Problem 75 - University Physics with Modern Physics

Question

Many of the stars in the sky are actually binary stars, in which two stars orbit about their common center of mass. If the orbital speeds of the stars are high enough, the motion of the stars can be detected by the Doppler shifts of the light they emit. Stars for which this is the case are called spectruscopic binary stars. Figure 37.30 (next page) shows the simplest case of a spectroscopic binary star: two identical stars, each with mass \(m,\) orbiting their center of mass in a circle of radius \(R\) . The plane of the stars' orbits is edge-on to the line of sight of an observer on the earth. (a) The light produced by heated hydrogen gas in a laboratory on the earth has a frequency of \(4.568110 \times 10^{14} \mathrm{Hz}\) . In the light received from the stars by a tele- scope on the earth, hydrogen light is observed to vary in frequency between \(4.567710 \times 10^{14} \mathrm{Hz}\) and \(4.568910 \times 10^{14} \mathrm{Hz}\) . Determine whether the binary star system as a whole is moving toward or away from the earth, the speed of this motion, and the orbital speeds of the stars. (Hint: The speeds involved are much less than \(c,\) so you may use the approximate result \(\Delta f|f=u| c\) given in Section \(37.6 . )\) (b) The light from each star in the binary system varies from its maximum frequency to its minimum frequency and back again in 11.0 days. Determine the orbital radius \(R\) and the mass \(m\) of each star. Give your answer for \(m\) in kilograms and as a multiple of the mass of the sun, \(1.99 \times 10^{30} \mathrm{kg}\) . Compare the value of \(R\) to the distance from the earth to the sun, \(1.50 \times 10^{11} \mathrm{m}\) . (This technique is actually used in astronomy to determine the masses of stars. In practice, the problem is more complicated because the two stars in a binary system are usually not identical, the orbits are usually not circular, and the plane of the orbits is usually tilted with respect to the line of sight from the earth.)

Step-by-Step Solution

Answer

The binary star system is moving towards Earth at 39.3 m/s. Each star's speed is 26.2 km/s, the orbital radius is \(3.96 \times 10^{10}\) m, and each star's mass is \(3.2 \times 10^{30}\) kg.

Step 1: Calculate the Difference in Frequency

The given frequency of hydrogen light in the laboratory is \(f_0 = 4.568110 \times 10^{14} \text{Hz}\). The observed frequencies from the stars vary between \(f_{min} = 4.567710 \times 10^{14} \text{Hz}\) and \(f_{max} = 4.568910 \times 10^{14} \text{Hz}\). The change in frequency \(\Delta f\) is given by \(\Delta f = f_{max} - f_{min} = 4.568910 \times 10^{14} - 4.567710 \times 10^{14} = 0.001200 \times 10^{14} \text{Hz}\).

Step 2: Determine the Direction of the Binary System

Since the maximum observed frequency \(f_{max}\) is greater than \(f_0\), the binary system is moving towards the Earth. Doppler shift results in increased frequency when the source is approaching. Hence, the system moves towards Earth.

Step 3: Calculate the System's Radial Velocity

Using \(\Delta f / f = u / c\), solve for the speed of the system \(u\):\[\frac{0.000600 \times 10^{14}}{4.568110 \times 10^{14}} = \frac{u}{3 \times 10^8}\]This gives:\[u = \frac{0.000600 \times 3 \times 10^8}{4.568110} = 39.3 \text{m/s}\]

Step 4: Calculate the Orbital Speed of Each Star

Since the stars are identical and orbit around their center of mass, we use the maximum frequency change to find the orbital speed \(v\):\[\frac{\Delta f'}{f} = \frac{v}{c}\], where \(\Delta f' = \frac{4.568910 - 4.568110}{2} \times 10^{14}\).The equation becomes:\[v = \frac{0.000400 \times 10^{14} \times 3 \times 10^8}{4.568110} = 26.2 \text{km/s}\]

Step 5: Determine the Orbital Radius \(R\)

The period \(T\) is given as 11 days = 950400 seconds. Use the formula for circular motion:\[ R = \frac{vT}{2\pi}\]This gives:\[R = \frac{26.2 \times 10^3 \times 950400}{2 \pi} = 3.96 \times 10^{10} \text{m}\], which shows the radius size relative to Earth's distance to the Sun.

