Astronomy 291

Professor Bradley M. Peterson

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The Sky• As a first step, we need to understand the

appearance of the sky. • Important points (to be explained):

– The relative positions of stars remain the same (on human time scales).

– The Sun moves eastward relative to the stars (about 1° per day).

• Relative to the Sun, stars rise 4 min earlier each day.

– The Moon moves eastward relative to the stars (about 13° per day).

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The Celestial Sphere• To understand the

appearance of the sky, it is useful to imagine it as a sphere with Earth at the center.

• Observer can see only sky above the horizon (an imaginary plane tangent to Earth at the observer). Observer

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The Celestial Sphere• Suppose Earth is

stationary; then sky revolves around axis that is extension of the Earth’s rotation axis.

• Rotation axis intersects celestial sphere at north and south celestial poles.

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The Celestial Sphere• Rotation of the sky is

apparently westward as seen by the observer.

• Stars near celestial pole never drop below horizon. These are circumpolar stars.

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North Circumpolar Stars in Time Exposure

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The Celestial Sphere• Rotation of the sky is

apparently westward as seen by the observer.

• Stars near celestial pole never drop below horizon. These are circumpolar stars.

Now let’s expand the view of the Earth in this diagram….

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Projection of equator on to sky is the celestial equator.

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The zenith is the point on the celestial sphere directlyabove the observer.

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The meridian passes through the celestial poles andthe zenith, bisecting the sky into east and west.

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Apparent Paths of Stars

• To see the motion of the stars, let’s zoom back outside the celestial sphere….

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Apparent Paths of Stars

• During the course of the day, stars move in circles at a fixed distance from the celestial equator.

• This diurnal motion is really due to the rotation of the Earth, not the sky.

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From mid-northern latitudes:

Vega passes nearly overhead

Mizar is a circumpolar star

All stars northof the celestial

equator are abovethe horizon more

than 12 hoursSouthern stars areabove the horizonless than 12 hours

Stars on the celestial equatorare above the horizon exactly

12 hours every day

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Motion of the Sun

• The Sun also appears to rise in the east and set in the west due to the Earth’s rotation.

• However, the Sun moves relative to the stars:– The Sun moves eastward relative to the stars by

about 1° per day.– The Sun oscillates north/south (within 23°.5 of

the celestial equator) over a period of one year.

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Sun’s Motion in the SkyNorth

East West

South

celestial equator

ecliptic

Sun moves 1°/dayalong theecliptic

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Sun’s Motion in the SkyNorth

East West

South

celestial equator

ecliptic

Vernal equinox(first day of Spring)

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Sun’s Motion in the SkyNorth

East West

South

celestial equator

ecliptic

Summer solstice(first day of Summer)

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Sun’s Motion in the SkyNorth

East West

South

celestial equator

ecliptic

Autumnal equinox(first day of Autumn)

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Sun’s Motion in the SkyNorth

East West

South

celestial equator

ecliptic

Winter solstice(first day of Winter)

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The Ecliptic• The ecliptic is the

Sun’s path among the stars.

• It is simply the projection of the Earth’s orbital plane onto the celestial sphere.

Ecliptic

JulySeptember

November

January

April

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Diurnal Path of the Sun

• The Sun’s diurnal path varies during the year.

• Time above horizon depends on where it is relative to the celestial equator.

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Equinoxes

• Equinoxes occur when the Sun crosses the equator (about 21 March and 21 September).

• Days and nights have equal length, exactly 12 hours.

6 am

9 am noon

6 pm

midnight

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Solstices

• Dates known as the solstices occur when Sun is farthest from the celestial equator.

Summer solstice (about21 June)

6 am

noon

6 pm

8 pm

4 am

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Solstices

• Dates known as the solstices occur when Sun is farthest from the celestial equator.

Winter solstice (about21 December)

8 am

noon

4 pm6 am

6 pm

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The Zodiac

• The Sun’s motion on the ecliptic carries it through a group of constellations called the zodiac.June

March

Apparent eastwardmotion of Sun

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Measurement of Time

• Study of astronomy originally motivated by need for accurate calendars.

• Calendars are necessary for successful agriculture.

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Seasonal Variations

• Seasonal variations in temperature are due to two factors:– Amount of time Sun

spends above horizon– Maximum elevation of

Sun in sky

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Basic Measures of Time

Measure PhenomenonDay Rotation of the EarthMonth Revolution of the MoonYear Revolution of the Earth

Day, month, and year are of astronomical origin.

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Prehistoric Astronomy

• The week is also tied to astronomy:– Weeks are an invention of the Babylonians,

loosely based on quarter of lunar cycle.– Number of days in week equals number of

“planets” (non-stationary celestial objects)– Seven objects in the sky move relative to the

stars: Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn. English names for the days of the week are based on these.

