and more than a hundred of references about him.
Raul Nunes
5.
19
EPILOGUE
At every crossway on the road that leads to the future,
each progressive spirit is opposed by thousands appointed to guard the past.
Maurice Maeterlingk’s Our Social Duty
Although Classical Mechanics is usually assumed to be a completely finished chapter in Improvements
Physics, the new deduction of the so-called Kepler’s equations shown in this paper may
be scientifically interesting because it uses only kinematical concepts. This new viewpoint
is intended to be more intelligible, more plausible and more consistent than the traditional
approach, with no loss of descriptive and predictive powers. With the unique introduction
of the concept of maccel, the renowned Kepler’s equations can be mathematically deduced
with greater economy of concepts and assumptions, greater simplicity in the theoretical
formulation and wider applicability to a variety of natural phenomena. Although the
purpose of this paper is not to be an exhaustive study on Kepler’s equations, a lot of
related informations and references is here furnished in order to help other persons to
understand, compare and evaluate the value and advantages of this novel approach.
An inevitable comparison is relatively to the Newtonian approach to gravitational Comparisons
phenomena in virtue of its remarkable empirical success through the last three centuries.
Although concepts derived from different theories rarely can be considered as completely
equivalent — the exact meaning and perfect understanding of any concept depends on the
context where is defined — maccel would sometimes be regarded as numerically quasi equal to the negative product of the Newtonian gravitational constant G by the gravitational mass of a body situated at the origin of the reference frame. But, differently of
the Newtonian approach which considers the accelerations of the equation (12) as actual
effects caused by ’attractions’ (or ’repulsions’) directed to the origin of the reference frame,
in the Nunesian approach, they may be physically interpreted as mere consequences of the
relative movements between the extremities of the position vector. In fact, neither the
existence of two (or even more) attractive bodies nor the usual Newtonian theoretical
approximation by means of point-like bodies have been here required or assumed. Maccel
is simply a singular attribute of every position vector and may be physically thought of
as being the resultant from previous (by-contact) interactions. It remains unaltered the
possibility of eventual employment of that accelerations to produce some useful work,
depending only on the ingenuity of the inventors. Moreover, this innovative conception
seems be a comprehensive solution to three very old enigmas associated to gravity, namely,
apparent action-at-a-distance, shielding impossibility and observed instantaneousness.
The author believes that anybody well informed about Astrophysics and its intriguing
questions immediately will perceive the deep and broad implications of his contribution.
For example, on the contrary of Einsteinian approach, the introduction of the concept
of maccel turns unnecessary any appeal to a never-detected propagation of gravity. On
the other hand, the observed gravitational bending of light rays will need an alternative
justification — much probably based on refraction phenomenon. Above all, the concept of
maccel clearly constitutes an innovative perspective on the called ’dark matter’ and ’dark
