Resonance Articles

391 Pages·20.39 MB·English

by  Thanu Padmanabhan

GENERAL I ARTICLE
Planets Move in Circles !
A Different View of Orbits
T Padmanabhan
The orbits of planets, or any other bodies moving under an
inverse square law force, can be understood with fresh
insight using the idea of velocity space. Surprisingly, a
particle moving on an ellipse or even a hyperbola still
moves on a circle in this space. Other aspects of orbits
such as conservation laws are discussed.
T Padmanabhan works at
the Inter University
Centre for Astronomy and Yes, it is true. And no, it is not the cheap trick of tilting the paper
Astrophysics (IUCAA) at to see an ellipse as a circle. The trick, as you will see, is a bit more
Pune. His research
sophisticated. It turns out that the trajectory of a particle, moving
interests are in the area of
under the attractive inverse square law force, is a circle (or part of
cosmology, in particular
the formation of large a circle) in the velocity space (The high-tech name for the path in
scale structures in the velocity space is hodograph). The proof is quite straightforward.
universe, a subject on
Start with the text book result that, for particles moving under
which he has written two
any central force f (r) ~, the angular momentum J= r x p is
books. The other area in
which he works is the conserved. Here r is the position vector, p is the linear
r
interface between gravity momentum and is the unit vector in the direction of r. This
and quantum mechanics.
implies, among other things, that the motion is confined to the
He writes extensively for
plane perpendicular to J. Let us introduce in this plane the polar
general readers, on topics
ranging over physics, coordinates (r, 8) and the cartesian coordinates {X, y). The
mathematics, and just conservation law for J implies
plain brain teasers.
de
-- = constantlr 2 == hr 2, (1)
d,t
which is equivalent to Kepler's second law, since (r28/2)= h/2 is
the area swept by the radius vector in unit time. Newton's laws
of motion give
dv dv
m _x = fer) cos 8; m 2 =f(r) sin 8. (2)
dt dt
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GENERAL I ARTICLE
Di viding (2) by (1) we get The high-tech
name for the path
m dvx = fer) r2 cos e; m dvy = fer) r2 sin e . (3) in velocity space is
de h de h hodograph.
The miracle is now in sight for the inverse square law force, for
whichf(r) r2 is a constant. For planetary motion we can set it
to fer) r2 = - GMm and write the resulting equations as
dv -GM dv -GM.
x = - cose; --L = __ Sine. (4)
de h de h
Integrating these equations, with the initial conditions Vx (e=O)
=0; Vy (8 = 0) = u, squaring and adding, we get the equation to
the hodograph:
v2 +(v - u + GM/h)2 = (GM/h)2 (5)
x y
which is a circle with center at (0, u - GM/h) and radius GM/h.
So you see, planets do move in circles!
Some thought shows that the structure depends vitally on the
ratio between u and GM/h, motivating one to introduce a
quantity e by defining (u - GM/h) =e (GM/h). The geometrical
meaning of e is clear from Figure 1. If e = 0, i.e, if we had chosen
the initial conditions such that u = GM/h, then the center of the
hodograph is at the origin of the velocity space and the
magnitude of the velocity remains constant. Writing h=ur, we
get u2 = GM/r2 leading to a circular orbit in the real space as
well. When 0 <e < 1, the origin of the velocity space is inside the
circle of the hodograph. As the particle moves the magnitude of
the velocity changes between a maximum of (1 +e) (GM/h) and
a minimum of (1-e) (GM/h). When e=l, the origin of velocity
space is at the circumference of the hodograph and the
magnitude of the velocity vanishes at this point. In this case, the
particle goes from a finite distance of closest approach, to
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GENERAL I ARTICLE
ellipse
-~-----l~-----1-~ V'&
infinity, reaching infinity with zero speed. Clearly,e = 1 implies Figure 1 Velocity Space:
The x and y components
u2=2GM/rinitial which is just the text book condition for escape
of the velocity vector v are
velocity and a parabolic orbit.
