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1B11 Orbits
Before we begin our review of the Solar System, this section introduces the basics of orbits.
conjunction (full)
superior conjunction
(full)
inferior conjunction
(new)
opposition (full)
orbit of superior planet
Earth’s orbit
orbit of inferior planet
1B11 Sidereal Period
The sidereal period is the time taken for a planet to complete one orbit with respect to the stars.
1B11 Synodic period
The synodic period is the time taken for a planet to return to the same position relative to the Sun, as seen from the Earth.
orbit of superior planet
Earth’s orbit P4
P2
P2
P1
P3
P3
1B11 Kepler’s Laws
1. The orbit of a planet is an ellipse with the Sun at one focus (1609).
2. The radius vector joining the planet to the Sun sweeps out equal areas in equal times (1609).
3. The squares of the orbital periods of the planets are proportional to the cubes of the semi-major axes of their orbits (1619).
Johannes Kepler (1571-1630)
1B11 Ellipses
a
b Fae
rr1
a = semi-major axis
b = semi-minor axis
e = eccentricity
= “true anomaly”
Equation of an ellipse: r + r1 = constant = 2a
The eccentricity:
2
222
a
bae
ecosθ1
e1ar
2
and the relation
between r and :
1B11 Kepler’s First Law
The orbit of a planet is an ellipse with the Sun at one focus.
F2F1
major axis
min
or a
xis
aphelion perihelion
1B11 Kepler’s Second LawThe radius vector joining the planet to the Sun sweeps out equal areas in equal times.
A
B
C
D
At perihelion, the planet moves at its
fastest
At aphelion, it travels at its most slow
1B11 Kepler’s Third Law
The squares of the orbital periods of the planets are proportional to the cubes of the semi-major axes of their orbits. 32 aT
Planet Period, T (years)
T2 Distance, a (AU)
a3 T2/a3
Mercury 0.24 0.0058 0.39 0.059 0.97
Venus 0.62 0.38 0.72 0.37 1.0
Earth 1.0 1.0 1.0 1.0 1.0
Mars 1.9 3.6 1.5 3.4 1.1
Jupiter 12 140 5.2 140 1.0
Saturn 29 840 9.5 860 0.98
1B11 Newton and Kepler
Earth
2EarthEarth
Sun
2SunSun
cen r
vm
r
vmF
Their centrifugal forces must be balanced:
centre of mass
The Sun and the Earth rotate about each other, around their common centre of gravity. rSun + rEarth = a
rSun rEarth
1B11 Newton and Kepler
Earth2
2EarthEarth
Sun2
2Sun
2Sun
rT
r4m
rT
r4m 2
T
r2v
The velocity v may also be written in terms of the radius r and period T:
Substituting:
Sun
Earth
Earth
Sun
m
m
r
rWhich leaves:
1B11 Newton and Kepler
a = rSun + rEarth, so
rEarth = a – rSun, and :
SunSun
EarthSun ra
m
mr
EarthSun
EarthSun mm
amr
So:
2EarthSun
2SunSun
2
cen a
mGm
T
rm4πF
And:
1B11 Newton and Kepler
2Earth
2Sun
2
a
Gm
T
r4π
2Earth
EarthSun2
Earth2
a
Gm
mmT
am4π
GmmT
a4π
EarthSun2
32
1B11 Newton and Kepler
3
EarthSun
22 a
mmG
4πT
And finally:
which is Newton’s form of Kepler’s Third Law.
Notice that the “constant” isn’t strictly constant for every planet, because each planet’s mass will be different. But since the mass of the Sun is so large, it is true to first order.
1B11 Kepler’s Second Law
A quick reminder…
A
B
C
D
At perihelion, the planet moves at its
fastest
At aphelion, it travels at its most slow
1B11 Orbits
Planet moves from P to Q in time t through angle .
F
P
Q
v
vt
v = orbital velocity at P vt = transverse component of v
FPQ has area A where A = ½ r (vtt) and A/t = ½ vtr
(assuming the ellipticity e is low, ie it’s almost a circle)
1B11 Orbits
So since A/t = ½ vtr, as t -> 0,
dA/dt = ½ vtr
But vt ~ rd/dt = r, where is the angular velocity - so
dA/dt = ½ r2
Moment of inertia, I = mr2 = r2 (for unit mass)
dA/dt = ½ I½HWhere H is the angular momentum per unit mass. Since H is
conserveddA/dt = constant ie Keplers 2nd Law
1B11 Orbits
2
H
dt
dA dt
2
HdA
P2
HπabA
rvr
vrIωH t
t2
Pr
ab2
r
Hv t
Now: So integrating over the orbit:
Therefore:
Since:
We have:
1B11 Orbits
At perihelion:
e)a(1aear
vv perit
e)P(1
bvperi
2 )e(1ab 222
e)(1e)(1e)P(1
avperi
2
e1
e1
P
avperi
2
therefore where
therefore
and
1B11 Orbits
Similarly, for aphelion:
e1
e1
P
avap
2
For the Earth, a = 1AU = 1.496 x 108 km
P = 1 year = 3.156 x 107 seconds
e = 0.0167
Therefore vperi = 30.3 km/s
and vap = 29.3 km/s
1B11 Masses from orbits
For a body (eg a moon) in orbit around a much larger body (a planet), if you know the period of rotation of the moon, T, and its distance from the planet, a, you can calculate the mass of the planet from Newton’s version of Kepler’s Third Law.
