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“First day, I’d tell the class all I knew,” he told The Saturday Evening Post, “and there was nothin’ left to say for the rest o’ the semester.” ­ Andy Griffith on teaching
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Orbital Velocity in the Solar System
•
You're set for life if you remember that the Earth orbits the
Sun at a speed of 30 km/s.
v orbital
v orbital = 30 km/ s
√
√
GM
=
R
1
R A.U
Where R is expressed in astronomical units
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Orbital Velocity in the Solar System
•
Derivation of this scaling ... basically multiply by “1” at various
locations and extract the equation for Earth's orbital speed
(the 30 km/s). Mo is the mass of the Sun. Re is the Earth's
distance from the Sun.
v orbital =
v orbital = 30 km / s
√
1
R A.U
√
Mo
GM
Mo
=
Re
R
Re
Mass of the star compared to the Sun
√ √
30 km/s
where R is expressed in astronomical units
GM o
Re
M /Mo
R/ Re
Distance from star in astronomical units
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How Can it be Practical to Deliver Your Own
Interplanetary Launch Platform to the Surface of
the Moon???
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Escape Velocity
•
Make an object's energy positive.
•
That is, kinetic energy exceeds gravitational potential/binding
energy.
1
2
K.E. = m v
2
GM m
P.E. =
R
Equating the two gives you the “escape velocity”
•
•
•
•
v escape =
√
2 GM
R
Escape velocity from the surface is sqrt(2) larger than orbital
velocity.
moriginal
Δ v = v exhaust ln
m payload
mo
=e
mp
Δv
v ex
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The Oberth Effect
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Oberth: Energy vs. Momentum
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The Celestial Sphere
From our perspective on Earth the stars appear embedded on a
distant 2-dimensional surface – the Celestial Sphere.
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The Celestial Sphere
Although we know better, it is helpful to use this construct to think
about how we see the night sky from Earth.
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Local Perspective: Altitude, Azimuth, and Zenith
Altitude
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A Personal Perspective: Horizon and Zenith
Horizon
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The Horizon vs. the Celestial Sphere
Each individual observer has their own personal local horizon.
➔
➔
In simplest terms this horizon is a flat plane tangent to the Earth at the
observer's location.
The giant observer below is misleading. For an observer of proper size
the Earth would block ½ of the sky, defining the horizon.
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The Key to Understanding the Night Sky
For a given observer the Earth blocks ½ of the sky at any instant.
●
The key is understanding which half.... which depends on
➔
The observer's location on the Earth
➔
The time of day/night (which way the Earth is turned relative to the sky)
➔
The time of year is also important as it determines which part of the sky
is washed out by daylight (or, said another way, which part of the sky
you are facing at midnight).
d
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A Personal Perspective: Horizon and Zenith
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Sunset
What Time is It?
To the Sun
Midnight
Noon
Sunrise
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Reference Points on the Celestial Sphere
Extend the Earth's poles and equator onto the sky and you have
defined the celestial poles and celestial equator.
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The Celestial Poles
The “North Celestial Pole” lies overhead for an observer at the
North Pole and on the horizon for an observer on the Equator
➔
The altitude of the pole equals your latitude.
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The Celestial Poles
The “North Celestial Pole” lies overhead for an observer at the
North Pole and on the horizon for an observer on the Equator
➔
The altitude of the pole equals your latitude.
To Pole
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The Celestial Poles
The “North Celestial Pole” lies overhead for an observer at the
North Pole and on the horizon for an observer on the Equator
➔
The altitude of the pole equals your latitude.
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The Celestial Poles
The “North Celestial Pole” lies overhead for an observer at the
North Pole and on the horizon for an observer on the Equator
➔
The altitude of the pole equals your latitude.
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Polaris
In the Northern Hemisphere there is a star, not all that bright,
near the North Celestial Pole.
➔
It resides at the end of the handle of the “Little Dipper” and is called
Polaris (for good reason – at least for now)
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The Celestial Poles
The rotating Earth makes it look like the Celestial Sphere is
spinning about the celestial poles.
http://www.atscope.com.au/BRO/warpedsky.html
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In the Southern Hemisphere
there is no good pole star (at
present).
Polaris
Note that there are some
stars (near the pole) that
never set below the horizon “Circumpolar Stars”
●
●
For an observer at the
North or South pole
every star is circumpolar.
At the Equator there are
no circumpolar stars
Given the altitude of the
pole, circumpolar stars have
declinations between 90 and
90-lat degrees.
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A Personal Perspective: Horizon and Zenith
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The Celestial Equator in the Sky
The Celestial Equator is the locus of all points lying 90 degrees
from the celestial pole.
●
●
It is a great circle around the celestial sphere.
Since the celestial sphere “turns” around the poles. The celestial
equator is a fixed reference line in the sky (rotating over itself).
➔
➔
The celestial equator runs from the horizon due east, up in the sky
(90-lat) degrees and back down to the horizon due west.
Stars “above” the celestial equator have positive declination (at least as
seen from Charlottesville).
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The Sun and the Celestial Sphere
As the Earth orbits the Sun we seen the Sun in different locations
against the backdrop of stars.
The Earth reaches the same location in its orbit on the same
calendar date each year.
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The Sun and the Celestial Sphere
Said another way, the Sun finds itself fixed at a different location
on the celestial sphere each day (more or less) – as a result on
that day it behaves like any other given star, following a path
dictated by the rotation of the Earth.
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The Sun and the Celestial Sphere
As the Earth orbits the Sun we seen the Sun in different locations
against the backdrop of stars.
The set of constellations through which the Sun passes is called
the Zodiac.
●
The Sun lies in front of your “birthsign” constellation on your
birthday.
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Solstices and Equinoxes
The Sun does not lie on
the celestial equator but
follows a path inclined by
23 ½ degrees.
The path crosses the
celestial equator at 2
points (the vernal and
autumnal equinox) marking
the instant of the beginning
of Spring and Fall
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How You See the Sun's Motion Through a Year
●
Since the Sun is sometimes 23 ½ degrees above the Celestial
Equator, sometimes 23 ½ degrees below, and sometimes right on
the Equator the Sun's behavior is different as the Celestial Sphere
turns.
➔
Remember that day by day the Sun occupies a slightly different location
on the celestial sphere, but it is the turning of the celestial sphere that
dictates its daily motion.
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How You See the Sun's Motion Through a Year
●
In the Summer, the Sun is well north of the celestial equator and
behaves more like a star near the north celestial pole (more like a
circumpolar star) – so it is above the horizon much more than 12
hours.
➔
●
At very northerly latitudes the Sun actually can be circumpolar.
In the Winter, the Sun is well south of the celestial equator. It
behaves more like one of those southern stars that barely makes it
above the horizon – short days.
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Consequences for the Seasons
●
Note that the Seasons are reversed between the Northern and
Southern hemispheres. It is Summer in January in Brazil.
Views from the Sun at the Winter (left) and Summer (right) solstice
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