1. Hohmann Transfer Orbit
PHYS Astronomy
Homework Set #3
9/11/17
Due 9/18/17
1. The eccentricity of a comet is Its semimajor axis is 17.8 AU. What is its distance from the Sun at perihelion? Aphelion?
2. An asteroid has a mean orbital radius of 4.12 AU. What is its period?
3. Comet Hale-Bopp has an orbital period of 2537 years. What is its average distance from the Sun?
4. What is the equation of time on May 21? (show your work)
5. What is the equation of time on August 15? (show your work)
6. What is the approximate difference between Universal Time and Ephemeris Time today (2017)?
7. How long does it take to reach Mars, in the most efficient orbit - called the "Hohmann Transfer Orbit” (ignore the ellipticity of Mar’s and Earth’s orbits). (Worth double.)
Hohmann Transfer Orbit

2. Astronomical Time Periods
PHYS Astronomy
Astronomical Time Periods

3. Definitions of a Day Sidereal Day
PHYS Astronomy
Definitions of a Day
Sidereal Day
Time from one transit of a star across the meridian to the next.
Related to the Stars
Apparent Solar Day -
Time from one transit of sun across the meridian to the next.
From one high noon to the next
Related to the sun
Mean Solar Day
Time between successive transits of mean sun.
Average of apparent solar days over one year.
Defined to be 24 Hours

4. PHYS Astronomy
Sidereal Day
Sidereal - “related to the stars” - the time it takes for any star to make a circuit of the sky - about 23 hours 56 minutes. Measure of the Earth’s rotation - varies about 1 second in 45,000 years. Today defined relative to an ensemble of extra-galactic radio sources.

5. PHYS Astronomy
Solar Day
The time it takes for the Sun to make one circuit around the local sky - length varies over course of year (up to 25 seconds longer or shorter) but averages 24 hours.

6. Why is a Sidereal Day Shorter than a Solar Day?
PHYS Astronomy
Why is a Sidereal Day Shorter than a Solar Day?
One full rotation represents a sidereal day - but while orbiting the Sun, Earth travels in its orbit (about 1 degree per day). So the Earth must rotate slightly farther to point back at the Sun - solar day.

7. PHYS Astronomy
Mean Solar Day
Length of solar day varies over course of year - averages about 24 hours - mean solar day.
Two reasons for variance.:
1. Earth's orbit is not a perfect circle - it’s an ellipse - Earth moves faster when it is nearest the Sun and slower when it is farthest from the Sun.
2. Earth's axial tilt - the Sun appears to move at an angle to equator during the year - apparently moves fast or slow depending on whether it is apparently far from or close to the equator. Apparent solar days are shorter in March and September than they are in June or December.
Solar day may differ from a mean solar day by as much as nearly 22 s shorter to nearly 29 s longer. Because many of these long or short days occur in succession, the difference builds up to as much as nearly 17 minutes early or a little over 14 minutes late. Discrepancy called Equation of Time - leads to analemma - shape of Sun’s yearly path in the sky.

8. PHYS Astronomy
The Analemma
Images of the Sun at same time at day intervals over the course of a year:
Altitude variation due to annual north-south oscillation of the Sun’s declination angle - axial tilt
East-west spread due to variation in Sun’s motion wrt to the stars - axial tilt and ellipticity
Distortion of figure 8 due to ellipticity of Earth’s orbit.
In northern hemisphere - if Sun’s position east of where your watch indicates it would be, Equation of Time is negative. If Sun to the west, the Equation of Time is positive.

9. PHYS Astronomy
The analemma shows where the sun is at any given time of day, on any day of the year - it rises, crosses the sky, and sets. From northern latitudes, the analemma slants upward and to the left at "sunrise" and upward and to the right at "sunset". (The analemma is vertical at local noon.)
Why do the earliest sunset, latest sunrise, and shortest day of the year occur on different dates?

