220
Parallax, Angular Size, and Angular Resolution
4.6 Exercise Solutions
1. From Eq. 4.3, parallax angle (deg) = 57.3◦
parallax angle = 10�� = 10�� ×
�
1 deg
3600��
�
�
baseline
object distance
�
= 2.78 × 10−3 deg
baseline = 2RE = 12,756 km
�
�
baseline
parallax angle (deg) = 57.3
, so
object distance
�
�
baseline
◦
object distance = 57.3
parallax angle (deg)
�
�
12, 756 km
◦
= 57.3
2.78 × 10−3 deg
◦
= 2.63 × 108 km (ans.)
(this is about 1.75 AU)
2. In this case, parallax angle = 1�� ×
�
1 deg
3600��
�
= 2.78 × 10−4 deg
Baseline is the same as in the previous exercise: 12,756 km
�
�
baseline
object distance = 57.3◦
as before, so
parallax angle (deg)
�
�
12, 756 km
= 57.3◦
2.78 × 10−4 deg
= 2.63 × 109 km (ans.)
(this is about 17.5 AU)
3. In this case, baseline = 2 AU = 3 × 108 km
�
baseline
For 10 parallax angle: object distance = 57.3
parallax angle (deg)
�
�
3 × 108 km
◦
= 57.3
2.78 × 10−3 deg
��
◦
= 6.18 × 1012 km (ans.)
�
4.6 Exercise Solutions
For 10�� parallax angle: object distance = 57.3◦
�
3 × 108 km
2.78 × 10−4 deg
= 6.18 × 1013 km (ans.)
4. From Eq. 4.8, angular size (deg) = 57.3◦
�
221
�
physical size
distance
�
physical size = 1000 × 2RSun = 1000 × 2(6.96 × 105 km)
= 1.39 × 109 km
distance = 650 light years = 650 light years ×
= 6.15 × 1015 km
angular size (deg) = 57.3◦
�
�
9.46 × 1012 km
1 light year
1.39 × 109 km
6.15 × 1015 km
= 1.3 × 10−5 deg (ans.)
�
�
�
λ(µm)
5. From Eq. 4.14, angular resolution (arcsec) = 0.25 aperture
(m)
�
�
−9
1 × 10 m
λ = 550 nm = 550 nm
= 5.50 × 10−7 m
1 nm
aperture = 4 mm = 4 × 10−3 m
�
= 0.550 × 10−6 m = 0.550 µm
�
0.550
angular resolution (arcsec) = 0.25
4 × 10−3
�
= 34.4 arcsec (ans.)
6. In this case, aperture = 9 mm = 9 × 10−3 m
wavelength, λ, is the same as before: 0.550 µm
�
0.550
angular resolution (arcsec) = 0.25
9 × 10−3
�
= 15.3 arcsec (ans.)