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--- courses:ast201:1 [2023/10/02 00:33] – [5.1 Stellar aberration] 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 =====
@@ Line 84: / Line 94: @@
 ==== - 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
+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 $\gamma=(1-v^2/c^2)^{-1/2}$ and $v$ is the velocity of the observer. The velocity components of light in the rest and moving frame are $(u_x,u_y)$ and $(u_x',u_y')$, respectively and $c^2 = u_x^2 + u_y^2$. So
+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
 $$ \tan\phi = \frac{c\sin\theta}{\gamma(c\cos\theta+v)} = \frac{\sin\theta}{\gamma(v/c+\cos\theta)}. $$