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Chapter 5: Problem 6

Binary stars are pairs of stars that orbit each other. The period \(p\) of such a pair is the time, in years, required for a single orbit. The separation \(s\) between such a pair is measured in seconds of arc. The parallax angle \(a\) (also in seconds of arc) for any stellar object is the angle of its apparent movement as the Earth moves through one half of its orbit around the sun. Astronomers can calculate the total mass \(M\) of a binary system using $$ M=s^{3} a^{-3} p^{-2} . $$ Here \(M\) is the number of solar masses. a. Alpha Centauri, the nearest star to the sun, is in fact a binary star. The separation of the pair is \(s=17.6\) seconds of arc, its parallax angle is \(a=0.76\) second of arc, and the period of the pair is \(80.1\) years. What is the mass of the Alpha Centauri pair? b. How would the mass change if the separation angle were doubled but parallax and period remained the same as for the Alpha Centauri system? c. How would the mass change if the parallax angle were doubled but separation and period remained the same? d. How would the mass change if the period doubled but parallax angle and separation remained the same?

Short Answer

Expert verified
a. Mass is approximately 1.872 solar masses. b. Mass becomes 8 times greater. c. Mass becomes one-eighth. d. Mass becomes one-fourth.

Step by step solution

01

Understanding the Mass Formula

The given formula for the total mass \( M \) of a binary star system is \( M = s^3 a^{-3} p^{-2} \). This means that the mass depends on the separation \( s \), parallax angle \( a \), and period \( p \), where \( s \) affects the equation through its cube, \( a \) through the inverse cube, and \( p \) through the inverse square.
02

Calculate the Mass of Alpha Centauri

Substitute the given values into the formula to find the mass of the Alpha Centauri pair. Here, \( s = 17.6 \), \( a = 0.76 \), and \( p = 80.1 \). Compute \( M = (17.6)^3 (0.76)^{-3} (80.1)^{-2} \).
03

Simplify Calculations for Alpha Centauri

Calculate each component individually:- \( s^3 = (17.6)^3 = 5451.776 \)- \( a^{-3} = (0.76)^{-3} = \frac{1}{0.76^3} \approx 2.207 \)- \( p^{-2} = (80.1)^{-2} = \frac{1}{80.1^2} \approx 0.000156 \)Multiply these values to get the mass: \( M \approx 5451.776 \times 2.207 \times 0.000156 \approx 1.872 \).
04

Consider the Effect of Doubling the Separation Angle

If the separation \( s \) is doubled (\( 2s = 35.2 \)), the new mass \( M' \) is calculated using \( M' = (2s)^3 a^{-3} p^{-2} = 8s^3 a^{-3} p^{-2} = 8M \). Thus, the mass becomes 8 times the original mass.
05

Consider the Effect of Doubling the Parallax Angle

If the parallax angle \( a \) is doubled (\( 2a = 1.52 \)), the new mass \( M' \) is \( M' = s^3 (2a)^{-3} p^{-2} = \frac{1}{8}s^3 a^{-3} p^{-2} = \frac{1}{8}M \). Thus, the mass becomes one-eighth of the original mass.
06

Consider the Effect of Doubling the Period

If the period \( p \) is doubled (\( 2p = 160.2 \)), the new mass \( M' \) is \( M' = s^3 a^{-3} (2p)^{-2} = \frac{1}{4}s^3 a^{-3} p^{-2} = \frac{1}{4}M \). Thus, the mass becomes one-fourth of the original mass.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Calculation
The mass calculation of binary stars is a fundamental task in astrophysics. Binary stars, which are systems of two stars orbiting a common center of mass, enable astronomers to determine stellar masses through their observable attributes. To calculate the total mass of a binary star system, we utilize the formula: \[ M = s^3 a^{-3} p^{-2} \] Here, the variables represent:
  • The separation \(s\), which stands for the angular distance between the two stars, measured in seconds of arc.
  • The parallax angle \(a\), also measured in seconds of arc, that gives the apparent shift of a star due to Earth's movement around the Sun.
  • The period \(p\), giving the time taken for one complete orbit by the binary star, measured in years.
By incorporating these values, astronomers can derive the mass \(M\) in terms of solar masses. This provides vital insights into the physical characteristics and dynamics of binary systems.
Stellar Parallax
Stellar parallax is an essential method for understanding the distance of stars from Earth. It involves measuring the apparent shift of a star when observed from two different positions of Earth's orbit, which are six months apart. This shift is a result of Earth's annual motion around the Sun. The smaller the parallax angle \(a\), the farther away the star is. In binary star systems, the parallax angle helps determine not only distance but also aids in mass calculations. The parallax angle appears in the mass equation as an inverse cube: \(a^{-3}\). Therefore, any change in the parallax angle significantly impacts the calculated mass. For instance, doubling the parallax angle leads to a reduction of the mass to one-eighth, due to the inverse cube relationship, as shown by the math: \[ M' = s^3 (2a)^{-3} p^{-2} = \frac{1}{8}M \] Understanding stellar parallax allows astronomers to assess distances in space effectively and contributes to precise calculations of binary star masses.
Orbital Period
The orbital period \(p\) is an essential factor when studying binary star systems. It represents the time it takes for the two stars to complete one full orbit around their common center of mass. The period is typically measured in years. Its importance in mass calculation comes from its presence in the formula for binary star mass, appearing as an inverse square: \(p^{-2}\). This means any increase in the orbital period decreases the calculated mass. For example, if the period is doubled, the mass will be reduced to one-fourth of its original value: \[ M' = s^3 a^{-3} (2p)^{-2} = \frac{1}{4}M \] The orbital period reflects the gravitational dynamics between the stars and their masses, thereby serving as a crucial input for estimating the mass of the binary system.
Separation Angle
The separation angle \(s\) in binary star systems refers to the angular distance between the binary stars. It is typically measured in seconds of arc and provides critical information about the configuration and dynamics of the binary system. The mass equation includes the separation angle as a cubic factor: \(s^3\). As a result, changes in the separation angle have a profound effect on the mass calculation. For example, if the separation angle is doubled, the mass becomes eight times the original value, because the equation now includes \((2s)^3\) which equals \(8s^3\): \[ M' = (2s)^3 a^{-3} p^{-2} = 8M \] Understanding the role of the separation angle helps astronomers gather insights about the distance between stars in a binary system, aiding in accurate mass calculations necessary for studying these stellar structures.

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Most popular questions from this chapter

According to Newton's law of gravitation, the gravitational attraction between two massive objects such as planets or asteroids is proportional to \(d^{-2}\), where \(d\) is the distance between the centers of the objects. Specifically, the gravitational force \(F\) between such objects is given by \(F=c d^{-2}\), where \(d\) is the distance between their centers. The value of the constant \(c\) depends on the masses of the two objects and on the universal gravitational constant. a. Suppose the force of gravity is causing two large asteroids to move toward each other. What is the effect on the gravitational force if the distance between their centers is halved? What is the effect on the gravitational force if the distance between their centers is reduced to one-quarter of its original value? b. Suppose that for a certain pair of asteroids whose centers are 300 kilometers apart, the gravitational force is \(2,000,000\) newtons. (One newton is about one-quarter of a pound.) What is the value of \(c\) ? Find the gravitational force if the distance between the centers of these asteroids is 800 kilometers. c. Using the value of \(c\) you found in part \(b\), make a graph of gravitational force versus distance between the centers of the asteroids for distances from 0 to 1000 kilometers. What happens to the gravitational force when the asteroids are close together? What happens to the gravitational force when the asteroids are far apart?

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