Transcript Celestial Mechanics Fun with Kepler and Newton

Positions on the Celestial Sphere
• How to locate
(and track)
objects from a
spinning, orbiting
platform in
space….
• Homework 1 due Tuesday Jan 15
Pointing and Tracking Astrophysical Objects
Frisco Peak Telescope Control Panel
Frisco Peak Telescope – Pointing and
Tracking Precision less than 10 arcsec
(Angle subtended by a penny at
roughly 200 yards)
1 radian=206,264.98 arcsec
Positions on the Celestial Sphere
The Altitude-Azimuth Coordinate System
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Coordinate system based on observers
local horizon
Zenith - point directly above the
observer
North - direction to north celestial pole
NCP projected onto the plane tangent
to the earth at the observer’s location
h: altitude - angle measured from the
horizon to the object along a great
circle that passes the object and the
zenith
z: zenith distance - is the angle
measured from the zenith to the object
z+h=90
A: azimuth - is the angle measured
along the horizon eastward from north
to the great circle used for the measure
of the altitude
Equatorial Coordinate System
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Coordinate system based on a projection of
the Earth’s geographic poles and equator
onto the celestial sphere
Celestial Poles - Points on celestial sphere
intersected by line coincident with Earth’s
rotation axis. North and South.
Celestial Equator - Projection of Earth’s
Equator onto the the celestial sphere.
: Declination - The latitudinal angle
measured from the celestial equator to the
object along a great circle that passes
through the object and the celestial poles.
Objects north of the celestial equator have
declinations in the range from 0 to +90
degrees. Objects south of the celestial
equator have declinations in the range from
0 to -90 degrees.
: Right Ascension - The longitudinal angle
measures the angle of how far east an object
lies from the Vernal Equinox.
: Vernal Equinox - Point on celestial equator
where the path of the sun crosses the
celestial equator.
Positions on the Celestial Sphere
Meridian
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Meridian- Imaginary line through the Zenith and North Celestial Pole extended along a great circle
on the celestial sphere to the northern and southern horizon
Positions on the Celestial Sphere
Hour Angle and Local Sidereal Time
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Local Sidereal Time (LST) - The amount of time that has elapsed since the Vernal
Equinox (point) last traversed the local meridian.
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Hour Angle- The hour angle (HA) of an object is equal to the difference between the
current local sidereal time (LST) and the right ascension (α) of that object:
Precession of the Equinoxes
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Due to slow wobble of Earth’s rotation axis
caused by Earth’s non-spherical shape and
gravitational interaction with the Sun and
planets
Precession period is 25,770 years
Pole will sweep out a circle on the sky with
angular diameter of nearly 47 degrees!!!
Impacts the values of RA and DEC of objects
over time ==> Epochs. The change in values
of RA and DEC from their values in the year
2000.0 are given by:
M = 1.2812323 T + 0.0003879 T 2 + 0.0000101 T 3
N = 0.556753 T - 0.0001185 T 2 - 0.0000116 T 3
T = (t - 2000.0) /100
Even More Astronomical Coordinate Systems…
Other coordinate systems may be
more appropriate …
For instance mapping objects in the
milky way is best done using galactic
coordinates
Conversions between coordinate
systems are possible…with a little
math!!!
Changes in the Sky
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Coordinates continuously changing in
alt-az system for all celestial objects
(except geo-stationary satellites)
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Earth’s rotation
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Earth’s orbit about Sun
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Proper motion of objects
– The moon
– Planets
– Asteroids
– Comets
– Satellites….
Milky Way From Frisco
Peak. Paul Ricketts 
Earth’s Rotation
• Earth’s rotation is responsible for
the “rapid” motion of objects
through the sky
Mud springs point 2 hour exposure
of NCP. Paul Ricketts
Tilt of Earth’s Axis
• The rotation axis
of the Earth is
tilted at 23.5
degrees from the
normal of the
Ecliptic plane.
