1B11
Foundations of Astronomy
Orbits
Liz Puchnarewicz
emp@mssl.ucl.ac.uk
www.ucl.ac.uk/webct
www.mssl.ucl.ac.uk/
1B11 Orbits
Before we begin our review of the Solar System, this section
introduces the basics of orbits.
conjunction (full)
orbit of
superior
superior
planet
conjunction
(full)
Earth’s
orbit
orbit of
inferior
planet
inferior
conjunction
(new)
opposition (full)
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
P2
Earth’s
orbit
P4
P3
P3
P2
P1
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
r1
a
ae
b
a = semi-major axis
r
F
q
b = semi-minor axis
e = eccentricity
q = “true anomaly”
Equation of an ellipse: r + r1 = constant = 2a
The eccentricity:
a b
e 
2
a
2
2
2
and the
relation
between r
and q:


a 1 e
r
1 ecosθ
2
1B11 Kepler’s First Law
aphelion
minor axis
The orbit of a planet is an ellipse with the Sun at one focus.
perihelion
major axis
F2
F1
1B11 Kepler’s Second Law
The radius vector joining the planet to the Sun sweeps out
equal areas in equal times.
At perihelion, the
planet moves at its
fastest
C
D
B
At aphelion, it
travels at its most
slow
A
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.
T a
2
3
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
Planet
1B11 Newton and Kepler
centre of mass
rSun
rEarth
The Sun and the Earth rotate about each other, around
their common centre of gravity.
rSun + rEarth = a
Their centrifugal forces must be balanced:
2
Fcen
mSun v Sun
mEarth v Earth


rSun
rEarth
2
1B11 Newton and Kepler
The velocity v may also be
written in terms of the radius r
and period T:
Substituting:
2r
v
T
mSun 4 rSun
mEarth 4 rEarth

2
2
T rSun
T rEarth
2
Which leaves:
2
2
rSun mEarth

rEarth mSun
2
1B11 Newton and Kepler
a = rSun + rEarth,
so rEarth = a – rSun, and :
So:
rSun
mEartha

mSun  mEarth 
2
And:
Fcen
rSun
mEarth
a  rSun 

mSun
4π mSunrSun GmSunmEarth


2
2
T
a
1B11 Newton and Kepler
2
4π rSun GmEarth

2
2
T
a
2
4π mEartha
GmEarth

2
2
T mSun  mEarth 
a
2
3
4π a
G
2
T mSun  mEarth 
1B11 Newton and Kepler
And finally:
2
4π
3
T 
a
GmSun  mEarth 
2
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…
At perihelion, the
planet moves at its
fastest
C
D
B
At aphelion, it
travels at its most
slow
A
1B11 Orbits
P
v
vt
Q
Dq
F
Planet moves from P to Q in time Dt
through angle Dq.
v = orbital velocity at P
vt = transverse component of v
DFPQ has area DA where DA = ½ r (vtDt) and DA/Dt = ½ vtr
(assuming the ellipticity e is low, ie it’s almost a circle)
1B11 Orbits
So since DA/Dt = ½ vtr, as Dt -> 0,
dA/dt = ½ vtr
But vt ~ rdq/dt = rw , where w is the angular velocity - so
dA/dt = ½ r2w
Moment of inertia, I = mr2 = r2 (for unit mass)
dA/dt = ½ Iw  ½ H
Where H is the angular momentum per unit mass. Since H is
conserved
dA/dt = constant ie Keplers 2nd Law
1B11 Orbits
Now:
dA H

dt 2
Therefore:
Since:
So integrating over
the orbit:
H
 dA  2  dt
H
A  πab  P
2
vt
H  Iω  r
 v tr
r
2
We have:
H 2ab
vt  
r
Pr
1B11 Orbits
At perihelion:
v t  v peri
r  a  ae  a(1  e)
therefore
v peri
2b

P(1 e)
therefore
v peri
and
where
b2  a2 (1 e2 )
2a

(1 e) (1 e)
P(1 e)
v peri
2a 1 e

P 1 e
1B11 Orbits
Similarly, for aphelion:
v ap
2a 1 e

P 1 e
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 = 42/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) = (42/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
m 2.
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
period
secondary
eclipse
primary eclipse
time
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.
ecliptic
5.2O
N
S
N
S
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
- G2V star
Mercury
Venus
Terrestrial planets
Earth
Mars
[Asteroid Belt]
Jupiter
Giant (gaseous)
Saturn
planets and
Uranus
moons
Neptune
Icy
Pluto
Planetessimals
Comets