Transcript ASTR2100

2. The Celestial Sphere
Goals:
1. Gain familiarity with the basic coordinate systems used in astronomy.
2. Tackle simple problems in practical
astronomy involving timekeeping and
star positions.
3. Examine the timekeeping systems in
current use.
Astronomical Co-ordinate Systems:
All co-ordinate systems constructed on spheres
are defined by a fundamental great circle (FGC)
and a reference point (RP) on the FGC.
All co-ordinates are angles measured:
(i) between great circles perpendicular to the
FGC, or
(ii) between small circles parallel to the FGC.
The FGC has two poles, and the RP is defined in
a variety of ways, which accounts for slight
differences from one system to another.
Terrestrial Co-ordinates.
FGC = Earth’s equator: poles
the North Pole and South Pole.
RP = crossing point of equator
by the Greenwich meridian.
Co-ordinates:
Longitude = angle measured east and west from
the Greenwich meridian. Longitude meridians
are great circles.
Latitude = measured north and south (not plus or
minus) from the Equator. Latitude parallels are
small circles.
Examples: Halifax. 63º 36'.0 W, 44º 36'.0 N
Vancouver. 123º 04'.2 W, 49º 09'.0 N
Horizon System.
FGC = horizon, with poles the zenith and nadir.
RP = north point.
Co-ordinates:
azimuth = angle measured
through east from 0º to 360.
Azimuth circles are great
circles.
altitude = measured from
horizon towards zenith (positive)
or nadir (negative) from +90º to
–90º. Alternate: zenith distance,
z = 90º – altitude.
Meridian = NS line running through zenith.
Prime vertical = EW line running through zenith.
Use. Airport runways are designated by azimuth
 10°, i.e. runway 32 aligns along azimuth 320°
magnetic. Air mass, for correction of photometry,
is calculated from zenith distance z.
Example Question. In what directions does
runway 05/23 run?
Solution: Given the name of the runway, 05/23, it
must run along azimuth directions 50°
(northeast) and 230° (southwest), which are 180°
apart.
Equatorial System.
FGC = celestial equator (CE, projection on the
sky of Earth’s equator), with poles the north and
south celestial poles, NCP and SCP.
RP = intersection point of meridian with CE
(observer-oriented), or vernal equinox γ (skyoriented).
Co-ordinates:
declination = angle measured
north or south of CE from 0º
to +90 and 90° (δ).
hour angle = angle measured
west of meridian (HA), or
right ascension = angle measured
eastward from vernal equinox (RA).
From relations for angles associated with parallel
lines, 90° – θ = 90° – , i.e. θ = .
Star
Trails
1h exposure
The orbit of Earth about the Sun and the 23½°
obliquity of the ecliptic (its angle relative to the
normal to Earth’s orbital plane) give rise to
Earth’s seasons.
Solar insolation as a function of season.
A typical sky scene showing seasonal variations in
the Sun’s diurnal motion.
RA() and δ() during the year are defined by
the apparent motion of the Sun in the sky along
the ecliptic = Sun’s apparent path, and can be
calculated directly or from tables.
Solar day (24h) is rotation of Earth relative to Sun,
sidereal day (23h 56m) is rotation of Earth relative to stars.
HA and RA are measured in temporal units and
are equivalent to angles. On the celestial equator:
1h = 15°, 1m = 15', and 1s = 15″, with the equalities
changing by cos δ with increasing declination.
Because of their link to timekeeping, HA and RA
are tied directly to sidereal (star) time and
apparent solar time.
Sidereal time (SidT)= HA(γ)
Apparent solar time = HA() + 12h
Now, HA(γ) = HA(*) + RA(*) = HA() + RA()
Thus, SidT = HA() + RA()
= Apparent solar time  12h + RA()
By geometry and algebra, Sidereal Time = HA(γ)
But HA(γ) = HA(*) + RA(*)
So Sidereal Time = HA(*) + RA(*)
Useful values:
Vernal Equinox, March 20:
RA() = 0h, δ() = 0°
Summer Solstice, June 21:
RA() = 6h, δ() = +23½°
Autumnal Equinox, September 23:
RA() = 12h, δ() = 0°
Winter Solstice, December 22:
RA() = 18h, δ() = 23½°
The actual dates of the equinoxes and solstices
slowly change with time. They were March 25,
June 25, September 25, and December 25 when
Julius Caesar modified the original Roman
calendar system (Julian Calendar) in 46 BC.
annual insolation
An example.
1. Towards what directions on the co-ordinate
axes must an equatorial telescope be set in order
to point it towards Betelgeuse, RA = 5h 56m, δ =
+7° 25' at 5h sidereal time?
Solution:
By definition, SidT = HA(γ) = HA(*) + RA(*)
 5h = HA(*) + 5h 56m
And so HA(*) = 5h – 5h 56m = –0h 56m
The hour angle setting of the telescope should be
set to –0h 56m, i.e. 56m east of the meridian.
And the declination setting of the telescope should
be set to +7° 25'.
Another example.
2. Show that apparent solar time (AST) and
sidereal time (SidT) are identical on the date of
the autumnal equinox.
Solution:
By definition, SidT = HA(γ) = HA(*) + RA(*)
The Sun is also a star, so SidT = HA() + RA()
But AST = HA() + 12h
And RA() = 12h at the Autumnal Equinox.
So at the Autumnal Equinox,
SidT = HA() + RA() = HA() + 12h = AST
i.e. Sidereal Time and Apparent Solar Time are
identical on the date of the Autumnal Equinox.
3. When is the best time of year to observe the
stars of Orion, RA = 5½h?
