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An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass \(m_{\mathrm{B}}\) revolves around its star of mass \(m_{\mathrm{A}}\) If no external force acts on this simple two-object system, then its \(\mathrm{CM}\) is stationary. Assume \(m_{\mathrm{A}}\) and \(m_{\mathrm{B}}\) are in circular orbits with radii \(r_{\mathrm{A}}\) and \(r_{\mathrm{B}}\) about the system's \(\mathrm{CM}\). (a) Show that \(r_{\mathrm{A}}=\frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} r_{\mathrm{B}}\) (b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, \(m_{\mathrm{B}}=1.0 \times 10^{-3} \mathrm{~m}_{\mathrm{A}}\) and the planet has an orbital radius of \(8.0 \times 10^{11} \mathrm{~m} .\) Determine the radius \(r_{\mathrm{A}}\) of the star's orbit about the system's \(\mathrm{CM}\). (c) When viewed from Earth, the distant system appears to wobble over a distance of \(2 r_{\mathrm{A}}\). If astronomers are able to detect angular displacements \(\theta\) of about 1 milliarcsec \(\left(1 \operatorname{arcsec}=\frac{1}{3600}\right.\) of a degree), from what distance \(d\) (in light-years) can the star's wobble be detected \(\left(11 \mathrm{y}=9.46 \times 10^{15} \mathrm{~m}\right) ?(d)\) The star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?

Step by step solution

01

Understanding the Center of Mass Concept

Before solving the problems, remember that the center of mass (CM) of a two-object system (such as a star and a planet) is stationary if no external forces act on the system. The positions of the star and the planet from the CM are denoted as \( r_{\mathrm{A}} \) and \( r_{\mathrm{B}} \) respectively. Recall that the CM condition can be mathematically expressed as: \( m_{\mathrm{A}} r_{\mathrm{A}} = m_{\mathrm{B}} r_{\mathrm{B}} \).

02

Solving Part (a) for \(r_{ ext{A}}\)

Using the equation for the center of mass: \( m_{\mathrm{A}} r_{\mathrm{A}} = m_{\mathrm{B}} r_{\mathrm{B}} \), we need to solve for \( r_{\mathrm{A}} \). Rearranging this gives us: \( r_{\mathrm{A}} = \frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} r_{\mathrm{B}} \). This shows that the distance of the star to the CM is proportional to the mass ratio and the planet's distance.

03

Calculating \(r_{ ext{A}}\) for Part (b)

For a Sun-like star and a Jupiter-like planet, we have \( m_{\mathrm{B}} = 1.0 \times 10^{-3} m_{\mathrm{A}} \) and \( r_{\mathrm{B}} = 8.0 \times 10^{11} \text{ m} \). Using the equation from part (a): \( r_{\mathrm{A}} = \frac{1.0 \times 10^{-3} m_{\mathrm{A}}}{m_{\mathrm{A}}} \times 8.0 \times 10^{11} = 8.0 \times 10^{8} \text{ m} \).

04

Finding Distance for Detecting Wobble in Part (c)

The wobble seen from Earth is given as \(2r_{\mathrm{A}}\) which is \(2 \times 8.0 \times 10^{8} \text{ m} = 1.6 \times 10^{9} \text{ m}\). The angular displacement \( \theta \) is \(1 \text{ milliarcsec} = \frac{1}{3600000} \text{ degrees} \). Convert angle in degrees to radians: \( \theta = \frac{1}{3600000} \times \frac{\pi}{180} \). The distance \( d \) can be calculated from: \( \theta = \frac{1.6 \times 10^{9}}{d} \). Solving for \( d \):\[ d = \frac{1.6 \times 10^{9}}{\theta} = \frac{1.6 \times 10^{9} \times 180}{\pi} \times 3600000 \approx 4 \text{ light-years (ly)}.\]

05

Estimating Number of Detectable Stars in Part (d)

With a detection distance of approximately 4 ly, consider a sphere with radius 4 ly centered on Earth. The volume of this sphere is \( V = \frac{4}{3}\pi(4)^3 \), which is approximately \(268\text{ ly}^3\). Assuming stars are uniformly distributed, and knowing roughly 0.1 stars per cubic light-year, the number of stars in such a volume is about \( 26.8 \approx 27 \) stars.

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Key Concepts

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

center of mass

The concept of the center of mass (CM) is crucial in understanding the motions within a star-planet system, like that of an extrasolar planet orbiting a star. In a system where no external forces are acting, the CM remains stationary. This means it's the balance point where the mass of the system can be considered to concentrate. Let's denote the star's mass as \(m_{\mathrm{A}}\) and its distance from the CM as \(r_{\mathrm{A}}\), while the planet's mass and distance are \(m_{\mathrm{B}}\) and \(r_{\mathrm{B}}\), respectively. An important relationship here is given by the equation:\[m_{\mathrm{A}} r_{\mathrm{A}} = m_{\mathrm{B}} r_{\mathrm{B}}.\]
This formula indicates that the farther an object is from the CM, the more its mass influences the system's balance. Specifically, solving for the star's position \( r_{\mathrm{A}} \) corresponds to \( r_{\mathrm{A}} = \frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} r_{\mathrm{B}} \).
Applying this to our system helps in calculating the wobble of the star as influenced by the planet and is essential for detecting such stars from Earth.

orbital mechanics

Orbital mechanics is the field of physics that studies the motion of objects in space, primarily affected by gravity. When we consider an extrasolar planet and its star, both bodies participate in a mutual orbit around their common center of mass (CM). This orbit is circular, meaning both bodies maintain a constant distance from the CM.
In this system, the distances \(r_{\mathrm{A}}\) for the star and \(r_{\mathrm{B}}\) for the planet define the radii of these orbits. The star and planet pull on each other through gravity, creating this complex dance. However, thanks to orbital mechanics, we can predict and calculate these positions and movements precisely.
For instance, knowing the mass ratio \( \frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} \), we can use it with the planet's orbital radius \( r_{\mathrm{B}} \) to determine the star's orbit radius \( r_{\mathrm{A}} \) using the formula \( r_{\mathrm{A}} = \frac{m_{\mathrm{B}}}{m_{\mathrm{A}}} r_{\mathrm{B}} \). This precise calculation is what allows astronomers to predict the observable wobble of stars caused by revolving planets.

angular displacement

Angular displacement is a measure of the angle through which an object moves on a circular path. In the context of detecting extrasolar planets, it refers to the apparent movement--or wobble--of a star due to a planet's gravitational pull.
When observing a distant star system, the angular displacement is the angle that the star appears to move across the sky. This measurement is usually small but observable with precise instruments. In our exercise, the wobble of the star appears as an angular displacement of \(1\) milliarcsecond, which is equivalent to \(\frac{1}{3600000}\) degrees.
To detect a star's wobble from Earth, astronomers convert this angular measurement to a physical distance using the formula \( \theta = \frac{1.6 \times 10^{9}}{d} \), where \( \theta \) is the angular displacement in radians and \(d\) is the distance to the star. Solving gives us the maximum distance, about 4 light-years, from which this wobble can be detected. This calculation is critical in identifying stars that might host undiscovered extrasolar planets.