Step 6: Calculate Mass \(m\) of Each Star

The centripetal force is given by \(F = \frac{Gm^2}{4R^2}\), equal to gravitational attraction. Using \(v^2 = \frac{GM}{2R}\), solve for mass \(m\):\[ m = \frac{4\pi^2R^3}{GT^2}\]Insert values to find:\[ m = 3.2 \times 10^{30} \text{kg}\], which is about 1.6 times the mass of the Sun.

Key Concepts

Spectroscopic Binary Stars

In the vastness of the night sky, many stars we see as individual points of light are actually part of what is known as binary star systems. When two stars orbit around a common center of mass, they are referred to as binary stars. Among these, spectroscopic binary stars are particularly interesting. They are called "spectroscopic" because their presence is detected through the observation of spectral lines rather than direct visual clues. This typically happens when the two stars are too close together to be distinguished with a telescope.

In these systems, the stars' orbital motion causes shifts in the wavelength of light they emit due to the Doppler effect, which we will discuss further. These shifts provide astronomers with valuable insights into the characteristics of the binary system, such as the speed at which each star is moving in its orbit.

• Spectroscopic binaries are often identified by changes in the spectral lines.
• The motion of the stars causes these lines to periodically shift towards the red or blue end of the spectrum.

Studying these spectral lines allows astronomers to deduce the presence of a binary system and calculate important information about the stars involved.

Doppler Shift

The Doppler shift is a change in the frequency of a wave in relation to an observer moving relative to the source of the wave. In the context of spectroscopic binary stars, this effect is crucial for discovering and studying these systems. The Doppler shift is responsible for the observed changes in the frequency of light that reaches us from binaries.

When a star in a binary system moves towards Earth, its light waves are compressed, leading to a higher frequency or blue shift. Conversely, when the star moves away, the waves elongate, resulting in a lower frequency or red shift. This periodic change in the frequencies can reveal the motion of stars within binary systems.

• As stars orbit each other, the Doppler effect causes alternating shifts in the light's frequency.
• A blue shift indicates the star is moving closer, while a red shift means it is receding.

The Doppler shift in these systems allows astronomers to measure how fast stars are moving in their orbits and, in some cases, whether the system as a whole is moving relative to Earth.

Orbital Speed

Orbital speed is a key characteristic in understanding the dynamics of binary star systems. It refers to the speed at which a star moves around the center of mass in its orbit. For spectroscopic binary stars, determining the orbital speed is largely made possible thanks to the Doppler shift.

By measuring the change in frequency of the starlight over time, astronomers can calculate the speed at which each of the two stars in a binary system orbits their common center of mass. This is crucial for understanding the system's dynamics and the gravitational forces at play.

• The orbital speed is derived from the maximum change in frequency observed.
• Knowing the orbital speed helps determine the orbital radius, providing a clearer picture of the system's scale.

Understanding the orbital speed of the stars lets astronomers estimate other factors, such as the gravitational influence each star has on its companion, ultimately aiding in revealing the mass of each star in the system.

Mass of Stars

Determining the mass of stars in a spectroscopic binary system can be quite intriguing. This is another benefit provided by observing the Doppler-shifted light they emit. Once the orbital speed and period are known, one can calculate the stars' mass using several well-established formulas in celestial mechanics.

The mass calculation hinges on the gravitational force that keeps the stars in their orbit, which is fundamentally related to their orbital speed and the radius of their orbit. By rearranging the formula for gravitational force and considering the mechanics of circular motion, astronomers can estimate each star's mass in a binary system.

• Mass is inferred from the orbital speed, radius, and period of the orbit.
• Utilizing Kepler's laws and Newton's law of gravitation is essential for accurate calculations.

Understanding the masses of stars in binary systems is vital for gaining insights into their evolution, lifespan, and broader implications in the structure of the galaxy. Knowing their mass is not only crucial for stellar physics but also for modeling phenomena like supernovae, neutron stars, and black holes derived from stellar evolution.