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courses:ast201:4 [2023/11/05 00:57] – [3. Time] asadcourses:ast201:4 [2023/11/08 21:10] (current) – [2.1 Astronomical Unit (AU)] asad
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 where $\Delta t$ is the time it takes for a radio signal to come back to earth after getting reflected from Venus. 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|}}
- +
-{{ :courses:ast201:parallax-angle.png?nolink&150|}} +
- +
-The tangent of the parallactic angle +
- +
-$$ \tan p = \frac{a}{r} $$ +
- +
-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 +
- +
-$$ p = \frac{a}{r}. $$ +
- +
-If $a$ is in au and $p$ in arcsec, then $r$ is in parsec. +
- +
-1 parsec = 206265 au = $3.085678\times 10^{16}$ m = 3.261633 ly. +
- +
-{{:courses:ast201:plate-measuring-machine.png?nolink|}} +
- +
-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. +
- +
-HIPPARCOS has parallax uncertainties of 0.97 mas for around 118k stars brighter than $m_V=8.0$. +
- +
-Explore Gaia: https://www.esa.int/Science_Exploration/Space_Science/Gaia +
 ===== - Time ===== ===== - 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). 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).
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 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.  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 ==== ==== - 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
 +
 +$$ \beta = \frac{v_R}{c} = \frac{(z+1)^2-1}{(z+1)^2+1} $$
 +
  
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