Astronomy 217C e l e s t i a l M o t i o n s

o r F i n d i n g t h i n g s i n

t h e S k y

Motions of the HeavensThe largest part of the Celestial Motions are in fact reflections of the Earth’s own motion.Earth’s rotation produces the diurnal motion of the fixed star’s (360°/day).Earth’s revolution around the Sun causes the apparent movement of the Sun (1°/day).Precession of the Earth’s pole produces the precession of the equinoxes (1°/72 years) .

There are also small wobbles in the precession, called nutation, caused by the precession of the Moon’s orbit and influence of other planets.

EpochTo account for slow variations like the precession of the Pole and nutation, every astronomical coordinate has a timestamp. An Epoch is a moment in time used as a reference point for some time-varying astronomical quantity.The current standard Epoch is J2000.0 which marks January 1, 2000, 11:58:55.816 UT or 12:00:00 Terrestrial Time (TT).Terrestrial Time does not include leap seconds.Intervals from J2000.0 are counted in Julian days.Values for constants are define from J2000, e.g., Greenwich Sidereal Time of J2000 (GSTJ2000) = 280.46061837°.Star charts from older epoch’s like J1950.0 are still occasionally found.

Motions in the HeavensTrue motions are limited because very large distances turn large velocities into small angles.The orbital motions of the “wanderers”; planets, moons and other solar system bodies, are comparable (1-1000°/yr) to the apparent motion caused by the Earth.

The proper motions of the distant stars, true motions relative to the Solar System, are much smaller.

Stars do move relative to the Solar System, but only in unusual circumstances is this discernible in human time. But the effects can be spectacular.Mira, a red Giant in the constellation Cetus, 350 ly from Earth, is moving across the sky at 130 km/s (291,000 miles/hr). Mira is losing gas, leaving a tail 13 ly long behind it over 30,000 years.

Even the fastest proper motion, like Barnard’s star (1°/350 yrs), are small compared to the Earth’s motions.

Proper Motion

13 LY2°

How can we find a star in the sky?

We start with the local time and our location:UT (Nielsen Physics):

Latitude 35.956602°, Longitude −83.925548°

The process has two steps1) Find out what part of the sky is directly above you.

or Find the hour angle of the vernal equinox. or Compute your Local Sidereal Time

2) Convert the Right Ascension and Declination of the desired object into local Altitude and Azimuth.

Computing the Local Sidereal Time

1)Pick a local time.

2)Compute the Universal Time (UT).

3)Compute the number of solar days from the J2000 epoch, Noon, January 1 2000.

9 pm EDT, September 5th 2018

EDT = UT-4 hours ⇒ 1 am UT, September 6th 2018

18 years × 365 days/yr + 4 leap days + 249 days − 11 hr = 6822.541667 days

So, since the J2000 Epoch, the Earth revolved, relative to the Sun, a little more than 6812 and a half times.

Computing the Local Sidereal Time

4)Convert the number of solar days to sidereal days.

5)Greenwich Sidereal Time (GST) is the sidereal time at the Prime Meridian.GST° = 360° × Remainder of Sidereal Days + GSTJ2000.

6)Local Sidereal Time (LST) is GST+local longitude.

GST° = 360° × .221167 + 280.46061837° = 360.080610°

6822.541667 × (366.2422 / 365.2422) = 6841.221167 sidereal days

GST = .080731° ÷ 15°/hour = 00:00:19

LST = 360.080610° − 83.925548° = 276.155062° = 18:24:37 Objects with this Right Ascension are

at their Culmination at this time and place.

So What can I see Tonight?We are looking for objects with α ~ 18h and δ > 36-90 =

AntaresAntares, also know as α Scorpii, is a red supergiant star, perhaps 12 times the mass of the Sun, as a distance of ~550 ly.

Right ascension 16:29:24Declination -26° 25′ 55″

Converting CoordinatesTo really know where an object is in the sky, we need to convert from Right Ascension (α), Declination (δ) and Latitude (ℓ) to Altitude (a) and Azimuth (A).Calculate the Hour Angle, parallel to the Right Ascension, but centered on Zenith.