Solution:
The optimum time for observing any object is
when it lies on the observer’s meridian at local
midnight, which corresponds to 0h local apparent
solar time (LAST).
i.e. LAST = HA() + 12h = 0h (midnight)
So HA() = 0h  12h = 24h  12h = 12h
Orion is then on the meridian, so local sidereal
time = RA(*) = 5½h = HA() + RA()
 RA() = 5½hHA() = 5½h12h = 29½h12h = 17½h
The Sun is at RA = 17½h approximately one week
prior to the winter solstice, i.e. around Dec. 15.
Ordering the Planets Outwards
Object
Motion relative to the Stars
Stars
Saturn
Jupiter
Mars
Sun
Venus
Mercury
Moon
infinite
29.30 years
11.86 years
1.88 years
365¼ days
225 days
88 days
27½ days
Saturn governs the 1st hour of the 1st day, Jupiter
the 2nd hour, Mars the 3rd hour, etc., and Mars the
24th hour. The Sun then governs the 1st hour of
the 2nd day, the Moon the 1st hour of the 3rd day,
Mars the 1st hour of the 4th day, Mercury the 1st
hour of the 5th day, Jupiter the 1st hour of the 6th
day, and Venus the 1st hour of the 7th day. The
days of the week are therefore:
Saturn-day
Sun-day
Moon-day
Mars-day
Mercury-day
Jupiter-day
Venus-day
North American time zones
World time zones ─ note the peculiarities
Calendars.
Apparent solar time is defined by the passage of
the Sun across the sky, but civil time is more
closely related to the motion of the mean Sun, a
fictitious object, across the sky.
Mean solar time = HA(mean Sun) + 12h
The mean Sun differs from the true Sun in the
following way. The true Sun travels along the
ecliptic at a rate that varies according to the
distance of Earth from the Sun. The mean Sun
travels along the celestial equator at a uniform
rate.
Additional complications arise from the use of
time zones and daylight saving time.
The analemma represents the equation of time =
Apparent Solar Time – Mean Solar Time.
Length (days)
Julian
Gregorian
Month
Calendar
Calendar
January
31
31
February
29
28
March
31
31
April
30
30
May
31
31
June
30
30
Quintilus (July)
31
31
Sextilus (August) 30
31
September
31
30
October
30
31
November
31
30
December
30
31
Important Calendar Dates
Calendar Event Julian Modern
Vernal Equinox March 25
Summer Solstice June 25
Autumnal Equinox Sept. 25
Winter Solstice
Dec. 25
March 20
June 21
Sept. 23
Dec. 22
The year length varies according to the calendar
system, which has changed from lunar calendars,
through luni-solar calendars, to solar calendars,
such as the Julian Calendar, Gregorian Calendar,
and current modified Gregorian Calendar.
Variable star studies normally cite observations
according to the Julian Date, JD, measured as the
number of sequential days from noon, UT, on
January 1, 4713 BC (named by Joseph Scaliger
after his father Julius Scaliger), or, better yet,
HJD = Heliocentric Julian Date (corrected to the
barycentre of the solar system). Another term,
modified Julian Date, MJD = JD  2400000.5, is
occasionally used.
Ecliptic System.
FGC = ecliptic, with poles the north and south
ecliptic poles, NEP and SEP.
RP = vernal equinox γ.
Co-ordinates:
celestial (or ecliptic)
longitude, λ = angle
measured eastward
from γ from 0º to 360.
celestial (or ecliptic)
latitude, β = angle
measured from ecliptic.
The system is useful for studies of solar system
objects.
Galactic System.
FGC = Galactic equator (GE), defined by the
Milky Way, with poles the north and south
Galactic poles, NGP and SGP.
RP = direction to the Galactic centre (GC),
defined by Sgr A*.
Co-ordinates:
Galactic longitude, l =
angle measured
Eastward from GC
from 0º to 360.
Galactic latitude, b =
angle measured north or
south of GE from 0º to
+90 and 90°.
Precession of the Equinoxes.
Earth’s axis of rotation precesses relative to the
perpendicular to its orbit because of gravitational
influences by the Sun and Moon, but not in the
fashion implied by the Wikipedia figure below. The
sense of precession is
actually opposite the sense
of Earth’s rotation. The
period is ~25,725 years.
A top’s
precession.
Precession and
the location of
the NCP. Note
that the NCP
was near the
bright star
Thuban near
2700 BC, when
the pyramids
were built, and
was once near
Vega, a name
that means
“fallen.”
Astronomical Terminology
Zenith. The point in the sky directly overhead.
Nadir. The point directly beneath one’s feet.
Azimuth. A measurement of angle increasing from north
through east.
Altitude (astronomical). A measurement of angular
distance from the true horizon upwards.
Ecliptic. The great circle in the sky along which the Sun
appears to move because of Earth’s orbit about it.
Right Ascension. A celestial co-ordinate like longitude on
Earth, increasing eastwards.
Declination. A celestial co-ordinate like latitude on
Earth, measured from the celestial equator.
Celestial Equator. The projection on the celestial sphere
of Earth’s equator.
Celestial Sphere. The imaginary sphere centred on the
observer upon which the stars appear to be
projected.
Astronomical Terminology (continued)
Diurnal. = daily (once a day).
Insolation. The amount of sunlight falling on Earth’s
surface.
Constellation. A group of conspicuous stars designated
by ancient star gazers.
Zodiacal Constellation. A constellation lying in the band
of sky around the ecliptic, where the Moon and
planets are always found.
Solstice. Time of greatest or smallest declination for the
Sun.
Equinox. Time when the Sun crosses the celestial
equator. (Vernal = spring)
Stellar Aberration. The apparent displacement in a
star’s location in the sky of at most 20½ seconds of
arc resulting from Earth’s orbital motion about
the Sun at a speed of 30 km/s.