Chapter 3, Astrometry Video Solutions, An Introduction to Astronomy and Astrophysics

Problem 1

The relationship between Cartesian and spherical polar coordinates is given in Equation 3.3. (a) Express $x, y, z$ in terms of $r$ and the equatorial coordinates ( $\alpha, \delta$ ). (b) In an equatorial grid, the Cartesian coordinates of a star are $x=10.0, y=15.0, z=6.0$ in some chosen units. Determine its distance $r$ and the angular coordinates $\alpha$ and $\delta$. (c) Repeat (b) for $x=-10.0, y=-15.0, z=6.0$.

John Nicolle

Numerade Educator

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Problem 2

The proper motion of Barnard's star is $10.3^{\prime \prime}$ per year. It is located at a distance of 1.834 pc . Determine its transverse speed, $v_t$, in $\mathrm{Km} / \mathrm{s}$.

Victor Salazar

Numerade Educator

01:30

Problem 3

Consider a star that is moving away from us at a speed of $300 \mathrm{Km} / \mathrm{s}$. It emits radiation of wavelength 500 nm . Determine the Doppler shift, $\Delta \lambda$, and the observed wavelength, $\lambda$.

Mayukh Banik

Numerade Educator

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Problem 4

Let $\vec{\mu}$ represent the proper or angular velocity of a star. We define $\mu_\delta=\delta=d \delta / d t$ and $\mu_\alpha=\dot{\alpha}=d \alpha / d t$, where $\delta$ and $\alpha$ represent the Dec and RA of the star. Show that the components of the space velocity $\vec{v}$ are given by $v_\delta=r \mu_\delta$ and $v_\alpha=r \cos \delta \mu_\alpha$, where $r$ is the distance of the star. Hence prove Equation 3.7. Hint: First consider a small angular displacement, $\Delta \delta$, along the longitude in time $\Delta t$. Find the distance traveled and hence determine the velocity. Repeat this for a small displacement along the latitude. Note that the two displacements can be considered independently.

Andrew Eddins

Emory University

09:44

Problem 5

Recall that at the summer solstice, the Sun never sets at latitudes close to the North Pole. (a) Find the range of latitudes for which this is true. (b) At any latitude $l$ there exists a group of stars that always remain either above or below the horizon. These are called circumpolar stars. Find the range of declinations for these stars at latitude $l$. (c) Determine the names of a few bright circumpolar stars visible at your location. Locate them in the night sky and track their motion by observing them at different times.

Marybeth Senser

Numerade Educator

02:20

Problem 6

Using the precession rate of $50^{\prime \prime}$ per year, verify that the period of precession is roughly 26,000 years.

Sarah Mccrumb

Numerade Educator

Problem 7

Determine the inverse transformation from the ecliptic to equatorial coordinate system, that is, express the equatorial coordinates of a point in terms of its ecliptic coordinates.

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Problem 8

Determine the rate at which the equatorial coordinates of a source change due to precession. Start by differentiating the equations obtained in the previous exercise with respect to time. This gives us $d \delta / d t$ and $d \alpha / d t$ in terms of $\beta, \lambda, d \lambda / d t$ and the transformation angle $\theta=23^{\circ} 26^{\prime}$. Next, eliminate $\beta, \lambda$ in terms of $\delta$ and $\alpha$ to obtain the final result.

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Problem 9

Determine the transformation between the equatorial and galactic coordinate systems following the procedure explained in the text. The galactic pole is located at $\delta=27.13^{\circ}$ and $\alpha=192.86^{\circ}$. The galactic center is located at $\delta=-28.94^{\circ}$ and $\alpha=266.40^{\circ}$.

Andrew Eddins

Emory University