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Oberth: Energy vs. Momentum
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Oberth: Energy vs. Momentum
Energy is still conserved.
You are cashing in on the investment you made adding kinetic energy to the fuel itself while accelerating during launch.
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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
“Depth” is not apparent, but can be inferred
(and ultimately measured)
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Celestial Coordinates:Right Ascension and Declination
RA/Dec represents a “fixed” coordinate system on the sky in which every star has a celestial latitude and longitude measured relative to celestial sphere reference points which are extensions of Earth coordinates onto the sky.
Declination == Celestial Latitude● Measured in degrees (eg. dd:mm:ss.s or dd.dddddd)● Ranges from -90 to +90 just like latitude on Earth
Right Ascension (R.A.) == Celestial Longitude● Measured in hours around the sky from a “prime meridian”.
● hh:mm:ss.s (sometimes dd.dddddd)● Since there are 24 hours around the sky corresponding to 360
degrees, RA in hours converts to RA in degrees via a factor of 15.➔ RA 7.0 hours corresponds to RA 105 degrees
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Declination is the angular distance above (+) or below (-) the celestial equator.
Right Ascension is the angular distance “eastward” from an not-so-arbitrary reference point around the sphere.
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A Practical Example
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A Practical Example
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Viewing 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.
The celestial sphere is “infinite” is size but that wouldn’t fit here….
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Conversion to a Local Perspective: Altitude, Azimuth, and Zenith
Altitude
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Horizon and Zenith vs. R.A. and Dec
Horizon
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Horizon
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Altitude and Azimuth in the Context of RA/Dec
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Altitude and Azimuth in the Context of RA/Dec
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Transforming Between Local and Equatorial
sin (alt )=sin (dec)∗sin(lat )+ ¿ ¿
cos(az)=sin (dec)−sin(alt )∗sin (lat )
cos(alt )∗cos(lat )¿
cos(dec)∗cos(lat )∗cos(HA)
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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).
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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.
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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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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
Each star traces out a circle around the pole at its Declination.
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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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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.
“Daily” Motion of the Stars (and Sun)
As seen from the Northern Hemisphere….
• a star well south of the celestial equator may rise at H.A. = -3 hours (above horizon for only 6 hours)
• a star well north of the celestial equator may rise at H.A. = -9 hours (above horizon for 18 hours)
(circumpolar)
Celestial equator
“Hour Angle” and The Meridian Every line of celestial longitude is a meridian of longitude passing
through both poles, but we recognize the line of longitude, or simply the great circle line, running overhead as “THE” Meridian.
“Hour Angle” and The Meridian The Meridian divides the visible sky into Eastern and Western
celestial hemisphere. As a star crosses The Meridian it is said to “transit” A star is at its highest point in the sky when it transits.
“Hour Angle” and The Meridian
For a particular star Hour Angle measures the time, in hours, either before (East, or negative) or after (West, or positive) transit.
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Transforming Between Local and Equatorial
sin (alt )=sin (dec)∗sin(lat )+ ¿ ¿
cos(az)=sin (dec)−sin(alt )∗sin (lat )
cos(alt )∗cos(lat )¿
cos(dec)∗cos(lat )∗cos(HA)
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Transforming Between Local and Equatorial
sin (alt )=sin (dec)∗sin(lat )+ ¿ ¿
cos(az)=sin (dec)−sin(alt )∗sin (lat )
cos(alt )∗cos(lat )¿
cos(dec)∗cos(lat )∗cos(HA)
Hour Angle
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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 and the analog of the Earth’s Equator.
● 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 (R.A., Dec) on the celestial sphere each day. As a result, on that day it behaves like any other star, following a path dictated by the rotation of the Earth.
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The Sun and the Celestial Sphere
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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To the Sun
Sunrise
Sunset
MidnightNoon
What Time is It?
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Solstices and Equinoxes
The Sun’s path does not follow the celestial equator but is inclined by 23 ½ degrees (due to the “obliquity” of the Earth).
The inclined Solar path intersects the celestial equator at 2 points (the vernal and autumnal equinox).
When the Sun arrives at these locations it marks the instant of the beginning of Spring and Fall.
