Download AST111, Lecture 1b

AST111, Lecture 1b
❑ Measurements of bodies
in the solar system
(overview continued)
❑ Orbital elements
Planetary properties (continued):
Measuring Mass
• The orbital period of a moon about a planet
depends on the semi-major axis and on the
planet’s mass. Moons or satellites can be
used to estimate masses of planets or
planetesimals.
• Perturbations on other planets. Neptune was
discovered because it perturbs Uranus.
• Spacecraft tracking data. Doppler shifts can be
measured precisely, allowing direct
measurement of gravitational acceleration.
• Perturbations on nearby debris such as rings.
Planetary Size
• Actual sizes can be estimated from angular sizes once the distance
to an object has been estimated from its orbit. Remember that
Kepler’s third law relates the orbit period to it semi-major axis.
However, angular resolution from most earth ground based
observations is limited by atmospheric seeing. (~1”)
• Radar echoes. Intensity drops as 1/r4 so only nearby objects can
be studied.
• Occultations of a bright star. Requires accurate predictions,
however gives good size estimates, particularly if more than
one occultation of bright stars occur. An occultation is when an
object blocks the light from an other object.
• Using albedos. The albedo is the reflectance or the fraction of light
which is reflected from the object. Once you know how far
away from the sun the object is, and its distance, you can
estimate how bright it is. By comparing brightnesses at
different wavelengths it is possible to constrain the
uncertainties in the albedo. If you know the albedo accurately
then you can estimate the size of the object accurately.
Rotation
Measurements:
• Observe markings on the object’s surface and track
them in images at different times.
• If the planet has a magnetic field, charged particles
will rotate with the planet. Radio signals from the
charged particles should have same periodicity as
the rotation of the object.
• Observing a lightcurve. Some of the surface may have
higher or lower albedo, so the amount of sunlight
reflected from the object will vary depending upon
which side is facing the observer.
The rotation period of most objects is between a few
hours and a week, with the exception of Mercury and
Venus (59 and 243 days, respectively).
Rotation – (continued)
• The angle between the planet’s axis of rotation and the vector
which is perpendicular to the ecliptic plane, is called the
obliquity. The ecliptic plane is the plane containing the planets.
• Six of the nine planets rotate in the same direction as they orbit
the sun. We say they rotate in the prograde direction. They
typically have obliquities < 30o
• Pluto and Uranus are tilted with rotation axis nearly in the
ecliptic plane. Obliquities are ~90o .
• Venus has a retrograde rotation, obliquity > 90o. Venus’s
obliquity is 177o.
• Satellites often rotate at the same period as their orbital period.
For example, we always see the same side of the moon
because the moon rotates so that the same side always faces
the Earth. We say the moon is tidally locked with the Earth.
Shape
Measurement:
• Direct imaging.
• Stellar occultations.
• Analysis of radar echoes.
Self-gravity tends to produce
bodies of spherical shape.
Material strength maintains
shape irregularities which
may be produced by
accretion, impacts, or
geological processes.
Typically bodies which are
larger than ~200km are
spherical.
Mars and Phobos
Temperature
• The equilibrium temperature can be estimated from the
energy balance of that absorbed compared to that
emitted. For many bodies the total energy absorbed
by the sun balances that emitted thermally. However,
many planets also have internal heat sources.
• There can be diurnal (daily), latitudinal and seasonal
variations in temperature. Radiation transfer through
the atmosphere can be complex (such as happens
where there is a greenhouse effect).
• Measurements can be taken in situ or by spectroscopic
study of the atmosphere and surface.
Magnetic field
• Magnetic fields are produced by moving charges.
Currents moving through a solid usually decay quickly.
• Large planetary magnetic fields are produced by a dynamo
which requires a moving fluid core. The giant planets,
Earth and Mercury have magnetic fields generated
from their interiors.
• Smaller ones can be produced by remnant
ferromagnetism. Mars and the Moon have localized
“crustal fields.”
• Magnetic fields can also be produced by the interaction
between the object’s ionosphere and the solar wind.
(Venus and comets).
Atmosphere composition and structure
• The planets and some satellites are surrounded by gaseous
atmospheres. These are measured spectroscopically at a
variety of wavelengths. Occultations also give
information about the vertical structure.
• Gas giants tend to have mostly H2, He. Venus has a thick
CO2 atmosphere. Earth’s atmosphere is nitrogen and
oxygen. Mars has a tenuous CO2 atmosphere. Titan has
a dense nitrogen atmosphere containing many organics.
Pluto and Triton have tenuous nitrogen atmospheres.
Mercury and the moon have extremely tenuous
atmospheres of atomic atoms (Mercury) and He and Ar
(the moon).
Bulk Composition
• The bulk composition must be inferred
indirectly. Constraints include the size and
mass of the object, structure and
composition of the atmosphere and the
observed effective temperature.
• Models as a function of radius can be
constructed which attempt to match these
constraints.
• Gravitational, magnetic and seismic
measurements can also provide constraints.
Orbits of the planets and asteroids
Inner solar system
Outer solar system
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Orbits
and
Spins of
Planets
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Orbits: Kepler’s laws
❑ All planets move along elliptical paths with the Sun at
one focus.