energy’ problems recorded in large astronomical systems. It is probable these theoretical
deficiencies may be satisfactorily removed as the gravitational masses are alternatively
interpreted as maccels. Additionally, since the mathematical definition of maccel given by
(9) is completely independent of dynamical concepts such as masses, charges, and the like,
there exist certainly many other applications for it, in different Physics-related areas.
20
Expectations
Kepler’s Equations
The author expects optimistically to receive, from the esteemed readers, a number of
constructive criticisms, interesting commentaries and also sympathetic encouragements.
Be sure that everything will be duly appreciated and, as much as possible, incorporated
in future writings. For example, the detailed demonstrations of integrals presented in the
appendixes aim to facilitate the understanding of the text by graduated students, while
the approximated symmetry of their solutions (particularly in terms of correspondence
among circular and hyperbolic functions) aim to satisfy mnemonic and aesthetic guidelines
specified by a friend of him. Above all, he thinks that the contents of his work should
be appreciated with enough rationality and impartiality. As emphasized by Richard P.
Feynmann [18]: ”Most people find it surprising that in science there is no interest in
the background of the author of an idea or in his motive in expounding it. You listen,
and if it sounds like a thing worth trying, a thing that could be tried, is different, and
is not obviously contrary to something observed before, it gets exciting and worthwhile.”
In truth, Science is an adaptive process and its adaptability is the principal guarantee of
its permanent utility for the humanity.
oooOOOooo
Raul Nunes
21
APPENDIXES
A
SOLVING THE INTEGRAL (Qr2 −2Mr−J2 )−1/2 r dr
R
This appendix proves that the solutions of the integral
Z r
r dr
p
I=
,
2
ri
Qr − 2Mr − J2
(23)
where r 6= 0 and <ri , Q, M, J > are four given constants, can be expressed by the next
formulas, according to the sign of Q:
p
−2Mr − J2
(−Mr + J2 )
when Q = 0 ⇒ I= −K= =
(24)
3M2
!
p
Qr2 − 2Mr − J2
M
−Qr + M
when Q < 0 ⇒ I< −K< =
+
cos−1 p 2
(25)
3/2
Q
(-Q)
M + QJ2
!
p
Qr − M
Qr2 − 2Mr − J2
M
−1
p
when Q > 0 ⇒ I> −K> =
(26)
+ 3/2 cosh
Q
Q
M2 + QJ2
where < K= , K< , K> > are the habitual constants of integration associated to the rightside expressions (substituting r by ri ), and where the possibilities involving complex numbers have been left out.
When Q = 0 — In this subcase, the integral (23) takes the form
Z r
r dr
p
I= =
,
ri
−2Mr − J2
where the value of the radicand (−2Mr − J2 ) is assumed to be greater than zero. In
consequence, M 6= 0, otherwise the remainder expression (−J2 ) would not be greater than
zero. The integral can then be solved by using the auxiliary variable
α2 ≡ −2Mr − J2
whence r = −(α2 + J2 )/2M and dr = −α dα/M. Thus, its solution is immediate:
Z α
Z α
1
1
−(α2 + J2 ) −α dα
=
(α2 + J2 ) dα ,
I= =
2M
M
α
2M2 αi
αi
p
where αi = −2Mri − J2 . So, it results
I= =
1 2
2
2
2
2 α(α + 3J ) − αi (αi + 3J ) .
6M
Now, returning to the former variables, the solution given by (24) is obtained:
p
−2Mr − J2
I= −K= =
(−Mr + J2 )
3M2
where
q
−2Mri − J2
K= = −
(−Mri + J2 ) .
3M2
22
Kepler’s Equations
When Q < 0 — In this subcase, the integral (23) takes the form
r
Z
I< =
ri
r dr
,
q
−Q2< r2 − 2Mr − J2
where Q2< ≡ −Q > 0 and the value of the radicand (−Q2< r2 − 2Mr − J2 ) is assumed to be
