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--- courses:ast201:4 [2023/11/01 00:32] – [1.3 Equatorial system] asad
+++ courses:ast201:4 [2023/11/08 21:10] (current) – [2.1 Astronomical Unit (AU)] asad
@@ Line 71: / Line 71: @@
 The relation between the equatorial and horizon systems is shown above. It assumes an observer located at a latitude of $60$ degrees north on Earth. Both spheres show the horizon, equator and observer's meridian, the north celestial pole P and the zenith T. Panel (a) shows the diurnal path of a circumpolar star that never sets and the path of a star that never rises for the observer at the center of the horizon.
-Altitude of NCP = Geodetic latitude of observer.
+The following equations can all be derived geometrically from the above diagram.
+  - Altitude of NCP = Geodetic latitude of observer
+  - Object HA = Meridian RA - Object RA
+  - Sidereal day = Time between upper meridian transits of the March equinox
+  - Sidereal time = Object RA on the upper meridian
+  - Object HA = Sidereal time now - sidereal time when the object culminates
-Object HA = Meridian RA - Object RA
+=== - Precession and nutation ===
+The equatorial system is not inertial, it is accelerated due to the long-term general **precession** and short-term oscillatory **nutation**.
-Sidereal day = Time between upper meridian transits of the March equinox
+{{:courses:ast201:precession.png?nolink&500|}}
-Sidereal time = Object RA on the upper meridian
+As shown above, the north ecliptic pole remains fixed with respect to the stars, but the north celestial pole rotates in a small circle around the ecliptic pole. The precessional circle has a radius of around 23 deg, the same as the obliquity of Earth's axis. Obliquity is the angle between the celestial and ecliptic axes.
-Object HA = Sidereal time now - sidereal time when the object culminates
+NCP completes one orbit in 26,000 years moving at an almost constant speed. So the celestial equator moves at an almost constant speed westward toward ecliptic at a rate of $5029.097$ arcsec (1.4 deg) per century, i. e. almost $50''$ per year.
-=== - Measuring RA-DEC ===
+Hipparchus already knew about precssion. Polaris was not always so close to the NCP. Around 4500 years ago, Thuban was the pole star. After 12,000 years, Vega will be the pole star and Polaris will have a declination of $45$ degrees.
-Transit telescope / Meridian circle
-{{:courses:ast201:troughton-10-foot-transit-instrument-royal-observatory-greenwich.jpg?nolink|}}
+{{https://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Praezession.svg/640px-Praezession.svg.png?nolink&300}}
-{{:courses:ast201:atm_refr.png?nolink&400|}}
+Unlike the **lunisolar** precession, **planetary** precession causes the obliquity, i. e. distance between the ecliptic and celestial axes, to change. It results in an oscillation on top of the precessional circle as shown above. The obliquity oscillates between $22^\circ$ and $24^\circ$ once in 41,000 years. Currently the obliquity is decreasing by $47''$ per century, i. e. only $0.47''$ per year.
-=== - Precession, nutation and ICRS ===
+The actual reasons behind these motions lies in the physics of a spinning top. When you release a top, you exert a torque on the axis of the top and, hence, the axis rotates and oscillates. The details can be found in this [[uv:precession|Universe article]].
-{{:courses:ast201:precession.png?nolink|}}
-Celestial equator moves westward toward ecliptic at a rate of $5029.097$ arcsec (1.4 deg) per century.
+=== - Barycentric coordinates ===
+The equatorial system is highly **non-inertial**, has a lot of different accelerations. Four effects, specially, change the standard J2000 equatorial RA-DEC of an astronomical object: (1) precession, (2) nutation, (3) heliocentric stellar parallax, and (4) aberration of starlight. Parallax is caused by the changing position of Earth as it goes around the sun. Aberration is caused by the changing velocity of Earth. We have already talked about these 4 effects.
-Around 4500 years ago, Thuban was the pole star.
+In order create a more **inertial** frame of reference, IAU recommended [[uv:icrs|ICRS (International Celestial Reference System)]] in 1991. The origin of ICRS is the barycenter of the solar system. The equator still defines the fundamental plane, but the poles and the axis are determined by distant [[uv:quasar|quasars]].
-After 12,000 years, Vega will be the pole star.
+==== - Measuring RA-DEC ====
+RA and DEC are traditionally measured using **transit telescopes** (or **meridian circles**). A transit telescope always points toward an observer's meridian and rotates east-west. To determine declinations, first locate the celestial pole and take a circumpolar star. Then read the graduated circle twice: once when the star crosses the upper meridian, once when it crosses the lower meridian. The average of the two gives the location of NCP or SCP on your meridian circle. After this calibration, any other declination can be determined.
-Unlike the **lunisolar** precession, **planetary** precession causes the obliquity to decrease by $47$ arcsec per century.
+{{:courses:ast201:atm_refr.png?nolink&400|}}
-{{https://upload.wikimedia.org/wikipedia/commons/thumb/b/bb/Praezession.svg/640px-Praezession.svg.png?nolink&300}}
-International Celestial Reference System where the barycenter is the origin.
-Equator defines the fundamental plane, but the poles and axis are determined by distant quasars creating an **inertial reference frame**.
-==== - Other Systems ====
-North ecliptic pole, NEP: RA 18 hours, Dec = $90^\circ-\epsilon$ where $\epsilon$ is the **obliquity of the ecliptic**.
