# Oberth: Energy vs. Momentum

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1 1

2 2 The Oberth Effect

3 3 Oberth: Energy vs. Momentum

4 4 The Celestial Sphere From our perspective on Earth the stars appear embedded on a distant 2-dimensional surface the Celestial Sphere.

5 5 The Celestial Sphere Depth is not apparent, but can be inferred (and ultimately measured)

6 6 Celestial Coordinates Right Ascension and Declination A fixed coordinate system on the sky in which every star has a celestial latitude and longitude measured relative to celestial sphere reference points. 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 == Celestial Longitude Measured in hours around the sky from a prime meridian. hh:mm:ss.s (sometimes dd.dddddd)

7 7

8 8

9 9 A Practical Example

10 10 A Practical Example

11 11 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.

12 Conversion to a Local Perspective: Altitude, Azimuth, and Zenith Altitude 12

13 13 A Personal Perspective: Horizon and Zenith Horizon

14 14 A Personal Perspective: Horizon and Zenith Horizon

15 15 Altitude and Azimuth in the Context of RA/Dec

16 16 Altitude and Azimuth in the Context of RA/Dec

17 17

18 18 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.

19 19 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

20 20 Sunset What Time is It? To the Sun Midnight Noon Sunrise

21 21 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.

22 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.

23 23 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.

24 24 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.

25 25 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.

26 26 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.

27 Daily Motion of the Stars (and Sun) (circumpolar) Celestial equator 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)

28 28 The Celestial Poles The rotating Earth makes it look like the Celestial Sphere is spinning about the celestial poles. Each star traces out a circle around the pole at its Declination.

29 29 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)

30 30 In 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. Polaris

31 31 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 (90lat) degrees and back down to the horizon due west. Stars above the celestial equator have positive declination (at least as seen from Charlottesville).

32 32 Transforming Between Local and Equatorial sin (alt )=sin (dec) sin (lat)+ cos(dec) cos(lat ) cos( HA ) sin (dec) sin (alt) sin (lat ) cos(az)= cos(alt ) cos(lat )

33 33 Transforming Between Local and Equatorial sin (alt )=sin(dec) sin (lat )+ Hour Angle cos(dec) cos(lat ) cos( HA) sin (dec) sin (alt ) sin (lat ) cos(az)= cos(alt ) cos(lat )

34 35 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.

35 36 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.

36 37 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.

37 38 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.

38 39 Solstices and Equinoxes The location of the Vernal Equinox marks the celestial Prime Meridian 0h 0m 00.0s R.A

39 Daily Motion of the Stars (and Sun) (circumpolar) Celestial equator 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)

40 41 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.

41 42 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.

42 43 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

43 44

44 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.

45 46 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. While the Earth completes a rotation it moves 1/365th of the way around its orbit. It must turn for an extra 24 hours/ (= about 4 minutes) to get the Sun back to Noon The Solar Day, by definition, is exactly 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.

46 47 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. The sidereal clock (and thus the celestial sphere) runs fast by 3m 56s every 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.

47 48 Transforming Between Local and Equatorial sin (alt )=sin(dec) sin (lat )+ cos(dec) cos(lat ) cos( HA) sin (dec) sin (alt) sin (lat ) cos(az)= cos(alt ) cos(lat )

48 The Meridian The Meridian runs from due north on the horizon, through the zenith, to due south. As the sky rotates stars follow small circle paths, rising in the east, and reaching their highest point when the cross The Meridian.

49 50 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.

50 Oee Simple Coeeectoe/Defeitoe The curreet Sidereal Time equals the Hour Aegle of the Vereal Equieox It also equals the Right Asceesioe of the Meridiae. Celestial Sphere Review

51 52 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.

52 Convergence of Longitude at the Pole On Earth one degree of latitude (equivalent of declination) is 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.

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