Mmoon = mass of the moon
Mplanet = mass of the planet, and Mplanet >> Mmoon
G = Gravitational constant
So then: P2 = 42/GMplanet x a3
1B11 Masses of stars in binary systems
In visual binary stars, we can sometimes observe P and measure a if the distance to the binary is known.
We can then solve for the sum of the masses, ie:
(m1 + m2) = (42/G) + a3/P2
(P is typically tens of thousands of years)
If the stars have a high proper motion, the centre of mass moves in a straight line and a1 and a2 can be measured.
m1r1 = m2r2
In a few cases, can solve for m1 and m2.
1B11 Masses of stars spectroscopic binaries
Spectroscopic binaries are those binary systems which are identified by periodic red and blue shifts of spectral lines.
In general, the parameter (m1 + m2) can be calculated.
Sometimes the individual masses can be calculated.
1B11 Eclipses
Eclipses occur when one body passes directly in front of the line of sight from the observer to a second body. For example, a solar eclipse
absolutely not draw to scale!
1B11 Solar eclipses
Important facts:
The Moon’s orbit is inclined to the ecliptic by 5.2O, so an eclipse will only occur when the Moon is in the ecliptic plane.
The angular diameter of the Moon (which varies between 29.5 and 32.9arcmins) is very similar to that of the Sun (32 arcmins), which is why solar eclipses are so spectacular.
There are three types of eclipse –
Partial – the observer lies close to, but not on, the path of totality
Annular – the Moon is relatively distant from the Earth
1B11 Three types of eclipse
There are three types of eclipse –
Partial – the observer lies close to, but not on, the path of totality
Annular – the Moon is relatively distant from the Earth, so a ring of Sun appears around the Moon’s shadow.
Total – when the Moon’s and the Sun’s angular diameters match. At the point of totality, the Sun’s corona (its outer atmosphere) appears.
1B11 Lunar eclipses
When the Earth lies directly between the Sun and the Moon, a lunar eclipse occurs. From the Earth, we watch as the Earth’s shadow passes across the face of the Full Moon.
As seen from the Moon, the Earth has an angular diameter of 1O 22’, so there are no annular lunar eclipses.
The Earth’s shadow is not black however, light from the Earth’s atmosphere reaches the Moon during totality and we see this light reflected from the Moon. This light is red – the blue light has been scattered away by dust in the atmosphere.
In a typical lifetime, you should see about 50 lunar eclipses from any one location – solar eclipses are much more rare.
1B11 Eclipsing stars
If the orbital plane of a binary system lies close to, or along, our line of sight, then we will see changes in the lightcurve as the eclipses occur.
flux
time
period
primary eclipsesecondary eclipse
1B11 Transits
A transit is when a small body passes in front of a much larger one. We can observe transits of Mercury and Venus across our Sun, for example.
We also search for evidence of transits by extrasolar planets, passing in front of their local stars. The drop in flux is tiny, but measurable if the relative angular size of the planet is large enough, eg a Jupiter-like planet in close orbit (Mercury-ish).
For planets in our Solar System which have their own moons, eg Jupiter, we can also observe transits as a moon passes across their face.
1B11 Occultations
When one object completely obscures another, this is known as an occultation. So when the angular size of the Moon is equal to or larger than the Sun’s, the total solar eclipse is an occultation.
Stars are occulted by the Moon or by planets and asteroids. Lunar occultations occur at predictable times so can provide precise positions.
[Strictly speaking, an eclipse occurs when one body passes through the shadow of another.]
1B11 Lunar libration
The Moon rotates on its axis once a month, therefore it always keeps the same face pointed towards the Earth.
Well almost – the Moon’s orbit is elliptical and inclined to the ecliptic, so we do see “around” the Moon making more than 50% of its face visible in total.
N
S
N
Secliptic
Moon’s
orbit
5.2O
Libration occurs in longitude and latitude and adds up to a “wobble” of about 6O. It’s also
called “phase-locking”.
1B11 The Solar System
The Sun
Mercury
Venus
Earth
Mars
[Asteroid Belt]
Jupiter
Saturn
Uranus
Neptune
Pluto
Comets
- G2V star
Terrestrial planets
Giant (gaseous) planets and moons
Icy Planetessimals
