Final answer:
Right ascension is a coordinate for measuring the east-west positions of celestial bodies and can be expressed in units of angle or time. The star's right ascension is 45°. The altitude of the Sun along the meridian on the first day of summer for an observer at a latitude of 42 degrees north is 71.5°.
Step-by-step explanation:
Right ascension (RA) is like longitude, except that instead of Greenwich, the arbitrarily chosen point where we start counting is the vernal equinox, a point in the sky where the ecliptic (the Sun's path) crosses the celestial equator. RA can be expressed either in units of angle (degrees) or in units of time. This is because the celestial sphere appears to turn around Earth once a day as our planet turns on its axis. Thus the 360° of RA that it takes to go once around the celestial sphere can just as well be set equal to 24 hours. Then each 15° of arc is equal to 1 hour of time.
Given that the star has an hour angle of 3 hours (3h) when the local sidereal time is 8:15, we can convert the hour angle to its equivalent in degrees. Since each hour of time corresponds to 15° of arc, we multiply the given hour angle by 15 to get 45°. So, the right ascension of the star is 45°.
On the first day of summer for an observer at a latitude of 42 degrees north, the altitude of the Sun along the meridian (the meridional altitude) can be calculated using the formula:
Altitude = 90° - Latitude + Declination
Since it is the first day of summer, the declination of the Sun is the maximum value for the year, which is approximately +23.5°. Substituting the values into the formula, we have:
Altitude = 90° - 42° + 23.5° = 71.5°