The Sun’s motion in the sky
Seasonal Changes
•The Sun’s motion across the sky changes
over the course of the year. This is due
mainly to the tilt of the Earth’s rotation axis.
•The maximum angular height above the
horizon at a location in the Northern
Hemisphere is maximum on the Summer
Solstice and minimum on the Winter
Solstice. This results in sunlight being
spread over a larger area during Winter
causing a reduction in heating of the
surface of the Earth at Northern latitudes
during Winter.
• The values of the Sun’s RA and DEC
change over the year as shown in the
graph.
Earth’s Orbit about the Sun
Due to the Earth’s motion about the Sun :
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Line of sight to the sun sweeps through
the constellations. The sun apparently
moves through the constellations of the
zodiac along a path known as the Ecliptic
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The constellations that are visible each
night at the same time changes with the
season
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A given star will rise approximately 4
minutes earlier each day
The Ecliptic.The path of the sun through
the year in equatorial coordinates.
What is a day?
The period (sidereal) of earth’s revolution
about the sun is 365.26 solar days. The
earth moves about 1 around its orbit in
24 hours.
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Solar day
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Is defined as an average interval
of 24 hours between meridian
crossings of the Sun.
The earth actually rotates about its
axis by nearly 361 in one solar
day.
Sidereal day
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Time between consecutive
meridian crossings of a given star.
The earth rotates exactly 360
w.r.t the background stars in one
sidereal day = 23h 56m 4s
Annalemma
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The curve traced on the sky by
the position of the Sun at local
noon over the course of the
year.
The change in altitude
(declination) is the result of the
tilt of the earth’s rotation axis.
The change in hour azimuth
(hour angle) is due mainly to
the variation in the length of
the solar day resulting from the
Earth’s elliptical orbit.
For more info see.…
http://www.annalemma.com
Proper Motions
Celestial objects may
have their own proper
motion…including
stars, galaxies…
Radial motion is along
the line of sight.
Transverse motion is
the component of
motion perpendicular to
the line of sight to the
object.
Measurements of Time
• Gregorian Calendar - Calendars which label months and dates
for each year need to account for the fact that Earth’s revolution
period is not an integer number of days…--> Leap Years, Leap
Seconds,…
• Julian Date - Astronomers typically refer to the times when
observations were made in terms of elapsed time since some
specified zero time. The zero time that is used is noon on
January 1,4713 BC…JD0.0. The Julian date of J2000.0 is
JD2451545.0. Times other than noon Universal time are
specified as a fraction of a day. MJD=JD-2400000.5. MJD
begins at midnight instead of noon…
Celestial Sphere and Spherical Trigonometry
Law of Sines
sin a sin b sin c
=
=
sin A sin B sinC
Law of Cosines for sides
Side= arc of great circle
Angle=angle between sides
cosa = cosbcosc +sinbsinccos A
Law of Cosines for angles
cos A = -cosBcosC +sin BsinCcosa
Sum of angles of spherical triangle<180 degrees
Celestial Sphere and Spherical Trigonometry
Want to find opening angle Dq
Given D and D
Apply law of sines
sin(DJ ) sin[90 - (d + Dd )]
=
sin(Da )
sin f
sin(Da )cos(d + Dd ) = sin(DJ )sin f
cose » 1,sin e » e
Small angle approximation
Da = DJ
sin f
cos(d )
Apply law of cosines
cos[90 - (d + Dd )] = cos(90 - d )cos(Dq )+sin(90 - d )sin(Dq )cosf
Small angle approximation (again)
Dd = Dq cos f
Angular distance in terms of changes in RA and DEC
(Dq )2 » (Da cosd )2 + (Dd )2
Celestial Sphere and Spherical Trigonometry
from a rotating platform
http://en.wikipedia.org/wiki/Celestial_sphere
"A Compendium of Spherical Astronomy".
Planetarium/Telescope Control Software
Starry Night
Your Phone…
Stellarium