Planets are flattened by rotation'

1
Planets are flattened by rotation. They can be
represented roughly by ellipsoids. More
irregular bodies can be represented by tri-axial
ellipsoids, with two different equatorial radii.
2
Geocentric latitude is measured by the angle that
a line to the COM makes with the equatorial
plane. Geographic latitude is measured by the
angle that the surface normal makes with the
equatorial plane.
For a non-spherical body the idea of latitude
becomes ambiguous. Many auxiliary latitudes exist
such as conformal, authalic, rectifying,
geocentric, parametric (aka reduced) latitude.
Most of these are just mathematical
constructs. Geographic (aka geodetic) latitude is
what is used in most map projections. Most space
based measurements are in the geocentric form.
3
e.g. Albers Equal Area e.g. Mercator
e.g. Stereographic
Map projections convert from lat/lon to x/y There
are three categories of map projections Find x/y
co-ordinates by taking a line from the center of
projection through the spheres surface and
intersecting the map surface. There are probably
100s of map projections none of which work
perfectly in a global sense.
4
Solid planets are not perfectly smooth
ellipsoids, they are lumpy irregular objects. A
gravitational equipotential surface is used to
define the shape of the planet. Its known as the
GEOID. This surface coincides with mean sea level
in a best fit sense.
Shown here is a representation of the geoid in
meters relative to a best fit ellipsoid. This is
a horizontal surface even though it has lumps
and holes in it.
5
The geoid undulates slowly over long
distances. Mass excesses in the lithosphere
cause Lumps in the geoid. Mass deficits in the
lithosphere cause depressions in the
geoid. Remember though the geoid is by
definition always flat.
6
A horizontal datum is an ellipsoidal shape used
to calculate longitude and latitude.
Global maps use a horizontal datum that fits the
geoid as well as possible over the whole
globe. Local/regional maps use a local
horizontal datum which fits the geoid best in
that particular region.
7
Topography is always measured relative to the
geoid. i.e. it is the orthometric height or
height above sea-level This is just as well since
we'd like our definition of downhill to be the
direction that gravity acts.
Topography and geoid height are usually
correlated. The difference between the two is
sometimes called the admittance.
8
You might expect that near a mountain gravity
would increase due to all that extra mass,
however the increase is not as much as you would
expect. This is due to what is called isostatic
compensation.
The Airy explanation of isostasy is the familiar
iceberg effect. A rigid lithosphere floats upon a
vicious asthenosphere. (Note This is not the
same as crust vs. mantle, which is a
compositional distinction) The Pratt explanation
of isostasy relies upon differing columns of
lithosphere having different densities. Its very
hard to distinguish between these two
possibilities.
9
Gravity measurements need many corrections so
that you can be sure your comparing apples with
apples. Before anything happens you must correct
for the observation latitude since planets are
oblate and spinning.
Terrain corrections remove mountains and fill in
valleys, this is only done for the most
mountainous regions. A Bouguer correction removes
the effect of an assumed infinite plate of
material. The free-air correction removes the
effect of height, this is sometimes combined with
the Bouguer correction into one elevation
correction. P Q should now have the same
gravity value but they usually dont. The
difference is called a gravity anomaly.
10
How do you tell if something is compensated or
not?
A strong positive free-air anomaly with a weak
Bouguer anomaly can be interpreted as meaning a
structure is supported only by the strength of
the lithosphere i.e. no compensation. A weak
free-air anomaly with a strong negative Bouguer
anomaly indicates that the structure is
compensated.
11
Bouguer anomalies can be interpreted as
Density flucuations below the surface. OR A
constant density lithosphere that varies in
thickness. A) Mars topography B) Mars free-air
gravity anomaly C) Bouguer anomaly converted to
crustal thickness. Figure from Zuber et al.,
Science, 287, 1788-1793,2000.