INTRODUCTORY ASTRONOMY
Lecture 2. The Solar System; two-body problem.
The Solar System consists of the Sun, nine plan- ets, satellites, comets and asteroids.
On the lower panel the planets are shown on the same scale.
Kepler’s Laws.
The planets move in planes close to that of the ecliptic.
The planetary orbits can be described with the three laws of Kepler.
Kepler’s First Law.
The planet’s orbit is an ellipse with the Sun in one of the foci.
The formula of the orbit is a(1 − e2)
r = 1 − ecos ν ν = true anomaly; e = eccentricity
a = semi-major axis, expressed in Astronomical Units (i.e. for the Earth a = 1 A.U.)
Kepler’s Second Law.
The distance to the Sun multiplied by the orbital velocity is constant. Kepler for this using the area of the triangles, which is described by the radius per unit of time.
It is in fact conservation of angular momentum.
It is also the cause of the unequal duration of the seasons.
(Northern) fall + winter is about 6 days shorter than spring + summer; this was already known to Hipparchus (second century BC).
Third or ‘harmonic’ Kepler’s law.
The orbital period around the Sun (period) squared is proportional to the third power of the semi- major axis.
T2
a3 = 4π2
G(M + Mp) ≈ 4π2 GM
T = period M = mass of the Sun G = gravitational Mp = mass of the planet
constant
If we express the period in years and the semi-
For a circular orbit, we can already easily see this by setting the gravitational attraction equal to centrifugal force:
GMpM
r2 = MpV 2 r . So
V 2 = GM r The period is T = 2πr/V , so
T2
r3 = 4π2 GM
Distances and masses.
Distance determination.
Distance unit is the Astronomical Unit (A.U.) = the average distance from the Earth to the Sun.
• Through an asteroid, that comes close to the Earth. From position measurements determione the position relative to the Earth’s orbit. Then determine the distance from the parallax from two places on the Earth’s orbit. Earth.
• Sun transits of Mercury or Venus (precise tim- ing from beginning or end from two places on Earth gives the parallax).
• Timing of radar pulses reflected on planets.
• Of course known very accurately from space travel.
1 A.U. = 1.496 × 108km.
Mass determination.
• Earth: Determine the acceleration of gravity and G.
• Sun: Kepler’s Third Law.
• Moon: Measures the motion of the Earth around the common center of gravity using the planets. Now on the Moon or with an orbit around it.
• ’s Planets with satellites: Kepler’s Third Law.
• ’s Planets without satelliets: Disturbances on other planetary orbits.
• Satellites: Measure radius and take ρ like the Moon.
• Now, of course, via positions and orbits of space probes.
M⊕ = 5.97 × 1024 kg ; M = 1.99 × 1030 kg
Orbits in space and the sky.
The orbits can be characterized using six orbital elements:
• The orbital plane follows from:
– Ω; the longitude (along the ecliptic) of the as- cending node (intersection of orbital plane and ecliptic, where the planet goes from south to north of the ecliptic).
– i; the inclination (angle between the orbital plane and the ecliptic).
• The size and shape of the orbit with:
– a; the half-long axis.
– e; the eccentricity.
• The ori¨entation of the orbit in the orbital plane:
– ω; the angle between the direction of the as- cending node and that of the perihelion.
• The position of the planet in orbit with:
– t◦; a time of perihelion passage.
Of course, the orbital period T is not an inde- pendent parameter according to Kepler’s Third Law.
Note: The (astronomical) longitude is measured along the ecliptic from the vernal equinox Υ (the position of the Sun on 21 March, or the as- cending node of the Sun, where the ecliptic and equator intersect).
As seen from Earth, the planets describe orbits on the sky that include loops.
For example, for Jupiter, if the Earth goes from A to E and Jupiter from a to e, then we see this as a loop in the sky.
This is called the retrograde motion.
It reflects the fact that the Earth also orbits the Sun.
Retrograde motion occurs during opposition for the outer planets and (inferior) conjunction for the inner planets.
The orbital period T, which appears in Kepler’s laws is the “real” orbital time.
This is also called sidereal period (sidereal refer- ring to the fact that the planet from the Sun is then exactly back on the same place relative to the stars).
The synodic period is measured relative to the Earth, when the planet and the Earth gave re- turned to has the same configuration.