10. PHYS Astronomy
Latest sunrise occurs when the last part of the analemma rises - this doesn’t happen when the sun is "at the bottom" of the analemma (December 21st) but a few days later (January 3th). Earliest sunset occurs when the first part of the analemma sets below the western horizon. Again, occurs a few days (December 8th) before the sun reaches "the bottom" of the analemma (December 21st). Same reasoning for solstice.

11. Often printed on globes and…
PHYS Astronomy
Often printed on globes and…
Turned vertical, the analemma can be used to determine solar declination and Equation of Time
sundials

12. Kepler’s Laws of Planetary Motion
PHYS Astronomy
Kepler’s Laws of Planetary Motion
Law 1: Law of elliptical orbits
Each planet moves in an elliptical orbit.
Law 2: Law of areas
The imaginary line connecting any planet to the sun sweeps over equal areas of the ellipse in equal intervals of time.
Law 3: Law of periods
The square of any planet's period of orbital revolution is proportional to the cube of its mean distance from the sun.

13. PHYS Astronomy
Kepler’s First Law
The orbit of each planet around the Sun is an ellipse with the Sun at one focus

14. Drawing a circle. (b) Drawing an ellipse. (c) Eccentricity
PHYS Astronomy
Drawing a circle.
(b) Drawing an ellipse.
(c) Eccentricity
describes how much an ellipse deviates from a perfect circle -ratio between distance from the center of the ellipse to the focus of the ellipse and the semi-major axis.

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Kepler’s Second Law
As a planet moves around its orbit, it sweeps out equal areas in equal times.

17. Area and Time Animation
PHYS Astronomy
Area and Time Animation

18. PHYS Astronomy
Kepler’s Third Law
The square of any planet's period, P, of orbital revolution is proportional to the cube of its mean distance, r, from the sun. I.e., the more distant a planet, the slower it moves on average. Example: For earth, r E= 1 AU, PE = 1 year. For Mars, r M= 1.52 AU, PM = 1.88 years

19. A plot of the cube of the average planetary distance vs the square
PHYS Astronomy
A plot of the cube of the average
planetary distance vs the square
of the orbital period is a straight
line
The average orbital speed is
inversely proportional to the
average distance from the sun

20. Planet Semimajor Axis (1010 m) Period (T yr) T2/a3 (10-34 y2/m3)
PHYS Astronomy
Planet
Semimajor
Axis (1010 m)
Period
(T yr)
T2/a3
(10-34 y2/m3)
Mercury
5.79
0.241
2.99
Venus
10.8
0.615
3.00
Earth
15.0 1
2.96
Mars
22.8
1.88
2.98
Jupiter
77.8
11.9
3.01
Saturn
143
29.5
Uranus
287 84
Neptune
450
165
Pluto
590
248

21. PHYS Astronomy
Note:
- Kepler's third law needs to be modified when the orbiting body's mass is not negligible compared to the mass of the body being orbited.
- Kepler's laws assume a two-body system - particularly bad approximation in the case of the Earth-Sun-Moon system
for calculations of the Moon's orbit, Kepler's laws are far less accurate than the empirical method invented by
Ptolemy.
- Kepler's laws do not consider the emission of radiation or relativity
- Because electrical forces, like gravity, obey an inverse square law, Kepler's laws also apply to bodies interacting electrically.
Kepler did not understand why his laws were correct - Isaac Newton discovered the answer more than fifty years later. For instance, the second law also a statement of conservation of angular momentum.

