1B11 Foundations of Astronomy Orbits

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1B11 Foundations of AstronomyOrbits

• Liz Puchnarewicz
• emp_at_mssl.ucl.ac.uk
• www.ucl.ac.uk/webct
• www.mssl.ucl.ac.uk/

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1B11 Orbits
Before we begin our review of the Solar System,
this section introduces the basics of orbits.
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1B11 Sidereal Period
The sidereal period is the time taken for a
planet to complete one orbit with respect to the
stars.
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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.
Earths orbit
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1B11 Keplers Laws

• The orbit of a planet is an ellipse with the Sun
  at one focus (1609).
• The radius vector joining the planet to the Sun
  sweeps out equal areas in equal times (1609).
• 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)
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1B11 Ellipses
a semi-major axis b semi-minor axis e
eccentricity q true anomaly
r1
r
a
q
F
ae
b
Equation of an ellipse r r1 constant 2a
The eccentricity
and the relation between r and q
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1B11 Keplers First Law
The orbit of a planet is an ellipse with the Sun
at one focus.
minor axis
perihelion
aphelion
major axis
F2
F1
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1B11 Keplers 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
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1B11 Keplers Third Law
The squares of the orbital periods of the planets
are proportional to the cubes of the semi-major
axes of their orbits.
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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
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1B11 Newton and Kepler
The velocity v may also be written in terms of
the radius r and period T
Substituting
Which leaves
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1B11 Newton and Kepler
a rSun rEarth, so rEarth a rSun,
and
So
And
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1B11 Newton and Kepler
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1B11 Newton and Kepler
And finally
which is Newtons form of Keplers Third
Law. Notice that the constant isnt strictly
constant for every planet, because each planets
mass will be different. But since the mass of the
Sun is so large, it is true to first order.
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1B11 Keplers 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
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1B11 Orbits
P
v
vt
F
Dq
Q
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
its almost a circle)
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1B11 Orbits
So since DA/Dt ½ vtr, as Dt -gt 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
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1B11 Orbits
Now
So integrating over the orbit
Therefore
Since
We have
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1B11 Orbits
At perihelion
where
therefore
therefore
and
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1B11 Orbits
Similarly, for aphelion
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
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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 Newtons version of Keplers Third
Law. Mmoon mass of the moon Mplanet mass of
the planet, and Mplanet gtgt Mmoon G
Gravitational constant So then P2
4p2/GMplanet x a3
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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) (4p2/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.
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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.
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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!
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1B11 Solar eclipses
Important facts The Moons 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
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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 Moons shadow. Total when the Moons and
the Suns angular diameters match. At the point
of totality, the Suns corona (its outer
atmosphere) appears.
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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 Earths 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 Earths shadow
is not black however, light from the Earths
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.
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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.
period
flux
secondary eclipse
primary eclipse
time
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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.
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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
Suns, 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.
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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 Moons 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.
5.2O
ecliptic
Libration occurs in longitude and latitude and
adds up to a wobble of about 6O. Its also
called phase-locking.
Moons orbit
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1B11 The Solar System
- G2V star
The Sun Mercury Venus Earth Mars Astero
id Belt Jupiter Saturn Uranus Neptune Pluto Comet
s
Terrestrial planets
Giant (gaseous) planets and moons
Icy Planetessimals