greater than zero. The integral can then be solved by using the auxiliary variables
ρ ≡ r + M/Q2<
and κ2 ≡ (M2 − Q2< J2 )/Q4<
and assuming that κ2 is a positive constant. In consequence, it results that
−Q2< r2 − 2Mr − J2 = Q2< (κ2 − ρ2 ) > 0
, r = ρ − M/Q2<
and dr = dρ
whence
Z
ρ
I< =
ρi
(ρ − M/Q2< ) dρ
1
q
=
Q<
Q2< (κ2 − ρ2 )
Z
ρ
ρi
M
ρ dρ
p
− 3
2
2
Q<
κ −ρ
Z
ρ
ρi
dρ
p
.
2
κ − ρ2
The solution of the first integral is immediate:
ρ
Z
ρ dρ
p
ρi
κ2 − ρ2
q
p
= − κ2 − ρ2 + κ2 − ρ2i ,
where ρi = ri + M/Q2< is a constant. But, in order to solve the second integral, let the
auxiliary variable β be defined as β ≡ cos−1 (ρ/κ), whenever (ρ/κ)2 < 1. So, its numerator
can
into dρ = −κ
p be transformed
p sinβ dβ because ρ = κ cosβ, and its denominator into
p
κ2 − ρ2 = κ 1 − (ρ/κ)2 = κ 1 − cos2 β = κ sinβ. Hence,
Z
ρ
ρi
Z
dρ
p
κ 2 − ρ2
β
=
βi
−κ sinβ dβ
=
κ sinβ
Z
β
−dβ = −β + βi ,
βi
where βi = cos−1 (ρi /κ) is another constant. So, it results
1
I< =
Q<
p
q
M
2
2
2
2
− κ − ρ + κ − ρi − 3 (−β + βi ) .
Q<
Now, returning to the former variables, the solution given by (25) is obtained:
p
Qr2 − 2Mr − J2
M
I< −K< =
+
cos−1
Q
(-Q)3/2
−Qr + M
p
M2 + QJ2
!
,
where
q
K< = −
Qri2 − 2Mri − J2
Q
M
−
cos−1
(-Q)3/2
−Qr + M
p i
M2 + QJ2
!
.
Raul Nunes
23
When Q > 0 — In this subcase, the integral (23) takes the form
Z
r
I> =
ri
r dr
q
2
Q>
r2
,
− 2Mr − J2
where Q2> ≡ Q > 0 and the value of the radicand (Q2> r2 − 2Mr − J2 ) is assumed to be
greater than zero. The integral can then be solved by using the auxiliary variables
ρ ≡ r − M/Q2>
and κ2 ≡ (M2 + Q2> J2 )/Q4> .
and assuming that κ2 is a positive constant. In consequence, it results that
Q2> r2 − 2Mr − J2 = Q2> (ρ2 − κ2 ) > 0
, r = ρ + M/Q2>
and dr = dρ
whence
Z
ρ
I> =
ρi
(ρ + M/Q2> ) dρ
1
q
=
2
Q
2
2
>
Q> (ρ − κ )
Z
ρ
ρi
ρ dρ
+
p
ρ2 − κ 2
M
Q3>
Z
ρ
ρi
dρ
p
ρ2 − κ 2
.
The solution of the first integral is immediate:
Z ρ
q
p
ρ dρ
p
= ρ2 − κ2 − ρ2i − κ2 ,
ρ2 − κ 2
ρi
where ρi = ri − M/Q2> is a constant. But, in order to solve the second integral, let the
auxiliary variable β be defined as β ≡ cosh−1 (ρ/κ), whenever ρ/κ > 1 . So, its numerator
can
into dρ = κp
sinhβ dβ because ρ = κ coshβ, and its denominator into
p be transformed
p
ρ2 − κ2 = κ (ρ/κ)2 − 1 = κ cosh2 β − 1 = κ sinhβ. Hence,
Z
ρ
ρi
Z
dρ
p
ρ2 − κ 2
β
=
βi
κ sinhβ dβ
=
κ sinhβ
β
Z
dβ = β − βi .
βi
where βi = cosh−1 (ρi /κ) is another constant. So, it results
I> =
q
1 p 2
M
ρ − κ2 − ρ2i − κ2 + 3 (β − βi ) .
Q>
Q>
Now, returning to the former variables, the solution given by (26) is obtained:
!
p
Qr2 − 2Mr − J2
M
Qr − M
−1
p
I> −K> =
+ 3/2 cosh
,
Q
Q
M2 + QJ2
where
K> = −
q
Qri2 − 2Mri − J2
Q
−
M
Q3/2
−1
cosh
Qr − M
p i
M2 + QJ2
!
.
24
Kepler’s Equations
B
SOLVING THE INTEGRAL
R
(1 + e cos θ)−2 dθ
This appendix proves that the solutions of the integral
Z θ
dθ
,
I=
(1
+
e
cos θ)2
θi
(27)
where the initial angle θi and the non-negative eccentricity e are given constants, can be
expressed by the next formulas, according to the value of e:
for e = 1
for e < 1
tan3 θ2
" 6
1
e sin θ
2
⇒ I< −K< = −
−√
tan−1
1 − e2 1 + e cos θ
1 − e2
tan
2
⇒ I= −K= = +
θ
2
+
(28)
r
"
for e > 1
e sin θ
2
1
−√
⇒ I> −K> = + 2
tanh−1
e − 1 1 + e cos θ
e2 − 1
r
1−e
θ
tan
1+e
2
e−1
θ
tan
e+1
2
!#
(29)
!#
(30)
where h K= , K< , K> i are the habitual constants of integration associated to the
right-side integrals (replacing θ by θi ), and where the points at which the integrand is
indefinite are excluded (that is, when 1 + e cos θ = 0).
By means of the known transformations:

2 tan θ2
2t


sin
θ
=
=


θ
2

1 + t2
1 + tan 2





θ
1 − tan2 θ2
1 − t2
t ≡ tan
=⇒
cos
θ
=
=

2

1 + t2
1 + tan2 θ2







 dθ = 2 dt
1 + t2
(31)
the integral (27) can be transformed in the next rational expression
Z
θ
I=
θi
where ti = tan
θi
2
dθ
=
(1 + e cos θ)2
Z
t
ti
2(1 + t2 ) dt
,
[(1 + e) + (1 − e)t2 ]2
.
For e = 1 — In this subcase, the integrals (32) take the form
Z
θ
I= =
θi
whence
1
I= =
2
Z
t
ti
dθ
1
=
(1 + cos θ)2
2
1
dt +
2
Z
Z
t
ti
t2 dt =
t
(1 + t2 ) dt ,
ti
t3 − t3i
t − ti
+
2
6
Now, returning to the former variables, the solution given by (28) is obtained:
I= −K= = +
tan θ2
tan3
+
2
6
θ
2
(32)
Raul Nunes
where
K= = −
tan
2
θi
2
−
25
θi
2
tan3
6
.
For e < 1 — In this subcase, the integrals (32) take the form
Z
I< =
dθ
=
(1 + e cos θ)2
Z
2
2(1 + t2 ) dt
= 4
2
2
2
2
(b + a t )
a
Z
dt
+
2
(c + t2 )2
t2 dt
(c2 + t2 )2
Z
,
where a2 ≡ 1 − e > 0, b2 ≡ 1 + e > 0, and c ≡ b/a. Note that, aiming a better clarity in
the notation, the integration limits have been here omitted; but, in the final expression,
they will be recalled through a unique constant K< . Thus
Z
dt
1
= 2
(c2 + t2 )2
c
Then
I< =
1
(c2 + t2 ) − t2
dt = 2
(c2 + t2 )2
c
Z
Z
1
dt
− 2
c2 + t2
c
Z
t2 dt
.
(c2 + t2 )2
Z
Z
2 1
dt
1
t2 dt
+
1
−
.
a4 c2
c2 + t2
c2
(c2 + t2 )2
Now, the solution of the
R second integral
R can be found using the known method of integration by parts (that is, p dq = pq − q dp). Thus, let
and q ≡ (c2 + t2 )−1
p≡t
Then
Z
whence
1
t2 dt
=−
2
2
2
(c + t )
2
and dq = −(c2 + t2 )−2 (2t) dt .
dp = dt
t
2
c + t2
+
1
2
Z
c2
dt
.
+ t2
Therefore,
I< =
2
a4
1
c2 − 1
+
2
c
2c2
Z
dt
−
c2 + t2
c2 − 1
2c2
t
c2 + t2
.
At last, the remainder integral has the well known solution:
Z
1
t
dt
−1
=
tan
.
2
2
c +t
c
c
Then
I< = −
c2 − 1
a4 c2
t
+
2
c + t2
c2 + 1
a4 c3
tan−1
t
.
c
But,
2
2 2
c2 − 1
b
a
b − a2
−2e
=
−
−
1
=
−
=
and
a4 c2
a2
a4 b2
a4 b2
a4 b2
2
2
2
c +1
2
1
c +1
1
b
a
b2 + a2
=
=
=
+
1
= 3 3.
4
3
2
2
2
2
2
a c
a b
c
a b
a
b
a3 b3
a b
−
In consequence,
I< =
−2e
a4 b2
t
+
c2 + t2
2
a3 b3
−1
tan
t
.
c
But, according to (31),
−2e
t
−2e
t
−1
2et
1 + t2
=
=
a4 b2 c2 + t2
a2 b2 b2 + a2 t2
a2 b2 1 + t2 (1 + e) + (1 − e)t2
26
Kepler’s Equations
=
−1
−1
1 − t2
−1
e sin θ
e
sin
θ
1
+
e
=
1 − e2
1 + t2
1 − e2 1 + e cos θ
and
2
3
a b3
−1
tan
−1
=
1 − e2
−2
√
1 − e2
r
!
r
1−e
θ
tan
1+e
2
θ
a2
tan
2
b
2
!
r
1−e
θ
tan
.
1+e
2
−2
t
−1
=
tan−1
c
a2 b2
ab
tan−1
Therefore, the solution given by (29) has been obtained:
"
1
e sin θ
2
I< −K< = −
−√
tan−1
1 − e2 1 + e cos θ
1 − e2
!#
,
where
1
K< = +
1 − e2
"
e sin θi
2
−√
tan−1
1 + e cos θi
1 − e2
r
1−e
θ
tan i
1+e
2
!#
.
For e > 1 — In this subcase, the integrals (32) take the form
Z
I> =
dθ
=
(1 + e cos θ)2
Z
2(1 + t2 ) dt
2
= 4
2
2
2
2
(b − a t )
a
Z
dt
+
2
(c − t2 )2
t2 dt
(c2 − t2 )2
Z
,
where a2 ≡ e − 1 > 0, b2 ≡ e + 1 > 0, and c ≡ b/a. Note that, aiming a better clarity in
the notation, the integration limits have been here omitted; but, in the final expression,
they will be recalled through a unique constant K> . Thus
Z
1
dt
= 2
2
2
2
(c − t )
c
Then
I> =
Z
1
(c2 − t2 ) + t2
dt = 2
2
2
2
(c − t )
c
Z
1
dt
+ 2
2
2
c −t
c
Z
t2 dt
.
(c2 − t2 )2
Z
Z
2 1
dt
1
t2 dt
+
1
+
.
a4 c2
c2 − t2
c2
(c2 − t2 )2
Now, the solution of the
R second integral
R can be found using the known method of integration by parts (that is, p dq = p q − q dp). Thus, let
p≡t
and q ≡ (c2 − t2 )−1
Then
Z
whence
t2 dt
1
=
2
(c − t2 )2
2
dp = dt and dq = −(c2 − t2 )−2 (−2t) dt .
t
c2 − t2
−
1
2
Z
c2 + 1
2c2
dt
.
c2 − t2
Therefore,
2
I> = 4
a
1
c2 + 1
−
c2
2c2
Z
dt
+
2
c − t2
t
2
c − t2
.
At last, the remainder integral has the following solution:
Z
1/2
dt
1
c+t
1
1 + t/c
1
t
−1
=
ln
= ln
= tanh
.
c2 − t2
2c
c−t
c
1 − t/c
c
c
Raul Nunes
27
In fact,
1+X
if x = ln
1−X
1/2
Then
I> =
, then e2x =
c2 + 1
a4 c2
1+X
1−X
t
−
c2 − t2
 2x
e − Xe2x = 1 + X