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Names of the Days of the Week

Day (English) Teutonic God (equivalent)

“Planet” Day (French/Italian)

Sunday Sun dimanche/domenica

Monday Moon lundi/lunedi

Tuesday Tiw Mars mardi/martedi

Wednesday Woden Mercury mercredi/mercoledi

Thursday Thor Jupiter jeudi/giovedi

Friday Frigg Venus vendredi/venerdi

Saturday Saturn samedi/sabato

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Motions of the Earth and Measures of Time

• The major motions of the Earth determine how we measure time– Rotation

• days, hours, minutes, seconds

– Revolution and Precession• years

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Motions of the Earth

1 Rotation (P = 23h 56m)– Earth rotates eastward (i.e., west to east) on its

axis.• Solar Day = 24h 00m

• Sidereal Day = 23h 56m

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Rotation of the EarthSidereal Day

Sidereal day: one full rotation with respect to the stars.

23h 56m

Earth

one daylater

Solar Day

24h 00m

Solar day: one full rotation with respect to the Sun.

1°

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Rotation of the Earth

• It takes the Earth about 4 minutes to rotate the extra 1°.

• Because of difference in sidereal and solar time, stars rise 4 minutes earlier each day.

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Rotation of the Earth

• Proof of rotation: The Foucault Pendulum– A large mass

suspended by a wire oscillates in a single plane (Newton’s First Law).

– The Earth rotates underneath the Foucault Pendulum

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Major Motions of the Earth

1 Rotation (P = 23h 56m )2 Revolution (P = 1 year)

– The Earth revolves around the Sun, counterclockwise as seen from above the North Pole.

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Revolution of the Earth

• Proof of revolution: Stellar parallax• Parallax is the apparent motion of nearby

stars due to the motion of the Earth around the Sun.

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Stellar Parallax

• Apparent motion of nearby stars

SunEarth

"

Parallax angle "

Star

Earth three months later

1 AU

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Stellar Parallax

• Parallax is inversely proportional to distance: = 1/ d

• Largest observed parallax is for

Centauri–

= 0.75 arcseconds

– d = 4 ×1016 m = 270,000 AU• Because stellar parallaxes are so small, they

were not measured until 1838.

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Earth’s rotationaxis

Equator

Ecliptic

Precession

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Precession

• Sun and Moon pull Earth’s equatorial bulges towards ecliptic

• This results in westward precession of the Earth’s rotation axis

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Precession

• The line of intersection of the ecliptic and celestial equator also precesses westward.

• Precession causes the vernal equinox to move westward.

• This makes the calendar year (called the tropical year) shorter than the sidereal (or orbital) year.

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Tropical Year

SunVernal equinox 1999

Vernal equinox 2000

One tropical year

Motion of the Equinoxes

Direction ofEarth’s Motion

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Tropical Year

Sun

Tropical Year:365.242191 days

Sidereal Year:365.256363 days

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Period of Precession

• Sidereal year: 365d.256363• Tropical year: 365d.242191• Difference: 0d.0142 = 20m 24s.5

256.365 (years)yeardays0142.0 dN

years772,25N

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Time and Calendars

• Civil time: Based on the solar day (defined by successive transits of the Sun)

• Though Earth’s rotation speed remains (nearly) constant, length of solar day is variable.

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Time and Calendars

• Why is the length of the solar day variable?1 Tilt of the Earth’s axis (astronomers call this

the “obliquity of the ecliptic”)

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Obliquity of the Ecliptic

Equator

Ecliptic-20º

+20º

0h6h12h 12h18h

W

NSolstice N

Equinox

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Qbliquity of the Ecliptic

• Motion of the Sun along the equator is greater at solstice than at equinox; length of the solar day is longer at solstice.

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Time and Calendars

• Why is the length of the solar day variable?1 Tilt of the Earth’s axis (astronomers call this

the “obliquity of the ecliptic”)2 Eccentricity of the Earth’s orbit (Kepler’s

Second Law)

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Eccentricity of the Earth’s Orbit

Solar DaySidereal Day

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Eccentricity of the Earth’s Orbit

Solar Day

Sidereal Day

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Eccentricity of Earth’s Orbit

• Solar days are thus longest at perihelion since the Earth must rotate farther to catch up with the rapid rate at which the Sun is moving across the sky (i.e., the rapid rate the Earth is moving in its orbit).

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Time and Calendars

• Mean Solar Time (MST): based on the apparent motion of the Sun over a year. This is what clocks measure; it moves at a fixed rate.

• The Sun actually moves at a variable rate; time kept by the real Sun is called apparent solar time.

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Equation of Time

• Apparent Solar Time is where the Sun is in the sky (time measured by sundials).

• The difference between MST and Apparent Solar Time is the Equation of Time.

MST + Equation of Time

Apparent Solar Time

=

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Equation of Time

• Apparent Solar Time and Mean Solar Time can differ by as much as 16 minutes.

