1 “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.

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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.

Where R is expressed in astronomical units

vorbital = √GMRvorbital = 30 km / s √ 1

RA.U

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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.

where R is expressed in astronomical units

30 km/s Distance from star in astronomical units

Mass of the star compared to the Sun

vorbital = √GM M o

M o

RReRe

= √GM o

Re √M /M o

R /Re

vorbital = 30 km / s √ 1RA.U

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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.

Equating the two gives you the “escape velocity”

• Escape velocity from the surface is sqrt(2) larger than orbital velocity.

K.E. = 12mv2 P.E. = GMm

R

vescape = √ 2 GMR

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mom p

=eΔ vv exΔ v = vexhaust ln

moriginalm payload

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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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To the Sun

Sunrise

Sunset

MidnightNoon

What Time is It?

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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



To Pole

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The Celestial Poles



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The Celestial Poles



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PolarisIn 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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PolarisIn the Southern Hemisphere there is no good pole star (at present).

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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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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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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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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1 “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.

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