Let's solve each part step by step, with explanations. --- ## (a) What is the focal length of HST in mm? ### **Step 1: Use the f/ratio definition** The f/ratio (focal ratio) is defined as: \[ f/\# = \frac{\text{Focal Length}}{\text{Aperture Diameter}} \] Given: - \( f/\# = 24 \) - Aperture diameter \( D = 2.4\ \text{meters} = 2400\ \text{mm} \) ### **Step 2: Solve for focal length** \[ \text{Focal Length} = f/\# \times D = 24 \times 2400\ \text{mm} = 57,\!600\ \text{mm} \] **Explanation:** The focal length is just the f/ratio times the diameter in the same units. --- ## (b) What is the plate scale of HST in mm/arcsec? ### **Step 1: Plate scale formula** The plate scale tells us how much (in mm) one arcsecond of sky covers in the focal plane. The formula is: \[ \text{Plate Scale} = \frac{f \times \pi}{648000} \] where \(f\) is in mm and 648,000 is the number of arcseconds in 180 degrees. Alternatively, a commonly used version: \[ \text{Plate Scale (arcsec/mm)} = \frac{206265}{f} \] So, \[ \text{Plate Scale (mm/arcsec)} = \frac{f}{206265} \] ### **Step 2: Plug in the numbers** \[ \text{Plate Scale} = \frac{57,\!600\ \text{mm}}{206265} = 0.279\ \text{mm/arcsec} \] **Explanation:** This means that 1 arcsecond on the sky corresponds to 0.279 mm in the focal plane. --- ## (c) Can the Ring Nebula (230 arcsec × 230 arcsec) fit on a 61 mm × 61 mm CCD array?** ### **Step 1: Calculate the angular field covered by 61 mm** \[ \text{Angular size} = \frac{\text{CCD size (mm)}}{\text{Plate scale (mm/arcsec)}} \] \[ \text{Angular size} = \frac{61\ \text{mm}}{0.279\ \text{mm/arcsec}} \approx 218.6\ \text{arcsec} \] ### **Step 2: Compare with Nebula's size** - CCD covers \( 218.6'' \times 218.6'' \) - Nebula is \( 230'' \times 230'' \) So the nebula is slightly larger than the field of view. ### **Step 3: Conclusion** **Explanation:** The entire Ring Nebula (230'' across) would not fit on the 61 mm × 61 mm CCD array, since the array can only image about 218.6'' × 218.6'' at once. The nebula would be slightly clipped at the edges. --- ## **Final Answers (with explanations)** **(a)** The focal length of HST is **57,600 mm**. **(b)** The plate scale of HST is **0.279 mm/arcsec**. **(c)** The Ring Nebula (230'' × 230'') would **not fit** on a 61 mm × 61 mm CCD array, because the array only covers **218.6'' × 218.6''**. The nebula would be slightly larger than the field of view.