22. PHYS Astronomy
Equation of Time

23. Elliptical Orbit Effect
PHYS Astronomy
Elliptical Orbit Effect
Elliptical orbit means orbital velocity varies - (Kepler’s 2nd Law).
- greatest at perihelion - in January
- smallest at aphelion - in July
In January, Earth is traveling faster than average - at the end of 24 hours each Earth has rotated approximately 361 degrees

24. Superimpose the orbits - after 24 hours:
PHYS Astronomy
Superimpose the orbits - after 24 hours:
- sun appears to be directly overhead on Earth “A”.
- sun is NOT directly overhead on Earth “B” - has not quite rotated far
enough relative to the
sun.
Looking at your watch on Earth “B” at noon - appears that the sun's position slightly to the east. Error in time accumulates and the sun will continue to appear to move farther and farther east in the sky.
- error accumulates until April 2 - maximum offset to east
- from April 2 to July 3 - sun drifts back to west
- reaches maximum westward offset on October 2

25. PHYS Astronomy
mean anomaly (M) - the fraction of the orbital period that has elapsed since the last passage at periapsis (perehelion) z, expressed as an angle.
true anomaly (T) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse
eccentric anomaly (E) - the angle between the direction of periapsis z and the current position of an object on its orbit, projected onto the ellipse's circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse.

26. equal to 0 degrees at perihelion. equal to 180 degrees at aphelion.
PHYS Astronomy M T
true anomaly - angle (as seen from the Sun) between the Earth and the perihelion of the orbit of the Earth
equal to 0 degrees at perihelion.
equal to 180 degrees at aphelion.
mean anomaly - what the true anomaly would be if the Earth moved with constant speed along a perfectly circular orbit around the Sun in the same time.
equal to 0 at perihelion and 180 degrees at aphelion - only places true and mean anomalies are equal

27. (Assume no axial tilt in discussion)
PHYS Astronomy
Effect of Elliptical Orbit on Equation of Time
(Assume no axial tilt in discussion)
Effect equal to difference between mean anomaly and true anomaly
G is the gravitational constant, m is the planetary mass, a is the semimajor axis, t is the time from perihelion
(1)
(2)
e is the eccentricity, r is the orbiting body’s position vector
e is the eccentricity vector - points
toward perihelion with magnitude e=|e|
(3)
(4)

28. PHYS Astronomy
(5)
(6)

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32. (Assume no ellipticity in discussion)
PHYS Astronomy
Effect of Axial Tilt on Equation of Time
(Assume no ellipticity in discussion)
Path of the Sun between March 21 and March 22
If no axial tilt, motion of sun horizontal only - at noon every day, sun appears at highest point in the sky (culmination) only drifting slightly east against background of stars - represents Mean Sun that would travel on the celestial equator.
In reality, sun drifts slightly east and to the north (or south) depending on the season - represents True Sun that travels in the ecliptic.

33. View of True Sun (red) and Mean Sun (blue) over year
PHYS Astronomy
View of True Sun (red) and Mean Sun (blue) over year
- both travel at constant speed
- speed of projection on equatorial plane different
- in same position at vernal and autumnal equinoxes.
Looking at top view - start together but True Sun lags Mean Sun until May, then begins to catch up, catching it at summer solstice. True Sun appears in the sky to the right of Mean Sun between vernal equinox and summer solstice - culminates before the Mean Sun.

34. PHYS Astronomy
Close-ups - both suns moving at same speed around celestial sphere - only paths are different.
Near vernal equinox - True Sun has farther to travel to the same longitudinal position as the Mean Sun. Viewed from Earth - True Sun has drifted to the west.
View from the equator outside the celestial sphere - longitudinal lines added to the sphere.
Around the summer solstice - True Sun has caught up - moving in same direction horizontally. True Sun moving faster because longitudinal lines are closer together.

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38. PHYS Astronomy
Summation of the Effects of Tilt and Eccentricity on the Equation of Time

39. PHYS Astronomy
View From the Sun

40. Synodic Month vs Sidereal Month
PHYS Astronomy
Synodic Month vs Sidereal Month
Synodic month
- the time it takes for the moon to complete its cycle of phases - or to come back to the same position with respect to the Earth-Sun line - about 29 1/2 days
Sidereal month
- the time it takes for for the moon to complete one orbit relative to the position of the stars - about 27 1/3 days

