An Introduction to Modern Astrophysics

Bradley W. Carroll, Dale A. Ostlie

Chapter 1

The Celestial Sphere - all with Video Answers

Educators

MS

Chapter Questions

Problem 1

Derive the relationship between a planet's synodic period and its sidereal period (Eq. Consider both inferior and superior planets. $$1 / S=\left\{\begin{array}{ll}
1 / P-1 / P_{\oplus} & \text { (inferior) } \\
1 / P_{\oplus}-1 / P & \text { (superior) }
\end{array}\right.$$

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

Devise methods to determine the relative distances of each of the planets from the Sun given the information available to Copernicus (observable angles between the planets and the Sun, orbital configurations, and synodic periods.

MS

Marybeth Senser

Numerade Educator

Problem 3

(a) The observed orbital synodic periods of Venus and Mars are 583.9 days and 779.9 days, respectively. Calculate their sidereal periods.
(b) Which one of the superior planets has the shortest synodic period? Why?

Farhanul Hasan

Numerade Educator

Problem 4

List the right ascension and declination of the Sun when it is located at the vernal equinox, the summer solstice, the autumnal equinox, and the winter solstice.

Farhanul Hasan

Numerade Educator

Problem 5

(a) Referring to Fig. $12(\mathrm{a}),$ calculate the altitude of the Sun along the meridian on the first day of summer for an observer at a latitude of $42^{\circ}$ north.
FIGURE $12 \quad$ (a) The diurnal path of the Sun across the celestial sphere for an observer at latitude $L$ when the Sun is located at the vernal equinox (March), the summer solstice (June), the autumnal equinox (September), and the winter solstice (December). NCP and SCP designate the north and south celestial poles, respectively. The dots represent the location of the Sun at local noon on the approximate dates indicated.
(b) What is the maximum altitude of the Sun on the first day of winter at the same latitude?

Farhanul Hasan

Numerade Educator

Problem 6

(a) Circumpolar stars are stars that never set below the horizon of the local observer or stars that are never visible above the horizon. After sketching a diagram similar to Fig.12(a), calculate the range of declinations for these two groups of stars for an observer at the latitude $L$ FIGURE $12 \quad$ (a) The diurnal path of the Sun across the celestial sphere for an observer at latitude $L$ when the Sun is located at the vernal equinox (March), the summer solstice (June), the autumnal equinox (September), and the winter solstice (December). NCP and SCP designate the north and south celestial poles, respectively. The dots represent the location of the Sun at local noon on the approximate dates indicated.
(b) At what latitude(s) on Earth will the Sun never set when it is at the summer solstice?
(c) Is there any latitude on Earth where the Sun will never set when it is at the vernal equinox? If so, where?

MS

Marybeth Senser

Numerade Educator

Problem 7

(a) Determine the Julian date for 16: 15 UT on July 14,2006 . (Hint: Be sure to include any leap years in your calculation.)
(b) What is the corresponding modified Julian date?

Farhanul Hasan

Numerade Educator

Problem 8

Proxima Centauri ( $\alpha$ Centauri $\mathrm{C}$ ) is the closest star to the Sun and is a part of a triple star system. It has the epoch $\mathrm{J} 2000.0$ coordinates $(\alpha, \delta)=\left(14^{\mathrm{h}} 29^{\mathrm{m}} 42.95^{\mathrm{s}},-62^{\circ} 40^{\prime} 46.1^{\prime \prime}\right) .$ The brightest
member of the system, Alpha Centauri ( $\alpha$ Centauri A) has J2000.0 coordinates of $(\alpha, \delta)=$ $\left(14^{\mathrm{h}} 39^{\mathrm{m}} 36.50^{\mathrm{s}},-60^{\circ} 50^{\prime} 02.3^{\prime \prime}\right)$
(a) What is the angular separation of Proxima Centauri and Alpha Centauri?
(b) If the distance to Proxima Centauri is $4.0 \times 10^{16} \mathrm{m}$, how far is the star from Alpha Centauri?

Farhanul Hasan

Numerade Educator

Problem 9

(a) Using the information in Problem $8,$ precess the coordinates of Proxima Centauri to epoch $\mathrm{J} 2010.0$
(b) The proper motion of Proxima Centauri is $3.84^{\prime \prime} \mathrm{yr}^{-1}$ with the position angle $282^{\circ} .$ Calculate the change in $\alpha$ and $\delta$ due to proper motion between 2000.0 and 2010.0
(c) Which effect makes the largest contribution to changes in the coordinates of Proxima Centauri: precession or proper motion?

MS

Marybeth Senser

Numerade Educator

Problem 10

Which values of right ascension would be best for viewing by an observer at a latitude of $40^{\circ}$ in January?

Farhanul Hasan

Numerade Educator

Problem 11

Verify that Eq. ( 7) follows directly from the expression immediately preceding it. $$\Delta \delta=\Delta \theta \cos \phi$$

Farhanul Hasan

Numerade Educator