The Hyades has a proper motion of 0.07 arc sec/yr and appears \(26^{\circ}\) from its convergent point. The radial velocity is \(35 \mathrm{km} / \mathrm{s}\). (a) How far away is the cluster? (b) What is the actual speed of the cluster?

Proper motion is the apparent movement of a star or a star cluster across the sky, as observed from Earth, measured in arcseconds per year. This movement is due to the star's actual motion through space relative to the Sun. Proper motion does not include the motion along our line of sight, which is taken into account by radial velocity. In the Hyades cluster problem, the proper motion is given as 0.07 arcseconds per year. This means the cluster appears to move 0.07 arcseconds on the sky each year. To find the transverse velocity (the actual speed perpendicular to our line of sight), we need to know the distance to the cluster and use the transverse velocity formula: \[ v_t = r \times \text{proper motion} \] Here, proper motion needs to be converted into linear velocity.

Radial velocity is the speed at which an object moves toward or away from an observer along the line of sight. It is typically measured in kilometers per second (km/s). For the Hyades cluster, the radial velocity is given as 35 km/s. This is the component of the cluster's velocity directed along our line of sight. Along with the transverse velocity, radial velocity helps in determining the actual speed and motion of the cluster in space. To find the total speed, we will use the Pythagorean theorem: \[ v = \sqrt{v_t^2 + v_r^2} \] where \(v_t\) is the transverse velocity and \(v_r\) is the radial velocity.

Transverse velocity is the component of an object's velocity perpendicular to the observer's line of sight, given by the formula: \[ v_t = r \times \text{proper motion} \] Proper motion must be expressed in radians per year for this calculation. The transverse velocity can also be derived from the radial velocity and the angle of observation: \[ v_t = v_r \tan \theta \] In the problem, the angle \(\theta\) is given as 26 degrees, which needs to be converted to radians before calculating the transverse velocity. Once this is done, the transverse velocity can be calculated using the given radial velocity.

Angles are often provided in degrees but need to be converted to radians for many astronomical calculations. The conversion factor between degrees and radians is: \[ 1 \text{ degree} = \frac{\text{π}}{180} \text{ radians} \] For an angle of 26 degrees, this conversion is done using: \[ 26^{\text{degrees}} = 26 \times \frac{\text{π}}{180} = \frac{26\text{π}}{180} \text{ radians} \] This conversion is crucial for accurately determining quantities like transverse velocity in the exercise. The angle in radians is then used in various formulas, like the tangent function, to get precise results.