H = LST − αApply Spherical Trigonometrytjo ౚ > dpt ే dpt dpt Ψ , tjo tjo Ψubo ీ > tjo ే dpt dpt ే dpt tjo Ψ ѝ tjo dpt Ψ

Finding AntaresTo find Antares, we need to convert from Right Ascension (α = 247.351915), Declination (δ = -26° 25′ 55″) and Latitude (ℓ = 35.956602°) at LST (= 276.155062°) to Altitude (a) and Azimuth (A).

H=LST-α H = 276.155062°− 247.351915° = 28.803147°

sin a = 0.876280×0.895464×0.809462+-0.445135×0.587172

= 0.373796 ⇒ a = 21.95°

tan A = 0.525462 ⇒ A= 207.72° or 27.72°

sin a = cos H cos δ cos ℓ + sin δ sin ℓ

tan A = sin H cos δcos H cos δ sin ℓ − sin δ cos ℓ

tan A = 0.876280×0.895464×0.587172--0.445135×0.80              0.481802×0.895464

Finding MarsUnlike Antares, Mars moves significantly over the scale of days, so its α and δ are not constants. Instead we must consult a ephemeris which lists/calculates α and δ as a function of time.

year mo dy hr:mi Planet RA PlanetDec

2018-Sep-05 21:00 302.28823 -25.63932 2018-Sep-05 22:00 302.29323 -25.63582 2018-Sep-05 23:00 302.29826 -25.63232 2018-Sep-06 00:00 302.30331 -25.62882 2018-Sep-06 01:00 302.30838 -25.62531 2018-Sep-06 02:00 302.31348 -25.62179 2018-Sep-06 03:00 302.31860 -25.61827 2018-Sep-06 04:00 302.32374 -25.61474 2018-Sep-06 05:00 302.32891 -25.61121

Marking MarsFor Mars at 9 pm EDT (1am UTC), α = 20:09:14, δ = −25.62531°. Latitude (ℓ= 35.956602°) and LST (= 276.155062°) match our Antares calculation.

H = 276.155062°− 302.30838° = −26.153318°

= 0.401179 ⇒ a = 23.65°

tan A = −0.481550 ⇒ A= 154.29°or −25.71°

H = LST − α

sin a = 0.897618×0.901642×0.809462+-0.432484×0.587172sin a = cos H cos δ cos ℓ + sin δ sin ℓ

tan A = sin H cos δcos H cos δ sin ℓ − sin δ cos ℓ

tan A = 0.897618×0.901642×0.587172--0.432484×0.80              -0.440775×0.901642

So What Stars Can I See?You can figure this out based on 3 factors.1) Declination

For δ >90°− l, the star is circumpolar, for δ < l − 90°, it never rises.

2) The Season (or Hour Angle)Stars with 90°− l > δ > l − 90° are only visible part of the night. Their culmination occurs when H = 0 ⇒ LST = α.

3) The SunIf the Sun is up, it’s hard to see the stars.On vernal equinox, αSun = 0 and increases by 2 hours/month.

Estimating LSTWhile detailed calculations of LST are vital for telescopic observations, an estimate is useful for constellation finding.1) Calculate the solar right ascension, αSun.

2) Stars with αmid = αSun + 12 hours are overhead at midnight.

3) Correct LST for Current Time (if Daylight Savings Time, subtract 1 hour)

αSun = 0 + (Date - Vernal Equinox) × 2 hr/monthαSun = 5.0 months × 2 hr/month = 10.0 hours

αmid = 22.0 hours = LSTmid

LST = LSTmid − (Midnight − Time) −1 hour if Daylight TimeLST = 22.0 hours − 3 hours − 1 hour = 18.0 hours.

9 PM

Tonight

Antares

Mars

Saturn

10 PM11 PM

Next Time

What did the Greeks know?Read Chapter 2Turn in Homework #1