Section   18.2The Celestial Sphere

1. (Exercises 3 and 4) Calculate the distance to the star Vega if the parallax of this star has been measured to be 0.125 seconds of arc. The distance in parsecs is the reciprocal of the parallax as measured in seconds of arc so:

d = 1 / 0.125 s = 8 parsecs

In distance units of light-years this result would be:

d = 8 parsecs x 3.26 light-years/1 parsec = 26.1 light-years

Section   18.4Gravitational Collapse and Black Holes

2. (Exercises 11 and 12) What would be the mass of a star that collapsed to form a black hole with an event horizon radius of 30 km? Starting with R = 2GM/c², and converting 30 km to 30,000 m, we can solve for mass and get: M = R c² / 2 G = (30,000 m)(3.00 x 10⁸ m/s) ² / 2 (6.67 x 10^-11 N-m²/kg²) = 2.02 x 10³¹ kg Comparing this to the Sun's mass, which is given in the textbook as 2.0 x 10³⁰ kg, we see that the star that formed this black hole was about 10 times more massive than our Sun.


Section   18.7Cosmology

3. (Exercises 15 and 16) Determine the maximum age of the universe using an intermediate value for Hubble's constant of 75 km/s per million pc. Hubble's constant, found from the plot of recessional velocity as a function of distance, has units of one over time, so we must take the reciprocal of the constant to find the approximate age of the universe.

t = 1 / H = 1 / 75 km/s per 10⁶ pc = 0.013 s x 10⁶ pc/km

But this is not a true time. We must first cancel out the distance units and to do so they must be in the same units. In this problem we will use the notation that 1 million pc = 10⁶ pc. If we look in the textbook we will find that:

3.086 x 10¹⁹ km = 1 x 10⁶ pc

t = 0.013 s x 10⁶ pc/km x 3.086 x 10¹⁹ km per 10⁶ pc = 4.11 x 10¹⁷ seconds

As a practice exercise, convert this time from seconds into years as shown in this Section   of the textbook. The correct answer is 1.30 x 10¹⁰ years, so by using this method, the age of the universe is calculated to be about 13 billion years. If a value for Hubble's constant of 100 km/s per million pc is used in this calculation, as is done in Exercise 12 in the textbook, an age of about 10 billion years is obtained. Another example problem in the textbook calculates this age to be about 20 billion years using the other extreme for the estimated value of Hubble's constant. Although there is still some uncertainty in Hubble's constant, most scientists believe that the age of the universe is probably somewhere between these last two values.

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