❑ A line segment connecting the planet and the Sun
sweeps out area at a constant rate.
❑ The square of the planet’s orbital period about the Sun
Porb is proportional to the cube of its semi-major
axis, a.
2
Porb
∝ a3
Kepler’s laws can be derived from Newton’s laws of motion
and gravity in the case of two massive bodies under the sole
influence of gravity (we will do this next week).
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Orbital elements
• An orbit of a massless (very low mass) particle about
the Sun in 3D can be specified by 3 components of the
velocity and 3 components of the position at a
particular time. This is a Cartesian system.
• Alternatively we can describe the orbit in terms of
three parameters which determine the shape and
orientation of the orbit and three angles which
determines where the particle is along the orbit again
at a particular time. These six parameters are known
as orbital elements.
a semi-major axis
e eccentricity
f true anomaly
ω argument of periapse
i inclination
Ω longitude of ascending node
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Orbital elements
(continued)
16
• xx
Why the difference in
nomenclature?
For the two body problem, closed orbits are
elliptical.
• Periapse or perihelion is the point of closest
approach to the Sun or central object.
• Apoapse or aphelion is the most distant point
from the central object.
• The Sun lies at
a focal point
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Solar eclipses
• Moon’s orbit is inclined with
respect to the ecliptic
• Longitude of the ascending
node for the moon’s orbit
lines up with the EarthSun axis during an eclipse
• Moon’s orbit precesses fairly
quickly. Eclipses happens
about twice a year.
Orbital elements (continued)
Principal orbital elements
• a describes the size of the orbit
• e describes the shape
• i describes the tilt (inclination)
The angles that go with each of them
• f (true anomaly) describes the location of object in
the orbit.
• ϖ (longitude of periapse) or ω (argument of
perihelion) describe orientation of perihelion
• Ω (longitude of the ascending node) describes
position where orbit crosses the plane of the
ecliptic.
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Orbital elements (continued)
Principal orbital elements
• a units of length
– positive for an elliptical orbit, infinity for a parabolic
orbit, negative for a hyperbolic orbit.
• e unitless
– ranges from 0 to 1 for an elliptical orbit. Is equal to 1 for
a parabolic orbit. Greater than 1 for a hyperbolic orbit.
• i degrees or radians
– ranging from 0 to 180o (> 90o retrograde)
The angles
• f ϖ Ω degrees or radians
– ranging from 0 to 360o
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Nomenclature on angles
• angles are called arguments
• angles given with respect to perihelion or
periapse are called anomalies
• angles given with respect to the vernal
equinox (or line of sight) are called
longitudes.
• Slow variations are called secular variations.
Precession of the orbit is secular. Both ω
and Ω can precess
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Orbital elements (continued)
• Of the 6 orbital elements, 5 are slowly varying,
(constant when the system is 2 spherical bodies
alone under the sole force of gravitational
attraction).
• The true anomaly f, (or mean longitude M), is the only
element which varies quickly in time.
• Lots of conserved quantities!
• Perturbation theory (from purely Keplerian motion) is
appropriate
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The orbit
1
r∝
1 + e cos f
radius from
the Sun
Describes the ellipse
where f is the angle called
the true anomaly.
e is the eccentricity
You can figure out the
constant of
proportionality in terms
of the semi-major axis
and eccentricity
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The orbit
What values of true anomaly
give the maximum and
minimum radii?
What is maximum and
minimum radius in
terms of the semimajor axis?
1
r∝
1 + e cos f
1
r∝
1 + e cos f
The orbit
What values of true anomaly give the maximum and
minimum radii?
What is maximum and minimum radius in terms of
the semi-major axis?
ANSWERS:
rmax = A/(1-e) at f=0
rmin = A/(1+e) at f=π
r = A at f=±π/2
A
r=
1 + e cos f
Low eccentricity approximation
Eccentricity 0<e<1
(though unbound objects can be described with e>1)
1
⇠1
1+x
1
1
x
x + ...
Taylor series for small x
⇠ 1 + x + ...
rmax = A/(1-e) is approximately equal to A(1+e)
at low eccentricity (nearly circular orbit)
rmax = A/(1-e) at f=0
rmin = A/(1+e) at f=π
r = A at f=±π/2
Semi-major axis a
2a = rmax + rmin = 2A/(1-e2)
A = a(1-e2)
Semi-minor axis b
However b ≠ A because center of ellipse
is not at r=0
Who keeps track of orbits of planets and
planetsimals?
• Up to date information on web sites provided by the Minor Planet
Center and JPL’s Solar System Dynamics Group.
• MPL is a good site to get locations of comets and asteroids.
• If you find a new object you report your observations to this site
and they help you calculate the orbital elements. This allows
others to find the object again later. Minor Planet Electronic
Circulars Circulars (MPECs) are such reports.
• A list of positions on the sky (in RA and DEC) as a function of time
is known as an ephemeris.
• Many objects are being discovered as part of active on-going
searches (satellites of giant planets, Kuiper Belt objects, comets
and small asteroids).
• Orbital elements of recently discovered objects may not be very
well measured. Measurements over years are often required
before the elements are known to any precision.
• Spacecraft missions in order to be successful require accurate
predictions for orbits (e.g., think about the Cassini mission)
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