As seen from the starship Enterprise in the Star Trek television series and movies, stars appear to move across the sky due to the starship's motion. How fast would the Enterprise have to move in order for a star \(1 \mathrm{pc}\) away to appear to move \(1^{\circ}\) per second? (Hint: The speed of the star as seen from the Enterprise is the same as the speed of the Enterprise relative to the star.) How does this compare with the speed of light? Do you think the stars appear to move as seen from an orbiting space shuttle, which moves at about \(8 \mathrm{~km} / \mathrm{s}\) ?

Angular speed is a measure of how quickly an object moves through a particular angle. It's commonly used in astronomy to describe the apparent motion of celestial bodies as they move across the sky. The concept allows us to define the rate at which an object rotates or revolves around a fixed point or another object, such as a star seen from a spacecraft.

Imagine you're on a spinning roundabout; your angular speed would be the rate at which you're turning in circles, measured in angles per unit of time, typically degrees per second or radians per second. For example, if you complete a 360-degree revolution every minute, your angular speed would be 6 degrees per second (since there are 360 degrees in a full circle and 60 seconds in a minute).

Tangential speed, also known as linear speed, is the speed of an object moving along a circular path. It relates to angular speed but is expressed as the distance traveled per unit of time, like kilometers per hour or miles per hour. The further you are from the center of rotation, the greater your tangential speed will be, even if your angular speed remains the same.

Let's think about a figure skater spinning with outstretched arms. When the skater pulls in their arms and spins around the same spot, their angular speed increases, but so does their tangential speed, hence they spin faster. The relationship between tangential speed \( v \) and angular speed \( \omega \) can be described by the formula \( v = r\omega \), where \( r \) is the radius of the circle.

A parsec is a unit of distance used in astronomy to measure the vast spans between stars and galaxies. One parsec is equivalent to approximately 3.26 light-years or about \(3.086 \times 10^{13}\) kilometers. It's derived from the parallax of one arcsecond, hence the name 'parsec.'

When astronomers use the parsec, they're often measuring the distance to stars. The nearest star to our solar system, Proxima Centauri, is approximately 1.3 parsecs away. By using parsecs, scientists can make more accurate and meaningful measurements of space distances and the velocities of celestial objects.

The speed of light is a fundamental constant in the universe, clocking in at about \(299,792\) kilometers per second in a vacuum. This incredible speed is the cosmic speed limit; according to our current understanding of physics, nothing can travel faster than light in a vacuum.

In our everyday experience, light speed seems instantaneous, but over cosmic distances, the time light takes to travel becomes quite noticeable. For instance, it takes about 8 minutes for light to travel from the Sun to the Earth. Understanding light speed helps us not only travel conceptually across the vast universe but also provides a foundation for physics theories like Einstein's theory of relativity.

The small angle formula is a handy tool for astronomers, allowing them to relate the physical size of an object to its observed angular size when it's very far away. For small angles (measured in radians), the formula is approximately \(d = \theta D\), where \(d\) is the linear size of the object, \(\theta\) its angular size in radians, and \(D\) its distance from the observer. It's particularly useful for measuring the distance to stars or the size of objects within our solar system.

In the case of the starship Enterprise exercise, we use this formula to work out the apparent movement of a star 1 parsec away when the ship moves to cause the star's apparent position to change by 1 degree per second. While the Enterprise's calculated speed is impressive, it remains theoretical as it’s beyond our current technological capabilities.