Gravity

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Gravity

G mEarth m2 F
gravity on Earth ------------
r2
What about gravitational forces on the
Moon? mEarth 5.97e24 kg Earth on Moon
FEarth/GmMoon 4.05e13 Sun on
Moon FSun/GmMoon 8.84e13
FEarth/FSun 0.46
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Keplers First Law
Kepler I planetary orbits are ellipses with the
Sun at a focus
position of object in orbit
a (1 - e2) rSun --------------
1 e cos f f, true anomaly angle
between perihelion
and current position
eorbit (1 - b2minor/a2major)1/2
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Keplers Second Law
Kepler II a line between a planet and the Sun
sweeps out equal areas in equal times

dA/dt constant
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Keplers Third Law
Kepler III planetary orbital periods and
distances from the Sun are simply related
as long as you assume Solar System units
P2 (yr)
a3 (AU)
Newton it also works outside
of the Solar System
4p2a3
a3 P2
(yr) ---------------- or
Mtotal -------
G (m1 m2)
P2
solar
masses, AU, yrs Application stellar and
planetary masses need fractional mass, f, for
individual masses double dirty little secret
of exoplanet masses
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Orbital Elements
a semimajor axis size e eccentricity shape
i inclination tilt P orbital
period time T epoch of periastron a
date O longitude of ascending node flip
angle ? longitude of
periastron t?ist angle
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Flip Longitude of Ascending Node
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T?ist Longitude of Periastron
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Orbital Elements
a semimajor axis size e eccentricity shape
i inclination tilt P orbital
period time T epoch of periastron a
date O longitude of ascending node flip
angle ? longitude of
periastron twist angle equinox equinox of
date sets direction of equinox f fractional
mass a number Two observations will not yield
an orbit. Why? Each point has (position X,
position Y, time). There are 7 classical
unknowns, so you need a third point to give you 9
pieces of data to solve equations.
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Earths Orbit and Astrometry
a semimajor axis 1.00000261
AU e eccentricity 0.01671123 i inclination
0.00001531 deg
Must be careful to get Earths orbit correct when
measuring parallaxes or other object orbits using
astrometry.
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Scale of CCD Over Full Chip
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Scale of CCD Over Time
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Scale of CCD Which Reference?
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Rotation of CCD Good Case
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Rotation of CCD Bad Case
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Image Shapes
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Image Offsets Relative to 2MASS
all stars kept
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Image Offsets Relative to 2MASS
remove stars with largest offsets
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Image Offsets Relative to 2MASS
remove more stars divide into chip regions
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Centroid Errors RA
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Centroid Errors DEC
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Centroid Errors RA vs,. DEC
V filter observations worst I filter observations
best
DEC residuals tend to be worse than RA

• gt airmass sky position
• gt airmass observing tech.
• CCD readout!

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Astrometry Gotchas
Gotchas you can work on 1. detector
uniformity across chip 2. telescope/camera/detec
tor stability over time 3. image shape due to
seeing/optics/telescope tracking 4. filter
through which observations are made 5. model
Earths orbit accurately Gotchas given to you by
nature 6. object brightness 7. airmass of
object (DCR differential color refraction) 8.
number of reference stars 9. source
confusion 10. location in the sky vs. location of
telescope
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