plotted. The origin repre
sents zero velocity, and
When e > 1, the origin of velocity space is outside the hodograph
the circle gives the velocity
and Figure 2 shows the behaviour in this case. The maximum of the planet at different
velocity achieved by the particle is DB when the particle is at the times i.e it is the orbit in
point of closest approach in real space. The asymptotic velocities velocity space. The radius
vector in real space is
of the particle are DA and DC obtained by drawing the tangents
parallel to the tangent to
from D to the circle. From the figure it is clear that sin ~ = e-1•
this circle, because chan
During the unbound motion of the particle, the velocity vector
ges in velocity are parallel
traverses the part ABC. It is circles all the way! (Incidentally, can to the force which is
e
you find a physical situation in which the minor arc AC could be central. The angle turned
meaningful ?) by the tangent is thus the
same as that turned by the
radius vector. Real space
Gi ven the veloci ties, it is quite easy to get the real space tra jectories.
orbit at top right.
Knowing Vx (9) and Vy (9) from (4) one can find the kinetic
energy as a function of 9 and equate it to (E+GMm/r), thereby
recovering the conic sections. Except that, there is a more elegant
way of doing it.
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GENERAL I ARTICL!
hyperbDla
•
There is a theorem proved by Newton through (rather than in) Figure 2 Velocity space
representation of a
Principia, which states that 'anything that can be done by
hyperbolic orbit. Note that
calculus can be done by geometry' and our problem is no
the origin is now outside
exception. The geometrical derivation is quite simple: In a small
the circle. Only the arc ABC
ot,
time interval the magnitude of the velocity changes by ~V= is traversed. OB is the
ot
f (GM/r2) according to Newton's law. The angle changes by maximum velocity, attain
~e = (h/r2) ot from the conservation of angular momentum. ed at closest approach, 2cp
is the angle of scattering.
Dividing the two relations, we get
Real space orbit shown at
right.
~Ivl = GM (6)
~e h
But in velocity space ~V is the arc length and ~eis the angle of
turn and if the ratio between the two is a constant, then the curve
is a circle. So there you are.
To get the real space trajectory from the hodograph, we could
reason as follows: Consider the transverse velocity v at any
T
instant. This is clearly the component perpendicular to the
instantaneous radius vector. But in the central force problem,
the velocity change ~V is parallel to the radius vector. So v T is
also perpendicular to .i\v; or in other words, the v must be the
T
________, AAnAA,_ ______ _
VV
RESONANCE I September1996 v V V v 37
GENERAL I ARTICLE
component of velocity parallel to the radius vector in the velocity There is a theorem
space. Voila! From Figure 1, it is just proved by Newton
through (rather
GM GM GM h than in) Principia,
v = - + - e cos 9 = - (1 +e cos 9) = - (7)
T h h h r which states that
'anything that can
with the last relation following from the definition of angular be done by
momentum. One immediately sees the old friend - the conic calculus can be
section - with a latus rectum of l=h2/GM and eccentricity of e done by geometry'
(Good we didn't denote the ratio between (u - GM/h) and (GM/h)
by k or something!).
The elegance of geometry over calculus in the above analysis (or
anywhere for that matter, though lots of people disagree) is a bit
fake with calculus entering through the back door. But even with 1 This is a space obtained by
calculus, the more general way to think about the Kepler problem combining the three
coordinates xyz with three
is as follows. For any time-independent central force, we have
momentum components mv)(,
constancy of energy E and angular momentum J. Originally, a mv)" m vz. This is a good way of
particle moving in 3 space dimensions has a phase spacel which describing the current state of
the system. since one can use
is 6 dimensional. Conservation of the four quantities (E,J ' Jy'
= x this information to predict the
Jz) confines the motion to a region of 6 - 4 2 dimensions. The
future.
projection of the trajectory onto the xy-plane will, in general, fill
C/FtCLES OR NO CIRCL.ES •.•. As /..ON&
As w£'IlE IN oRBIT IN ~E
SPIJCE!
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GENERAL I ARTICLE
a two-dimensional region of space. That is, the orbit should fill a finite
region of the space in this plane, if there are no other conserved
quantities. But we are always taught that the bound motion is an ellipse
in the xy-plane, which is an one-dimensional curve. So, there must
exist yet another conserved quantity for the inverse square law force
which keeps the planet in one dimension rather than two. And indeed
there is, which provides a really nice way of solving the Kepler pro blem.