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Solstices and Equinoxes
The location of the Vernal Equinox marks the celestial Prime Meridian
● 0h 0m 00.0s R.A
“Daily” Motion of the Stars (and Sun)
As seen from the Northern Hemisphere….• a star well south of the celestial equator may rise at H.A. = -3 hours (above horizon
for only 6 hours)• a star well north of the celestial equator may rise at H.A. = -9 hours (above horizon
for 18 hours)
(circumpolar)
Celestial equator
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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 Northern Winter (left) and Summer (right) solstice
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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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Solar vs. Sidereal Time
The Sun rises and sets on a slightly different schedule than the stars.
● The difference arises from the changing perspective as the Earth orbits the Sun.
● During a 24-hour Solar Day the Earth moves 1/365th of the way around its orbit.
➔ It must turn for an extra 24 hours/365.25 (= about 4 minutes) to get the Sun back to “Noon”
The Solar Day, by definition, is exactly 24.0000 hours long and is the time from Noon until Noon.
● The Sidereal Day – defining the rising and setting of the stars - is 3m 56s shorter and represents the true rotation period of the Earth.
The Sidereal Difereece
Daily activity on Earth is keyed to the mean solar day for obvious reasons.
Astronomers, however, care how the Earth is turned relative to the stars.
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Solar vs. Sidereal Time
A Sidereal clock keeps star time – it keeps 24 hour time, but completes a cycle in 23h 56m 4s of Solar time
● By convention, the time on a Sidereal clock equals the meridian of Right Ascension that is overhead at the moment.
● At Noon on the Spring Equinox R.A.=00:00:00.0 is overhead by definition (because the Sun, by definition, is at the Vernal Equinox at the moment Spring begins).
● The sidereal clock (and thus the celestial sphere relative to the Sun) runs “fast” by 3m 56s every day.
➔ A given star sets 4 minutes earlier each day➔ This ~4 minutes a day accumulates to 2 hours in a month.➔ Today’s night sky seen at 11 p.m. will be identical to the night sky seen at
9 p.m. one month from now.➔ 2 hours a month x 12 months = 24 hours – back to square one
● 24 hours divided by 365 is 4 minutes.
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Hour Angle
The Hour Angle of a star is the time until (East) or since (West) it crosses or has crossed the meridian.
The Hour Angle is simply the Right Ascension of the star minus the current sidereal time.
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Right Ascension Review
Right Ascension ● If you point your finger at a particular Declination the declination
value remains unchanged, but Right Ascension ticks away as the sky (actually the Earth) rotates.
● Right Ascension is thus naturally measured in units of time – hh:mm:ss.s
➔ One hour of right ascension is 15 degrees➔ The sky rotates by at 15 arcseconds per second at the Equator➔ Since lines of RA converge toward the pole – 1 minute of RA spans a
different angle depending on Declination – a factor of cos(Dec) comes into play.
Right Ascension/Longitude needs an arbitrary zeropoint (Greenwich on Earth, the “First Point of Aries” on the sky).
This reference point is the intersection celestial equator and ecliptic at of the location of the Sun at the Spring Equinox.
Convergence of Longitude at the Pole
On Earth one degree of latitude (equivalent of declination) is 111.3 km at any latitude.
One degree of longitude is 111.3km * cos(latitude)
A minute of Right Ascension is 15 minutes of arc at the equator, but a smaller angle at higher latitudes.
Calculating Angular Distances for Small Angles
Distances on the surface of a sphere are tricky.
● Interior angles on triangles sum to greater than 180 degrees.● The Pythagorean Theorem does not apply in general.
● For small distances, however, the sphere is locally “flat” and Pythagorus rules.
● At least as long as you account for peculiarities of the coordinate system.
● Declination is the same everywhere on the sphere (good news). Don’t mess with Dec.● Right Ascension has to be converted to the appropriate angle.
● A factor of 15 to convert from hours to degrees.● A factor of cos(Dec) to account for convergence toward the poles.
● What Armstroeg Saw
● Luear Recoeeaissaece Orbiter
● MASCOT Laeder
