Import standard modules:

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import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
from IPython.display import HTML 
HTML('../style/course.css') #apply general CSS
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from IPython.display import HTML
HTML('../style/code_toggle.html')
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The raw code for this notebook is by default hidden for easier reading. To toggle on/off the raw code, click here.

3.1 Equatorial Coordinates (RA,DEC)

We can use a geographical coordinate system to uniquely identify a position on earth. We normally use the coordinates latitude $L_a$ (to measure north and south) and longitude $L_o$ (to measure east and west) to accomplish this. The equatorial coordinate system is depicted in Fig. 3.1.1 ⤵.

Figure 3.1.1: The geographical coordinates latitude $L_a$ and longitude $L_o$.

We also require a coordinate system to map the stars. For all intents and purposes we may think of our universe as being projected onto a sphere of arbitrary radius. This sphere surrounds the Earth and is known as the celestial sphere. This is not a true representation of our universe, but it is a very useful approximate astronomical construct. The celestial equator is obtained by projecting the equator of the earth onto the celestial sphere. The stars themselves do not move on the celestial sphere and therefore have a unique location on it. The Sun is an exception, it changes position in a periodic fashion during the year (as the Earth orbits around the Sun). The path it traverses on the celestial sphere is known as the ecliptic.

3.1.1 The NCP and SCP

The north celestial pole (NCP) is an important location on the celestial sphere and is obtained by projecting the north pole of the earth onto the celestial sphere. The star Polaris is very close to the NCP and serves as a reference when positioning a telescope.

The south celestial pole (SCP) is obtained in a similar way. The imaginary circle known as the celestial equator is in the same plane as the equator of the earth and is obtained by projecting the equator of the earth onto the celestial sphere. The southern hemisphere counterpart of Polaris is Sigma Octanis.

We use a specific point on the celestial equator from which we measure the location of all other celestial objects. This point is known as the first point of Aries ($\gamma$) or the vernal equinox. The vernal equinox is the point where the ecliptic intersects the celestial equator (south to north). We discuss the vernal equinox in more detail in $\S$ 3.2.2 ➞ .

Equatorial Coordinates

We use the equatorial coordinates to uniquely identify the location of celestial objects rotating with the celestial sphere around the SCP/NCP axis.

The Right Ascension $\alpha$ - We define the hour circle of an object as the circle on the celestial sphere that crosses the NCP and the object itself, while also perpendicularly intersecting with the celestial equator. The right ascension of an object is the angular distance between the vernal equinox and the hour circle of a celestial object measured along the celestial equator and is measured eastward. It is measured in Hours Minutes Seconds (e.g. $\alpha = 03^\text{h}13^\text{m}32.5^\text{s}$) and spans 360$\circ$ on the celestial sphere from $\alpha = 00^\text{h}00^\text{m}00^\text{s}$ (the coordinates of $\gamma$) to $\alpha = 23^\text{h}59^\text{m}59^\text{s}$.

The Declination $\delta$ - the declination of an object is the angular distance from the celestial equator measured along its hour circle (it is positive in the northern celestial hemisphere and negative in the southern celestial hemisphere). It is measured in Degrees Arcmin Arcsec (e.g. $\delta = -15^\circ23'44''$) which spans from $\delta = -90^\circ00'00''$ (SCP) to $+\delta = 90^\circ00'00''$ (NCP).

The equatorial coordinates are presented graphically in Fig. 3.1.2 ⤵ .

Warning: As for any spherical system, the Right Ascension of the NCP ($\delta=+90^ \circ$) and the SCP ($\delta=-90^ \circ$) are ill-defined. And a source close to the any celestial pole can have an unintuitive Right Ascension.

Figure 3.1.2: The equatorial coordinates $\alpha$ and $\delta$. The vernal equinox $\gamma$, the equatorial reference point is also depicted. The vernal equinox is the point where the ecliptic (the path the sun traverses over one year) intersects the celestial equator.

Warning: One arcminute of the declination axis (e.g. $00^\circ01'00''$) is not equal to one minute in right ascension axis (e.g. $00^\text{h}01^\text{m}00^\text{s}$).
Indeed, in RA, the 24$^\text{h}$ circle is mapped to a 360$^\circ$ circle meaning that 1 hour spans over a section of 15$^\circ$. And as 1$^\text{h}$ is 60$^\text{m}$, therefore 1$^\text{m}$ in RA correspond to $1^\text{m} = \frac{1^\text{h}}{60}=\frac{15^\circ}{60}=0.25'$.
You should be careful about this **factor of 4 difference between RA min and DEC arcmin** (i.e. $\text{RA} \; 00^\text{h}01^\text{m}00^\text{s}\neq \text{DEC} \; 00^\circ01'00''$)

Next: 3.2 Hour Angle (HA) and Local Sidereal Time (LST)

Future Additions:
  • add section on converting between ECEF and geodetic coordinates, see ecef.py ; introduce ITRF coordinates