To discover this last constant, consider the time derivative of the
r.
quantity p x J in any central force f(r) With a little bit of algebra, you
can show that
-d (px J) = - mf(r)r2 ~d~ (8)
dt dt
The miracle of inverse square force is again in sight: When
f (r)r2 = constant =-GMm, we find that the vector
GMm2
A=pxJ---r (9)
r
is conserved. But we needed only one more constant of motion, now we
have got three which will prevent the particle from moving at all! No,
it is not an overkill; one can easily show that A satisfies the following
relations:
A2 = 2 mp E + (GMm) 2; A· J = O. (10)
The first one tells you that the magnitude of A is fixed in terms of
other constants of motion and so is not independent; and the second
shows that A lies in the orbital plane. These two constraints reduce the
number of independent constants in A from 3 to 1, exactly what we
needed. It is this extra constant that keeps the planet on a sensible orbit
(i.e. a closed curve!). To find that orbit, we only have to take the dot
product of (9) with the radius vector r and use the identity r' (p X
J) = J. ( r x p) = p. This gives
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RESONANCE I September1996 39
GENERAL I ARTICLE
Stepping into Kepler's shoes
You must have read that Kepler analysed the astronomical data of Tycho Brahe and arrived at his laws
of planetary motion. Ever wondered how exactly he went about it? Remember that the observations are
made from the Earth which itself moves with an unknown trajectory! Suppose you were given the
angular positions of all the major astronomical objects over a long period of time, obtained from some
fixed location on Earth. This is roughly what Kepler had. How will you go about devising an algorithm
that will let you find the trajectories of the planets? Think about it!
= e
A . r Ar cos = ] 2 - GMm 2 r (11)
or, in more familiar form,
(12)
r
As a bonus we see that A is in the direction of the major axis of
the ellipse. One can also verify that the offset of the centre in the
hodograph, (GM/h) e, is equal to (A/h). Thus A also has a geo
metrical interpretation in the velocity space. It all goes to show
how special the inverse square law force is! Ifw e add a component
1Ir to the force, (which can arise if the central body is not a
3
sphere) ] andE are still conserved but not A. If the inverse cube
perturbation is small, it will make the direction of A slowly
change in space and we get a 'precessing' ellipse.
Address for Correspondence
T Padmanabhan
Inter-University Centre for Suggested Reading
Astronomy and AstrophYSiCS
•
Post Bag No.4, Ganeshkhind Sommerfeld A. Lectures on Theoretical Physics. Mechanics. Academic
Pune 411 007, India Press. YoU, p.40
• Rana N C and Joag P S. Classical Mechanics. Tata McGraw Hill. p.140
Poisson Mathematics
That Poisson liked teaching can be seen from his own words: -Life is made
beautiful by two things - studing mathematics and teaching W. ( From:
The Malhemalicallnlelligencer.VoI.l7.No.1. 1995 )
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4
Information and Announcements
Inter-University Centre for Astronomy and Astrophysics (IUCAA)
Starting as an idea in 1987 and dedicated to the nation in 1992, IUCAA is probably the
youngest astronomical institution in this country. Because of this youth, it is worthwhile to
go over the motivations and objectives of this institution as well as its structure and
achievemen ts.
Astronomy and astrophysics have come a long way in the last few decades and research in
these areas have become highly resource-dependent as regards manpower, observing facilities
and computing. If Indian Universities were to participate in this exciting adventure on an
individual basis, considerable investment would have been required in order to provide the
necessary resources to even a select set. It was, therefore, felt that a worthwhile beginning
could be the establishment of an Inter-University Centre in astronomy and astrophysics
which could coordinate the efforts to nucleate astronomy research in the Universities and act
as a field station for all the university staff in this country interested in astronomy and
astrophysics. The University Grants Commission (UGC) established IUCAA with this
motivation and has provided it with funding for facilities like computing, library, visitors
programmes and an optical telescope (under construction). It is felt that Indian Universities,
which are interested in developing astronomy and astrophysics activities of their own, could
use IUCAA as a launch pad.