 ex − Xex = e−x + Xe−x
⇒
ex − e−x = X(ex + e−x )


x
−x

 X = e −e

= tanh(x)
ex + e−x
c2 − 1
a4 c3
tanh−1
t
.
c
But,
2
2 2
b
a
b + a2
2e
c2 + 1
=
+
1
=
=
and
a4 c2
a2
a4 b2
a4 b2
a4 b2
2
c −1
−1 c2 − 1
−1 b2
a
−b2 + a2
−2
−
=
=
−
1
=
= 3 3.
4
3
2
2
2
2
2
a c
a b
c
a b
a
b
a3 b3
a b
In consequence,
I> =
2e
a4 b2
t
−
c2 − t2
2
a3 b3
tanh−1
t
.
c
But, according to (31),
2e
t
2e
t
1
2et
1 + t2
=
=
4
2
2
2
2
2
2
2
2
2
2
2
a b
c −t
a b
b −a t
a b
1+t
(1 + e) + (1 − e)t2
−1
1
1 − t2
e sin θ
1
= 2
e sin θ 1 + e
= 2
e −1
1 + t2
e − 1 1 + e cos θ
and
−
!
r
θ
1
−2
a2
t
−1
tan
tanh
=
tanh
c
a2 b2
ab
b2
2
!
r
1
e−1
θ
−2
√
= 2
tanh−1
tan
.
e −1
e+1
2
e2 − 1
2
a3 b3
−1
Therefore, the solution given by (30) has been obtained:
"
1
e sin θ
2
I> −K> = + 2
−√
tanh−1
2
e − 1 1 + e cos θ
e −1
r
e−1
θ
tan
e+1
2
!#
,
where
1
K> = − 2
e −1
"
e sin θi
2
−√
tanh−1
2
1 + e cos θi
e −1
r
e−1
θ
tan i
e+1
2
!#
.
28
Kepler’s Equations
References
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ISIS, 60, pp.75–90
[2] Aiton, E. J.: 1975, ”How Kepler discovered the elliptical orbit”,
The Mathematical Gazette, 59, 410, pp.250–260
[3] Aiton, E. J.: 1976, ”Johannes Kepler in the Light of Recent Research”,
History of Science, 14, pp.77–100
[4] Applebaum, W.: 1996, ”Keplerian Astronomy after Kepler: Researches and Problems”,
History of Science, 34, pp.451–504
[5] Baigrie, B. S.: 1987, ”Kepler’s Laws of Planetary Motion, before and after
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