41. Synodic vs Sidereal Month Animation
PHYS Astronomy
Synodic vs Sidereal Month Animation

42. Tropical vs Sidereal Year
PHYS Astronomy
Tropical vs Sidereal Year
Sidereal year
- time it takes for Earth to complete one orbit relative to the stars or the time for the Sun to return to the same position in respect to the stars - the orbital period of the Earth
Tropical year
- calendar year - time from spring equinox to spring equinox - about 20 minutes shorter than a sidereal year - because of precession. Changes not only the orientation of Earth’s axis but the location in Earth’s orbit of the seasons. 26,000 year precession period means that location of solstices and equinoxes among the stars shifts by 1/26,0000 around the orbit. 1/26,000th of a year is about 20 minutes.

43. After 100 years, in error by more than 3/4 day:
PHYS Astronomy
A tropical year is about days long - requires leap year every 4 years to keep solstices and equinoxes on same calendar date.
After 100 years, in error by more than 3/4 day:
100 tropical - (75 years + 25 leap years) = = days
Skip leap year every 100 years
After 400 years error of nearly a day:
400 tropical years - (304 regular years + 96 leap years)=146, ,096 = 0.876
So add a leap year every 400 years.
Earth’s rotation slowing - tidal drag - add about 29 leap seconds every 100 years.

44. PHYS Astronomy
Leap Seconds
Ephemeris Time - the average mean solar time between 1750 and 1890 (centered on 1820) - the period during which the observations on which Simon Newcomb's Tables of the Sun, which formed the basis of all astronomical ephemerides from 1900 through 1983, were performed.
Universal Time (UT) - timescale based on the rotation of the Earth. - a modern continuation of the Greenwich Mean Time (GMT), i.e., the mean solar time on the meridian of Greenwich, England.
Ephemeris Time and Universal Time are different because the Earth’s rotation is slowing down due primarily due to tidal friction
ΔT - the time difference obtained by subtracting Universal Time from Ephemeris Time.

45. one second =time for a cesium atom to make 9,192,631,770 vibrations
PHYS Astronomy
Initially, second defined as 1/86,400th of a mean solar day in the year Now use stable atomic clocks
one second =time for a cesium atom to make 9,192,631,770 vibrations
Rotation of Earth slowing down
- tidal friction, atmospheric circulation, internal effects, transfer of angular momentum to the Moon orbital motion, etc…
To a first approximation
- tidal forces slow Earth's rate of rotation by 2.3 ms/day/cy.
- melting of continental ice sheets at the end of the last ice age removed their tremendous weight
- allowed land under them to begin to isostatically rebound upward in the polar regions - continues to this day
- causes Earth's rate of rotation to speed up by 0.6 ms/day/cy.
The net tidal acceleration or the change in the length of the mean solar day is +1.7 ms/day/cy.

46. PHYS Astronomy

47. Actual T varies significantly because of aforementioned effects.
PHYS Astronomy
Actual T varies significantly because of aforementioned effects.
- all values of ΔT before 1955 depend on observations of the Moon, either via eclipses or occultations
- now, orientation of the Earth relative to an inertial reference frame formed by extra-galactic radio sources is used

48. PHYS Astronomy
Modern Time Scales
Temps Atomique International (TAI) or International Atomic Time - weighted average of the time kept by about 200 cesium atomic clocks in over 50 national laboratories worldwide - available since 1955
UT1 - timescale defined by the Earth's rotation
- computed by the International Earth Rotation and Reference Systems Service (IERS).
- TAI was defined such that TAI = UT1 on January 1, 1958.
Coordinated Universal Time (UTC) - basis for legal time worldwide, - TAI became the international standard on which UTC is based on January 1, 1972.
- UTC always differs from TAI by an integral number of seconds.
- in mid 2005, behind TAI by 32 seconds
- difference due to leap seconds periodically inserted into UTC to keep it from drifting more than 0.9 seconds from UT1
27 leap seconds since first leap second added in last was added on December 31, (Always added on June 30 or December 31)