For an outer planet, the synodical period is the time between successive oppositions; for inferior planets between successive (superior of inferior) conjunctions.
In the figure we see the Earth and an outer planet.
The average angular velocity of the Earth with respect to the Sun is 360◦/T⊕ and that of the planet 360◦/Tsid.
The relative angular velocity 360◦/Tsyn is the dif- ference between those two and a synodic period is then
1
Tsyn = 1
T⊕ − 1 Tsid.
For inner planets 1
Tsyn = 1
Tsid − 1 T⊕.
In practice, we express everything in years and then T⊕ = 1.
Geocentric and heliocentric worldview.
Pythagoras (sixth century BC) is the founder of the experimental science (experiments with pitches of sound; harmonic intervals; “Harmony of the spheres”).
Aristarchus (±310 – 230 BC) was the first to suggest that the Earth and planets orbit around the Sun. heliocentric world view).
The philosophy of Plato (427 – 348 BC) and Aristotle (384 – 322 BC) with the perfect mo- tion in straight lines or circles at uniform velocity in the ’superlunar’ and the imperfect nature of the Earth gave rise to the geocentric world view.
It culminated in the ‘model’ of the Solar System of Claudius Ptolemy (±100 – 170 AD).
In this case, the planets orbit the Earth along a deferent with a epicycle.
The eccentricity of the planet’s orbits reflects in this image from the fact, that the motion of the epicycle is uniform in angular velocity relative to the equant.
Remarkably, in real planetary orbits (with not too great eccentricity) the angular velocity from the focal point, where the Sun is not located, is indeed more or less constant.
For the outer planets, the deferent is the reflec- tion of the planet’s orbit and the epicycle the Earth’s orbit.
For the inner planets, it’s the other way around.
So the position in the epicycles for the outer planets must all be the same. while the centers of the epicycles for the inner planets must always be on the line Earth – Sun.
This is suspicious to us, but Ptolemy did not think in the form of such figure as the one above.
The model works, because planetary orbits in the sky are only directions and distances do not
Also, the Ptolemaic model does not explain the varying apparent brightness of the planets.
Copernicus (1473 – 1543) returned to the ‘Pla- tonic’ ideal picture. He was dissatisfied with the equant.
In fact, he used epicycles to represent the vary- ing velocity in an elliptical orbit.
At the same time he suggested (“because it was easier to calculate”), that the Sun would be in the center.
Copernicus was more a man of the Middle Ages and not the real innovator.
This was Johannes Kepler (1571 – 1630), who was the first to ask the question of what form the orbits of the planets actually is and deter-
mined it from observations rather than pre-conceived ideas
To this end, he used the precise observations of Tycho Brahe (1546 – 1601).
For an inner planet, you can find the orbit as in (A) but then with the Earth in different posi- tions.
For Mars (B) Kepler took position measurements on the sky, exactly an integer number of side- real periods apart. In the old model this is the period in the deferent.
Then the position P can be found from triangu- lation. Kepler did this for many different cases and then found his first two laws.
Then you can turn it around and find two posi- tions, where the Earth is at the same point in the orbit and thus determione the shape of the
Kepler found his “harmonic law” later, when he looke for harmony in the planetary system.
He returned to the ‘harmony of spheres’ of Pythago- ras, and assumed that, as it were, each planet
produces a tone proportional to its (angular) ve- locity in orbit.
Then the proportions turned out to be very close to harmonic, musical intervals.
Galileo Galilei (1564 – 1642) first used a tele- scope for scientific observations of the sky. In 1610 he found Jupiter’s satellites like a minia- ture planetary system and the phases of Venus.
This confirmed the heliocentric world view.
Isaac Newton (1642 – 1727) showed that the laws of Kepler directly follow from his gravita- tional theory.
Uranus was discovered by William Herschel in 1781 during his extensive observations of the sky.
Neptune was found in 1846, after Leverrier and Adams had predicted the position on the sky from the disturbances in Uranus’ orbit.
In 1610, Neptune was very close to Jupiter and Galileo had Neptune according to his drawings, also seen, but not recognized as a planet. John Herschel also failed to recognize it as a planet.
Pluto was discovered in 1930 by Tombaugh after an extensive search, but is no longer a planet.
The first ı (Ceres) was found on 1-1-1801 by Piazzi. Quickly followed by more. Now there are more than 10000 known and over 8000 named.
Law of Titius-Bode.
