Fix math typos and add a figure (#25)

* Fix a few typos preventing equations display. Add a figure for JWST trajectory

* Ugh dependencies

interplanetary-maneuvers/interplanetary-transfer-phasing.md

@@ -277,10 +277,10 @@ The wait times are shown in {numref}`tab:heliocentric-hohmann-wait-times`. The t
 
 | N | $t_{\text{wait}}$ (years) |
 |---|-----------------------|
-| 0 | {glue:text}`heliocentric-hohmann-t_wait_0` |
-| 1 | {glue:text}`heliocentric-hohmann-t_wait_1` |
-| 2 | {glue:text}`heliocentric-hohmann-t_wait_2` |
-| 3 | {glue:text}`heliocentric-hohmann-t_wait_3` |
+| 0 | {glue:text}`heliocentric-hohmann-t_wait_0:.4f` |
+| 1 | {glue:text}`heliocentric-hohmann-t_wait_1:.4f` |
+| 2 | {glue:text}`heliocentric-hohmann-t_wait_2:.4f` |
+| 3 | {glue:text}`heliocentric-hohmann-t_wait_3:.4f` |
 :::
 
 Clearly, the total mission time is dominated by the transfer time. This is because the synodic period of Venus relative to Neptune is quite small, at only {glue:text}`heliocentric-hohmann-T_syn:.2f` Earth years. Since Venus whips around the Sun, relative to Neptune, the same phase angle occurs relatively often.

the-n-body-problem/circular-restricted-three-body-problem.md

@@ -255,9 +255,9 @@ The characteristic length is the circular orbit radius, $r_{12}$. Using this, we
 :::{math}
 :label: eq:non-dim-r-vectors-cr3bp
 \begin{aligned}
- \vector{\rho} &= \frac{\vector{r}}{r_{12}} = x^_\uvec{\imath} + y^_\uvec{\jmath} + z^_\uvec{k} \\
- \vector{\sigma} &= \frac{\vector{r}_1}{r_{12}} = \left(x^_ + \pi_2\right)\uvec{\imath} + y^_\uvec{\jmath} + z^_\uvec{k} \\
- \vector{\psi} &= \frac{\vector{r}_2}{r_{12}} = \left(x^* - 1 + \pi_2\right)\uvec{\imath} + y^_\uvec{\jmath} + z^_\uvec{k}
+ \vector{\rho} &= \frac{\vector{r}}{r_{12}} = x^*\uvec{\imath} + y^*\uvec{\jmath} + z^*\uvec{k} \\
+ \vector{\sigma} &= \frac{\vector{r}_1}{r_{12}} = \left(x^* + \pi_2\right)\uvec{\imath} + y^*\uvec{\jmath} + z^*\uvec{k} \\
+ \vector{\psi} &= \frac{\vector{r}_2}{r_{12}} = \left(x^* - 1 + \pi_2\right)\uvec{\imath} + y^*\uvec{\jmath} + z^*uvec{k}
 \end{aligned}
 :::
 
@@ -281,7 +281,7 @@ where $\tau = t/t_C$. Making the terms on the right hand side of Eq. {eq}`eq:fiv
 
 :::{math}
 :label: eq:non-dim-five-term-accel-cr3bp
-\ddot{\vector{\rho}} = \left(\ddot{x}^* - 2\dot{y}^* - x^_\right)\uvec{\imath} + \left(\ddot{y}^_ + 2\dot{x}^* - y^_\right)\uvec{\jmath} + \ddot{z}^_\uvec{k}
+\ddot{\vector{\rho}} = \left(\ddot{x}^* - 2\dot{y}^* - x^*\right)\uvec{\imath} + \left(\ddot{y}^* + 2\dot{x}^* - y^_\right)\uvec{\jmath} + \ddot{z}^*\uvec{k}
 :::
 
 Now we have the non-dimensional inertial acceleration, we need to make Eq. {eq}`eq:vector-eom-cr3bp`, the equation of motion, non-dimensional. After a bunch of algebra, not shown here, we end up with:

the-n-body-problem/lagrange-points.md

@@ -499,7 +499,7 @@ The Trojan and the Greek asteroids are clusters of asteroids that have collected
 
 The collinear Lagrange points, $L_1$, $L_2$, and $L_3$ are all **saddle points** in {numref}`fig:pseudo-potential-energy-cr3bp`, meaning that the function increases when going in one axis, but decreases going in the other axis. This means that the three collinear Lagrange points are **unstable** and an object placed at one of those points, if perturbed, will diverge from the position.
 
-Nonetheless, these are quite useful points for observation of the solar system. Several satellites have been placed at the $L_1$ point of the Earth-Sun system for solar observation, and the James Webb Space Telescope is planned to launch to the $L_2$ of the Earth-Sun system sometime ~this year~ in 2022.
+Nonetheless, these are quite useful points for observation of the solar system. Several satellites have been placed at the $L_1$ point of the Earth-Sun system for solar observation, and the James Webb Space Telescope (JWST) is located at the $L_2$ point in the Earth-Sun system specifically to avoid sunlight interefering with observations.
 
 These satellites orbit around the unstable Lagrange points in a [Lissajous orbit](https://en.wikipedia.org/wiki/Lissajous_orbit). This type of orbit requires a very small amount of propulsion onboard the satellite to keep position, but the orbit can last for a very long time with only a little fuel. One example is the [Wilkinson Microwave Anisotropy Probe](https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe) (WMAP) which was sent to the $L_2$ point in the Earth-Sun system to study the [Cosmic microwave background](https://en.wikipedia.org/wiki/Cosmic_microwave_background). The trajectory of WMAP is shown in {numref}`fig:wmap-trajectory`.
 
@@ -509,4 +509,12 @@ These satellites orbit around the unstable Lagrange points in a [Lissajous orbit
 The trajectory of the [Wilkinson Microwave Anisotropy Probe](https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe) (WMAP) as viewed from Earth. Note the distance in the bottom of the animation, showing the satellite as approximately 1.5 million km from the earth. [Phoenix7777](https://commons.wikimedia.org/wiki/File:Animation_of_Wilkinson_Microwave_Anisotropy_Probe_trajectory_-_Viewd_from_Earth.gif), [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0), via Wikimedia Commons.
 :::
 
+Another example is the JWST, mentioned previously. JWST has a simpler [halo orbit](https://en.wikipedia.org/wiki/Halo_orbit) around $L_2$. The orbit of JWST is shown in {numref}`fig:jwst-trajectory`.
+
+:::{figure} ../images/jwst-trajectory.gif
+:name: fig:jwst-trajectory
+
+The trajectory of the [James Webb Space Telescope](https://en.wikipedia.org/wiki/James_Webb_Space_Telescope) (JWST) as viewed from above the ecliptic plane with Earth fixed. Note again the distance in the bottom of the animation, showing the satellite as approximately 1.5 million km from the earth. [Phoenix7777](https://commons.wikimedia.org/wiki/File:Animation_of_James_Webb_Space_Telescope_trajectory_-_Polar_view.gif), [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0), via Wikimedia Commons.
+:::
+
 $L_1$ and $L_2$ in the Earth-Sun system are about 1.5 million km towards the Sun and away from the Sun, starting at the Earth, respectively. $L_3$ lies on the other side of the Sun, and has long been the predicted location of a hidden planet, since it could not be observed from Earth prior to the advent of satellite observation. Now, of course, we know there is no planet at that location.