-North Galactic pole, NGP: $\alpha = 12:49:00$, $\delta=+27.4^\circ$ (equator and equinox of 1950)
 ===== - Distance measures =====
@@ Line 128: / Line 124: @@
 where $\Delta t$ is the time it takes for a radio signal to come back to earth after getting reflected from Venus.
-==== - Stellar parallax ====
+==== - Distance ladder ====
-{{:courses:ast201:parallactic-ellipse.png?nolink&700|}}
+{{:uv:distance-ladder.webp?nolink&600|}}
+===== - Time =====
+TAI: International Atomic Time defines 1 SI second as 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperﬁne levels of the ground state of the cesium-133 atom. Practical atomic clocks have a precision of about $2/10^{13}$. Two clocks located at two frames will have differing speed based on their relative velocities (special relativity) and their accelerations or local gravitational fields (general relativity).
-{{ :courses:ast201:parallax-angle.png?nolink&150|}}
+A **astronomical day** = 86,400 SI seconds. But historically we have used the sun, not atoms, for measuring time.
-The tangent of the parallactic angle
+{{:courses:ast201:solar-time.png?nolink&500|}}
-$$ \tan p = \frac{a}{r} $$
+This figure shows an Earth-sized clock from god's-eye-view from above the solar system. Figure (b) clearly shows that Local apparent solar time = HA of the Sun + 12 hrs. The apparent solar day varies from the astronomical day throughout a year because of Earth's obliquity of axis and ellipticity of orbit. So we introduce the
-where $r$ is the distance to the object and $a$ is in au. For $p\ll 1$ we can approximate $\tan p = \sin p = p$ and
+Local mean solar time = HA of the mean Sun + 12 hrs.
-$$ p = \frac{a}{r}. $$
+{{:courses:ast201:equation-of-time.png?nolink&400|}}
-If $a$ is in au and $p$ in arcsec, then $r$ is in parsec.
+Equation of time = local apparent solar time - local mean solar time. It takes on values up to $\pm 15$ minutes in a year.
-parsec = 206265 au = $3.085678\times 10^{16}$ m = 3.261633 ly.
+Universal time, UT = mean solar time at Greenwich.
-{{:courses:ast201:plate-measuring-machine.png?nolink|}}
+But even UT is not precise enough because Earth's precession is not that accurately known and also because its spin is not uniform. Variations in spin are caused by the tidal effects of the moon (monthly) and the sun (seasonal). Even the interaction between the core and the mantle of earth changes the spin on timescales of decades. Finally, tidal friction is slowing the spin rate and, hence, the mean solar day is getting longer by $1.5$ milliseconds per century, or $15$ $\mu$s per year.
-For nearby star, measurements of $p$ have uncertainties of 50 mas, but can be reduced to 5 mas. Only works for around 1000 stars closer than 20 pc.
+IERS maintains UTC: Coordinated Universal Time (UTC). UTC is close to UT, but it uses SI seconds instead of mean solar time. To keep pace with UT, UTC introduces **leap seconds** occasionally. The acceleration Earth's spin has a random component, so it is not possible to predict the need for leap seconds ahead of time. Between 1972 and 1998, a total of 22 leap seconds were added.
-HIPPARCOS has parallax uncertainties of 0.97 mas for around 118k stars brighter than $m_V=8.0$.
+Our legal time follows UTC but finds
-Explore Gaia: https://www.esa.int/Science_Exploration/Space_Science/Gaia
+Zone time = UTC + longitude correction for the zone.
-===== - Time =====
+The zones are usually $15^\circ$ wide in longitude; as the earth rotates that much in 1 hour. Computer networks synchronize clocks using standard protocols like ITS and ACTS.
-TAI: International Atomic Time:
-s = 9,192,631,770 periods of of the radiation corresponding to the transition between the two hyperﬁne levels of the ground state of the cesium-133 atom.
+Sidereal time = HA of the mean March equinox of date
-{{:courses:ast201:solar-time.png?nolink|}}
+which follows the UT. Astronomers count days continuously from a reference day as
-Local apparent solar time = HA of the Sun + 12 hrs.
+Julian date (JD) = number of elapsed UT days since 4713 BC January 1.5 (12 hrs UT).
-Local mean solar time = HA of the mean Sun + 12 hrs.
+In many cases, JD is given instead of UT date.
-Equation of time = local apparent solar time 2 local mean solar time.
-{{:courses:ast201:equation-of-time.png?nolink|}}
-Universal tine, UT = mean solar time at Greenwich.
+J2000.0 =  Julian epoch 2000.0 = 2000 Jan 1.5 UT = JD 2451545.0.
-UTC uses SI seconds.
 ===== - Motion =====
@@ Line 188: / Line 179: @@
 Measure the ICRS coordinate of a star in 10-year intervals. If the position changes, we know that the star actually moved because ICRS is not affected by parallax or precession.
 ==== - Radial velocity ====
+Christian Doppler gave a lecture on 25 May 1842 at the Royal Bohemian Scientific Society in Prague, Czechia.
+For small **redshifts** $z\ll 1$
+$$ \frac{\lambda-\lambda_0}{\lambda_0} = \frac{\Delta \lambda}{\lambda_0} = \frac{v_R}{c} = z $$
+{{:courses:ast201:spectrograph.png?nolink|}}
+Resolving power of a spectrograph
+$$ R = \frac{\lambda}{\delta\lambda} $$
+For large redshifts $z \gg 1$
+$$ z = \frac{\sqrt{1-\beta^2}}{1-\beta}-1 $$
+where
+$$ \beta = \frac{v_R}{c} = \frac{(z+1)^2-1}{(z+1)^2+1} $$