Looking for regularity in the planetary system Titius and Bode (1772) found, that the dimen- sions of the planet’s orbits followed a ‘simple’
series:
a = 0.4 + (0.3)2n,
with a in A.U. Well, then n = −∞ for Mercury, n = 0 for Venus, etc.
This is due to resonances. Mutual interactions bring planets into orbits with orbital times, which have a ratio of orbital time (and thus half of the major axis) that are quotients of small integers.
For example: Earth and Venus have rotation times of 1,000 and 0,615 years, almost in pro- portion 5 : 3. For Jupiter (11,862 years old) and Saturn (29,457 years) is the ratio 0,403 or almost exactly 5 : 2.
Computer experiments have confirmed this hy- pothesis.
These resonances also underlie Kepler’s variant of the Harmony of Spheres.
Creation of the planetary system.
Around the Sun a cloud of gas and especially dust arose, in which the most of the angular momentum ended up. That made it very flat-
In this the planets condensed, first as small pro- toplanets, that then fused together to form larger planets.
This explains why (1) the orbital planes of the planets lie are roughly in the same, (2) all go in the same direction in roughly circular orbits and (3) almost all the angular momentum is in the orbits of the planet (the Sun has 99.86% of the mass and only 2% of the angular moment).
The asteroids were probably formed in a collision of two (proto)planets in about the same orbit.
The Earth-Moon system similarly.
The age of the Solar System is approximately 4.5 × 109 years, measured using radioactive ele- ments.
Two-body Problem.
The following shows how the laws of Kepler follow from Newton’s gravitational theory and some related matters.
Take two masses m1 and m2. with position vec- tor ~r1 and ~r2.
Then there are two fundamental equations:
m1~r˙1 = −Gm1m2~r1 − ~r2.
r3 (1)
m2~r˙2 = −Gm1m2~r2 − ~r1.
r3 (2)
¨~r stands for d2~r/dt2.
Center of gravity.
The center of gravity is at R~ = m1~r1 + m2~r2
m1 + m2 ≡ m1~r1 + m2~r2 M
Add (1) and (2), then
m1¨~r1 + m2¨~r2 = 0 Integrate this twice, then
m1~r1 + m2~r2 ≡ M ~R = ~at +~b. (3)
So the center of gravity is in linear motion.
Co-moving coordinate system.
Now take the center of gravity as the origin, so that
m1~r1 + m2~r2 = 0 Then
¨~r1 = −Gm2~r1 − m2~r2
r3 = −GM~r1 r3
¨~r2 = −Gm1~r2 − m1~r1
r3 = −GM~r2 r3
So if ~r = ~r1 − ~r2 (the vector between the two bodies)
¨~r = −GM ~r
r3 (4)
We have three second-order differential equa- tions, so we will get in principle 6 integration constants.
Angular momentum.
Multiply (4) by ~r.
~r ר~r = −GM~r ×~r
r3 = 0
Integrate this equation
~r × ~r˙ = constant = ~h (5) You can see this by differentiating again and remembering that ~r˙ × ~r˙ = 0.
Equation (5) says that the angular momentum is conserved (Kepler’s second law) and that the motion is limited to the plane
~r ·~h = 0
Equation of the orbit.
We need the following equations from the vector analysis: ~a × (~b × ~c) = (~a · ~c) ·~b − (~a ·~b) · ~c and
~a ·~a˙ = aa˙ (because ~a ·~a = a2).
Start again with (4) and multiply by ~h.
~h ר~r = −GM
r3 (~h × ~r) = −GM
r3 (~r ×~r)˙ × ~r
= −GM
r3 {r2~r˙ − (~r · ~r)~˙ r}
= −GM
r3 {r2~r˙ − (rr)~˙ r}
= −GM ~r˙
r − r~˙r r2
!
= −GM d dt
~r
r = −GMdˆr dt Integrate this
~h × ~r˙ = −GMˆr − P~ (6)
To find outmore about P~ we multiply by ~h
~h · (~h × ~r) =˙ −GMˆr ·~h − P~ ·~h
The term on the left and the first one on the right are 0, so
0 = P~ ·~h (7)
So P~ is perpendicular to ~h and gives two further integration constants.
Now go back to (6) and multiply by ~r.