How does one translate this objective into a concrete plan of action? To begin with, IUCAA
has a small - but growing - core faculty, postdocs and research students all of whom are
engaged in basic research in A&A and participate in the teaching and developmental activities
of the institution. The research is expected to be of international standards in order to
maintain credibility as a national coordinating centre in activities related to astronomy and
astrophysics.
While in this activity IUCAA may be similar to several other research institutions in this
country, what sets it apart from other astronomy and astrophysics institutions is the fact that
it is an Inter-University Centre (IUC). As a part of the above role, it is important that it takes
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INF. & ANN.
a lead in·n urturing the
culture of astronomical
instrumentation and
manpower training in
astronomy and astro
physics in this country,
especially in the univer
sity sector. IUCAA
addresses this issue
through several intro
ductory and refresher
courses at the univer
sity and college tea
chers' level. It also
holds several national and international conferences which members of the university sector
are specially encouraged to attend. The participation of university faculty, in the national and
international conference held here, is significantly higher than in similar conferences held
elsewhere in India. IUCAA also org~nises lectures, seminars and minischools in university
campuses all over the country with core faculty and postdocs ofIUCAA participating actively.
About 3 -4 such programmes are organised each year.
IUCAA also has an extensive visitors programme aimed at closer interaction between the
members of the university sector as well as between the university faculty and other experts in
the field. It has nearly 500 visitors on an average per year and the university faculty are
specially encouraged to make use of the expertise of visitors from abroad or from other
institutions during their stay. To formalise the visitors activity in a systematic way, it has
introduced an Associateship Programme for university faculty. Among those who apply,
about 20 members of the university sector are selected every year as associates of IUCAA and
are invited to spend about 12 months in a period of three years during this associateship. Full
facilities are provided as regards travel, lodging etc. as a part of this programme so that their
visit can be as comfortable as possible. At present, IUCAA has seventy associates from about
forty different universities and colleges. It is hoped that the Associateship Programme will
eventually lead to nucleation of active astronomy and astrophysics research at different
universities in this country.
As another step in this direction, IUCAA plans to set up a few Regional Centres in astronomy
and astrophysics in different parts of the country, around already existing active research
groups. These centres will help introduce greater coherence among the university faculty in
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INF. & ANN.
the local region and will help them interact with IUCAA in a more coordinated and effective
manner.
Finally, all these efforts tacitly assume that there is significant awareness and interest in
astronomy and astrophysics in this country. While this is true to a certain extent, there is
definitely scope for improvement. One possible way of enhancing this awareness in the
community is through dedicated popular science programmes aimed at lay public and school
and college students. IUCAA conducts a programme for school students in astronomy and
astrophysics on two Saturdays each month and a project-oriented course during the summer
vacation. These two activities, which have now been going on for five years, have been
extremely successful and have drawn considerable amount of response from the schools in
Pune.
It should also be noted that, by and large, observational facilities in India are not of
international standard. To some extent this is because of inadequate training and motivation
at the grassroot level in astronomical observation. IUCAA is currently building its own
2 metre telescope, to be located at Giravili (about 80 kms away from Pune). This telescope
in addition to being used in active astronomy research - will also provide a hands-on
opportunity for the university community to learn the nuances of optical astronomy. Such
exposure and experience will foster, in the long run, progress towards cutting-edge research
in observational astronomy as well.
At present, IUCAA has 11 core faculty members, 11 students and 8 postdocs. The research
activities of this academic staff covers a wide span of subjects in different areas of A&A.
In cosmology, the focus of research in IUCAA is on the formation of large scale structures.
Over the last few decades, observational results have constrained theoretical models for
galaxy formation quite severely. This fact, as well as the rapid growth in computer technology
which allows one to simulate the nonlinear phases of evolution, have made the study of
structure formation a thrust area in international research. Both analytical modeling and
numerical simulation of the universe are carried out at IUCAA in order to understand
different facets of this problem.
In extra galactic astronomy, research concentrates around observations of quasar absorption
systems, their spectral analysis and the study of morphology of galaxies. Members of IUCAA
participate in international collaborations and do their observations either on telescopes
abroad or at different sites in India. The data analysis is almost entirely done at IUCAA.
Another major area in which IUCAA is involved in several international collaborations is in
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