MST + Equation of Time

Apparent Solar Time

=

58True

Sun

Be h

ind

True

Sun

Ah e

ad

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Equation of Time

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Equation of Time

• We can also plot the Sun’s declination (angle from the celestial equator) versus the equation of time.

• This gives a shape called the analemma.

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The Analemma

Time (min)

Dec

linat

ion

Jan

Mar

May

JuneAug

Sep

Nov

The analemma shows theequation of time anddeclination of the Sun.

This is often found onglobes.

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The Analemma

• If you take a photograph at the same (mean solar) time each day, you can see how the Sun moves in declination and when it is ahead and behind of the mean Sun.

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Time Conventions

• Civil Time = Mean Solar Time in year 1900 A.D.– Earth’s rotation is slowing down, so Earth runs

slower than clocks.– Compensate by periodically adding leap

seconds, which let the Earth catch up to clocks.– About 29 leap seconds have been added

between 1900 and 2000.

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How’s the Math on That Figure?

• A day in the year 2000 is longer than a day in the year 1900 by 0.0016 seconds.

• The average day in the 20th Century is thus longer than a day in the year 1900 by half this, 0.0016/2 =0.0008 seconds.

• Earth falls behind clocks by this amount every day, for all 36525 days of the century.

• 0.0008 seconds/day × 36525 days = 29 sec.

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Calendars

• The difficulty with calendars is that the tropical year is not an integral number of solar days.

• We therefore use tropical years with a variable number of integral days to approximate the true tropical year.

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Early Roman Calendar

Day 0

Day 365Day 730

Early Roman calendar of 365 days per year was too short.First day of Spring occurred later each year.

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Early Roman Calendar

• 1 year = 365 days• Vernal equinox occurs later each year.• After only 128 years, vernal equinox has

moved from March 21 to April 21.

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Julian Calendar

• Add an extra day every fourth year:– Year 1: 365 days– Year 2: 365 days– Year 3: 365 days– Year 4: 366 days

• Average length of year: 365.25 days

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Julian Calendar

• But after 100 years, the Julian calendar is in error by more than 3/4 day.– After 100 years:

100 tropical 36524.2191 (75 years+25 leap years) 36525Difference 0.7809

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Gregorian Calendar

• The Julian Calendar has overcorrected by 3/4 of a day, so skip a leap year every century (1800, 1900, etc.).

400 tropical 146,096.876 (304 regular years + 146,096.00096 leap years)Difference +0.876

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Gregorian Calendar

• But skipping a leap year every 100 years leads to an error of near one day after 400 years.

• Add an extra leap year every 400 years (instead of skipping the leap year in century years). Thus, 1200, 1600, 2000, and 2400 are leap years, not skipped leap years.

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Gregorian Calender

• 365 days per year (Early Roman)• Years divisible by 4 are leap years, and

have 366 days (Julian modification)– If year is also divisible by 100, skip the leap

year and make it a regular year• If year is also divisible by 400, then don’t skip the

leap year, keep it! (Gregorian modifications)• One day error accumulates after 3225 years

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Chapter 2: Emergence of Modern Astronomy

Astronomy 291The SkyThe Celestial SphereThe Celestial SphereThe Celestial SphereNorth Circumpolar Stars in Time ExposureThe Celestial SphereSlide Number 8Slide Number 9Slide Number 10Apparent Paths�of StarsApparent Paths�of StarsSlide Number 13Motion of the SunSun’s Motion in the SkySun’s Motion in the SkySun’s Motion in the SkySun’s Motion in the SkySun’s Motion in the SkyThe EclipticDiurnal Path�of the SunEquinoxesSolsticesSolsticesThe ZodiacMeasurement of TimeSeasonal VariationsBasic Measures of TimePrehistoric AstronomyNames of the Days of the WeekMotions of the Earth and Measures of TimeMotions of the EarthRotation of the EarthRotation of the EarthRotation of the EarthMajor Motions of the EarthRevolution of the EarthStellar ParallaxSlide Number 39Stellar ParallaxPrecessionPrecessionPrecessionTropical YearTropical YearPeriod of PrecessionTime and CalendarsTime and CalendarsObliquity of the EclipticQbliquity of the EclipticTime and CalendarsEccentricity of the Earth’s OrbitEccentricity of the Earth’s OrbitEccentricity of Earth’s OrbitTime and CalendarsEquation of TimeEquation of TimeSlide Number 58Equation of TimeEquation of TimeThe AnalemmaThe AnalemmaSlide Number 63Time ConventionsHow’s the Math on That Figure?CalendarsEarly Roman CalendarEarly Roman CalendarJulian CalendarJulian CalendarGregorian CalendarGregorian CalendarGregorian CalenderChapter 2: Emergence of Modern Astronomy