~
r · (~h × ~r) =˙ −GM~r · ˆr − P~ · ~r
−~h · (~r × ~r) =˙ −GM r − ~p · ~r
−~h ·~h = −h2 = −GM r − P~ · ~r Thus
h2
GM r = 1 + P~
GM · ~r
r = 1 + P~
GM · ˆr
Call the angle between rˆ and P ν~ (the true anomaly), so that P~ · ˆr = P cos ν. Then
h2
GM r = 1 + P
GM cos ν (8) This equation is the general form for a conical section (in polar coordinates and one focal point in the origin)
r = q
1 + ecos ν
So we see that the eccentricity e = P
GM
For an ellipse the following also applies h2
GM = q = a(1 − e2)
For a hyperbola h2
GM = q = a(e2 − 1) A parabola has e = 1.
In an elliptical planetary orbit, the perihelion dis- tance (the smallest distance from the Sun) is a(1 − e) and the aphelion distance (the largest) a(1 + e).
From (7) we see that ν = 0 the smallest r and that P~ indicates the direction of the perihelion.
The energy integral.
The total of kinetic and potential energy (the energy integral) is constant
V 2
2 − GM
r = C
The angular momentum equals the angular ve- locity (dν/dt) times the radius squared, so h = r2dν/dt. The tangential velocity then is
rdν
dt = h
r = h
q(1 + ecosν) = GM
h .(1 + ecos ν) The orbital equation for an elliptical orbit was
a
r(1 − e2) = h2
GM r = 1 + ecosν
Differentiate this and use dν/dt = h/r2, then the radial velocity is
r˙ = eGM h ν The total velocity then follows
V 2 =
GM h
2
(1 + e2 + 2ecos ν) Fill this in in the energy integral, then
C = −1 2
GM h
2
(1 − e2) = −GM 2a
This allows us to calculate the orbital velocity at any point:
V 2 = GM
2
r − 1 a
(9) For a parabolic orbit, C = 0 and for a hyperbolic orbit C = GM/2a.
In a circular orbit we have Vcirc2 = GM
r
At radius r the escape velocity is that in a parabolic orbit
Vescape2 = 2GM r
Kepler’s third law.
The position in an elliptical orbit followed from the true anomaly ν
r = a(1 − e2)
1 + ecos ν (10) Now define the eccentric anomaly E with
r = a(1 − ecos E) (11)
From the figure it can be deduced with semi- major axis a and semi-minor axis b = a
q
1 − e2, that
r2 = P D2 + DF2 =
b
aDQ
2
+ (CF − CD)2
=
b
aasinE
2
+ (ae − acos E)2
= a2(1 − ecos E)2
Furthermore, it can be calculated with (10) and (11) that
tan
ν 2
=
s1 + e
1 − e tan
E 2
(12) E is defined such that dE/dt ≥ 0, because E is an angle measured from the center of the orbit, such that it runs in the same direction as ν.
We had
~r × ~r˙ = ~h
Square this and insert in the energy integrally:
GM
2
r − 1 a
r2 − r2r˙2 = GM a(1 − e2) Differentiate (11)
r˙ = aedE
dt sin E and fill this in
dt =
s
a3
GM(1 − ecosE)dE
Integrate this over the full orbit
Z T
0 dt =
Z 2π 0
s
a3
GM(1 − ecosE)dE
T = 2π
s
a3 GM This is Kepler’s third law
T2
a3 = 4π2
G(m1 + m2). (13)
Kepler’s equation.
E does (just like ν) not vary uniformly with time.
Therefore, define the mean anomaly M, who does increase uniformly with time.
M = 2π
T (t −t◦) =
sGM
a3 (t −t◦) = n(t −t◦) (14) So here n is the average motion and t◦ a mo- ment of perihelion passage. Then
dt = 1
n(1 − ecos E)dE
Integrate this
n(t − t◦) = M = E − esin E (15) This is Kepler’s equation, which indicates with which construction Kepler worked.
Kepler calculated tables for various values of e, in which he could find E for any particalar values of M.
Orbit parameters.
We had 6 integration constants
• Three constants of the angular vector ~h. This defines the orbital plane and the size of the to- tal angular momentum (and thus the size of the orbit).
• Two constants with the vector P~. The direc- tion of this gives that of the perihelion and the size and eccentricity of the orbit.
• The sixth constant is a moment of perihelion passage t◦.
In practice, we replace these with the six orbital elements given earlier.
