Spherical Astronomy: Trigonometry and Stellar Applications

In this video, we do a short recap of the trigonometric functions and the trigonometric circle. Feel free to skip this video if you are already acquainted with them.

In this lesson, we note a few important formulas in planar trigonometry. You can skip this video as well if you think you know these formulas.

In this video we talk about the sphere itself, the lines (arcs) connecting 2 points on a sphere and the angles that can be formed.

In this lesson we introduce the notion of spherical triangle, mention its properties and note the fundamental formulas in spherical trigonometry.

Yet another important formula in spherical trigonometry, especially useful when there seems to be no way to apply neither the sine nor the cosine formula. It is a bit more difficult to remember, which is why, in this lesson, we are going to deduce it (it might be easier to demonstrate than to remember the final form).

Another feature of a spherical triangle we haven't mentioned so far - its area. This video teaches us how to calculate it as a function of the triangle's angles.

This lesson deals with a notion called solid angles, which describe the angular dimension of a spherical portion (calotte/cap).

In order to move on to discussing plane trajectories, we must first cover the most system of coordinates which we are going to be using: latitude and longitude.

This lecture presents the idea behind plane trajectories as well as an interesting common debate as a bonus.

In this video we learn to calculate angles, distances and flight times for different plane trajectories with the help of spherical triangles.

An interesting aspect of plane routes - the uppermost or lowermost point latitude a plane crosses.

Another interesting calculation of plane trajectories - a bit more challenging mathematically, but quite enjoyable once you've acquired the knack.

Let's get to the real astronomy!! In this lesson we are going to learn about the imaginary place on which all the stars in the sky seem to be fixed - the celestial sphere - a vital notion from now on, as all of our measurements will involve it in some way. This shall be fun!

Now that we are familiarized with the celestial sphere, let's find out what the most important lines are, as they will be needed for all of our future measurements. In this lecture we'll talk about the celestial equator, the ecliptic, the meridian, the horizon and others!

And lastly... the most important points of reference corresponding to those lines. About the celestial and ecliptic poles, cardinal points, zenith and nadir and more - this video teaches you just that!

In this lecture we'll be discussing about the first system of celestial coordinates and the most intuitive one - horizontal coordinates.

Next up - the equatorial coordinates - perhaps the most used system throughout this course and also throughout the entirety of astronomical measurements really.

Another coordinate used to describe the position of a celestial body relative to the meridian, giving rise to the hour coordinates (H and delta). It is also of utter importance when calculating what's called "sideral time".

Assume we only know the horizontal coordinates of a star and want to find out its equatorial coordinates to conduct further measurements - how do we do that? This video teaches you just that! (the same goes the other way round)

In this lecture we'll learn about the highest and lowest positions of a star for a given place of observation - the upper and lower culmination.

Moving on to something more familiar - sunrises and sunsets; what is the hour angle of stars the moment they touch the horizon?

In this short video we'll be looking into that remaining angle in the position triangle - what is it and how is it useful?

As a follow-up to sunrises and sunsets, this video teaches us how to determine the duration of such an event, depending on the parallactic angle. Check it out!

This video introduces the last set of coordinates that we are going to use - the ecliptic coordinates.

Another conversion - this time between ecliptic and equatorial coordinates.

For our first practical application, we'll figure out a method to calculate the distance between 2 stars, starting off with the angular distance.

Once the angular distance is known, we can find the linear distance between the stars with the help of some other notion called annual parallax (check the resources for a short debriefing on that).

In this lecture we are going to calculate how long can a star be seen above (or alternatively, below) the horizon. We'll also explain and exemplify stars' diurnal movement.

This lecture is focused on the way we calculate time - what you see on your watches and clocks is actually calculated with the help of astronomy!

In this lesson we are going to go over the set of equations needed to find out the legal time corresponding to the moment of star rise, star set, or any other known position of a star in the sky.

This video discusses a subject that is quite widely known and used, but you might not be too familiar with the mechanism behind it... so let's talk about twilights, shall we?!

Now that we are acquainted with twilights and the measurement of time, we can simply find how long there is light in the sky - in other words, how long does the astronomical night last? Something very similar can be done for any twilights or specific positions of the Sun.

Up until now, we've only considered the movement of stars due to the apparent rotation of the celestial sphere. In this section, however, we'll be talking about the motion induced by the stars themselves - i.e. how they move with respect to a stationary observer.

The proper motion of stars is basically the angular velocity at which they appear to be moving in the sky (over long periods of time), but what about linear velocities? How can we find the true individual speed of stars and what can we do with it? This video answers those questions!

With the help of what we've talked about in the last lesson, we can now carry out an interesting exercise - calculating when and where a star was or will be closest to us, considering Earth to be stationary.

For the last video of this course, I have prepared an interesting exercise, involving everything that we've learned so far, which sums up a famous ancient story - the story of 2 lovers, Altair and Vega (just like the stars), that are meant to unite at some distant point in time. Therefore, this lecture will focus on identifying the convergence point of two stars, and that of multiple stars in a stellar cluster.