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--- courses:ast201:1 [2023/10/01 00:07] – [4. Newton's intuition] asad
+++ courses:ast201:1 [2023/10/02 00:45] (current) – [4. Newton and Einstein] asad
@@ Line 72: / Line 72: @@
 Newton made gravity the physical reason behind planetary motions as opposed to Descartes' mysterious vortices. The full form his gravitational acceleration
-$$ g = \frac{GM}{r^2} $$
+$$ \vec{g} = \frac{GM}{r^2} \hat{r} $$
-and the gravitational potential
+where $G$ is the Newtonian gravitational constant, $M$ the mass of an object and $r$ is the distance at which the acceleration is being calculated. This $\vec{g}$ can be thought of also as a gravitational field around a massive object. The gravitational potential
 $$ U = - \frac{GMm}{r} $$
 for a small mass $m$ around a large mass $M$.
+The theory was later modified by Einstein as
+$$ G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu} $$
+where $G_{\mu\nu}$ is the [[bn:Einstein tensor]], $g_{\mu\nu}$ is the metric tensor, $T_{\mu\nu}$ is the stress-energy tensor, $\Lambda$ is the cosmological constant and
+$$ \kappa = \frac{8\pi G}{c^4} $$
+is the Einstein gravitational constant.
 ===== - Stellar aberration and parallax =====
+The heliocentric theory was finally proved by two observations: the observation of stellar aberration of $\gamma$ Draconis by James Bradley in 1727 and the observation of stellar parallax of Vega by von Struve.
+==== - Stellar aberration ====
+{{https://upload.wikimedia.org/wikipedia/commons/9/93/Simple_stellar_aberration_diagram.svg?nolink}}
+In this example of 2D frame moving only in the $x$-direction, the vertical component of the apparent speed of light ($u$) will not change, but the horizontal component will be modified due to the velocity of the Earth.
+$$ \tan\phi = \frac{u_y'}{u_x'} = \frac{u_y}{\gamma(u_x+v)} $$
+where $\gamma=(1-v^2/c^2)^{-1/2}$, $v$ is the velocity of the observer and $c$ is the magnitude of the velocity of light. The velocity components of light in the rest and moving frames are $(u_x,u_y)$ and $(u_x',u_y')$, respectively and $c^2 = u_x^2 + u_y^2$. So
+$$ \tan\phi = \frac{c\sin\theta}{\gamma(c\cos\theta+v)} = \frac{\sin\theta}{\gamma(v/c+\cos\theta)}. $$
+If $v \ll c$, $\gamma=1$. and if $\theta=90^\circ$ then
+$$ \tan(\theta-\phi) = \cot\phi = \frac{\gamma(v/c+\cos\theta)}{\sin\theta} = \frac{v}{c} $$
+and, finally, if $\theta-\phi$ is very small then **aberration** $\alpha = \theta-\phi \approx v/c$.
+Using google colab, calculate the factor $v/c$ considering $v$ to be the speed of earth around the sun which is $30$ km/s. You will see that $v/c\approx 20$ arcsec. So the maximum stellar aberration is 20 arcsec, but the actual measurement of aberration will vary from $-20$ to $+20$ arcsec for starlight coming at a right angle with the plane of the solar system.
+{{https://upload.wikimedia.org/wikipedia/commons/9/95/Aberrationseasons.svg?nolink}}
+The annual variation of aberration can be seen in the figure above. Around the time of March equinox, the Earth is travelling toward left in this figure, so starlight is also bent toward the left, but as the Earth approaches the June solstice, the aberration vanishes and then again increases as the Earth travels toward the September equinox. The aberration again goes to zero around the time of December solstice.
+[[https://colab.research.google.com/drive/1eveRRHSR3Peqwyl9lclL-ehMmOZ7l7h2?usp=sharing|{{:courses:ast201:stellar-aberration.png?nolink|}}]]
+This variation has been modelled above using a sine function with an amplitude of $20$ arcsec and a period of $2\pi$. The amplitude is not exactly $20$ and the sine curve is not vertically symmetric because the variation of the orbital velocity of the Earth in its elliptical orbit has been taken into account here. Click on the image to see the python code and the corresponding equations.
+Compare this model with the actual observations carried out by James Bradley in 1727 using the star $\gamma$ Draconis.
+{{https://upload.wikimedia.org/wikipedia/commons/6/62/Bradley%27s_observations_of_%CE%B3_Draconis_and_35_Camelopardalis_as_reduced_by_Busch.jpg?nolink&500}}
+You can see minimum aberration during the solstices and maximum aberration during the equinoxes as predicted. Think why!
+==== - Stellar parallax ====
+{{:courses:ast201:parallactic-ellipse.png?nolink&700|}}
+{{ :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.
+parsec = 206265 au = $3.085678\times 10^{16}$ m = 3.261633 ly.
+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.
+<gallery full>
+:courses:ast201:vonstruve.png Bessel 1838
+:courses:ast201:bessel.png von Struve 1837
+:courses:ast201:parallax-henderson.png Henderson 1837
+</gallery>
+HIPPARCOS has parallax uncertainties of 0.97 mas for around 118k stars brighter than $m_V=8.0$.
 ===== Further reading =====
+  - Bradley 1727, [[https://www.jstor.org/stable/103725|A Letter from the Reverend Mr. James Bradley]].
   - Reid & Menten, 2020, [[https://arxiv.org/pdf/2009.11913.pdf|The First Stellar Parallaxes Revisited]].
   - Warner, 2004, [[https://articles.adsabs.harvard.edu/pdf/2005IAUCo.196..198W|Thomas Henderson and α Centauri]].