Problem 47 - College Physics

Question

\(\bullet\) Detecting planets around other stars. Roughly 500 planets have so far been detected beyond our solar system. This is accomplished by looking for the effect the planet has on the star. The star is not truly stationary; instead, it and its planets orbit around the center of mass of the system. Astronomers can measure this wobble in the position of a star.(a) For a star with the mass and size of our sun and having a planet with five times the mass of Jupiter, where would the center of mass of this system be located, relative to the center of the star, if the distance from the star to the planet was the same as the distance from Jupiter to our sun? (Consult Appendix E.) (b) If the planet had earth's mass, where would the center of mass of the system be located if the planet was just as far from the star as the earth is from the sun? (c) In view of your results in parts (a) and (b), why is it much easier to detect stars having large planets rather than small ones?

Step-by-Step Solution

Answer

The center of mass shifts more noticeably with larger planets, making their detection easier compared to smaller planets like Earth.

Step 1: Understanding the Center of Mass Concept

The center of mass for a two-body system such as a star and a planet can be calculated using the formula for the center of mass of a two-object system along a line. The formula is given by \( R_{cm} = \frac{m_1 imes r_1 + m_2 imes r_2}{m_1 + m_2} \), where \(m_1\) is the mass of the star, \(m_2\) is the mass of the planet, \(r_1\) is the distance of the star from the center of mass, and \(r_2\) is the distance of the planet from the center of mass. Typically, \(r_2\) can be set as the distance between the star and the planet since we are finding \(R_{cm}\) from the star's center.

Step 2: Finding the Center of Mass for Part (a)

For a star with the mass of the sun \( (M_{sun}) \) and a planet with five times the mass of Jupiter \( (5M_{Jupiter}) \), positioned at the same distance as Jupiter from the sun \( (r_{Jupiter}) \): 1. Calculate the position of the center of mass relative to the star using the formula: \( R_{cm} = \frac{M_{sun} \times 0 + 5M_{Jupiter} \times r_{Jupiter}}{M_{sun} + 5M_{Jupiter}} \).2. Given that \( M_{Jupiter} \approx 0.001 M_{sun} \), substitute this into the equation: \( R_{cm} = \frac{5 \times 0.001M_{sun} \times r_{Jupiter}}{M_{sun} + 5 \times 0.001M_{sun}} \).3. Simplifying further: \( R_{cm} = \frac{0.005M_{sun} \times r_{Jupiter}}{1.005M_{sun}} \approx \frac{0.005}{1.005}r_{Jupiter} \approx 0.00498 r_{Jupiter} \).

Step 3: Finding the Center of Mass for Part (b)

For a star with the mass of the sun and a planet with the mass of Earth \( (M_{Earth}) \), located at the same distance as Earth from the sun \( (r_{Earth}) \):1. Use the center of mass formula: \( R_{cm} = \frac{M_{sun} \times 0 + M_{Earth} \times r_{Earth}}{M_{sun} + M_{Earth}} \).2. Substitute \( M_{Earth} \approx 0.000003 M_{sun} \): \( R_{cm} = \frac{0.000003 M_{sun} \times r_{Earth}}{M_{sun} + 0.000003 M_{sun}} \).3. This simplifies to: \( R_{cm} = \frac{0.000003}{1.000003}r_{Earth} \approx 0.000003 r_{Earth} \).

Step 4: Comparing Results and Drawing Conclusions

In part (a), the center of mass is located at about 0.00498 times the distance from the star to the planet. In part (b), it is about 0.000003 times the distance. The center of mass is much closer to the star in part (b), making the star's wobble much less noticeable when only small planets are present. Thus, larger planets cause more noticeable star movement, making detection easier.

Key Concepts

Center of Mass

The center of mass is a crucial concept in understanding how stars and planets interact in a system. In a two-body system, like a star and a planet, both bodies orbit around a common center of mass. This center is the point where the combined mass of the system is evenly distributed. What makes it particularly interesting is that it doesn't have to be located within either body; it could actually lie in the space between them.

To find the center of mass, you can use the formula:

• \( R_{cm} = \frac{m_1 \times r_1 + m_2 \times r_2}{m_1 + m_2} \)

Here, \(m_1\) and \(m_2\) are the masses, and \(r_1\) and \(r_2\) are their respective distances from the center of mass. This equation helps calculate the precise location of the center of mass by considering how the masses are distributed in relation to each other.

In practical terms, when observing stars and their planets, the center of mass shifts based on the mass and distance of each body, causing the star to wobble as it orbits this shared point.

Astrophysics

Astrophysics is the branch of science that explores the physical and chemical properties of celestial bodies. It's like the detective story of the universe. Astrophysicists investigate how stars, planets, and galaxies form and evolve, employing concepts from particle physics to gravitational dynamics.

A fundamental aspect of astrophysics is understanding how forces like gravity influence the motions and interactions of celestial objects. By observing these dynamics, scientists can infer a lot about the characteristics and behaviors of celestial bodies—including their masses, compositions, and distances from each other.

In the context of exoplanet detection, astrophysics comes into play through methods analyzing the movements of stars. This is because stars don't remain stationary as they orbit; they wobble due to the gravitational pull of orbiting planets. By studying this wobble, scientists can learn about the existence and attributes of exoplanets even if they cannot see them directly.

Gravitational Wobble

Gravitational wobble is a fascinating phenomenon that occurs when a star is influenced by the gravitational pull of nearby planets. This wobble is, in essence, a subtle movement or shift in the star's position as it orbits the system's center of mass alongside its planets.

The size of the wobble depends on several factors, including:

• The mass of the planet: A larger planet exerts a stronger gravitational force, producing a more pronounced wobble.
• The distance between the star and the planet: The farther apart they are, the larger the orbit and, consequently, the more noticeable the wobble.

Detecting this wobble involves precise measurements over time, usually through observable changes in the star's spectral lines as it moves slightly towards and away from Earth. By observing these shifts, scientists can infer the presence of a planet, estimate its mass, and determine its distance from the star.

Therefore, studying gravitational wobble provides critical insights into the discovery of planets beyond our solar system, contributing significantly to our understanding of the universe.

Planetary Mass and Distance

Understanding planetary mass and distance is vital when studying exoplanets and their detection. It affects how a planet influences its host star and how easily it can be detected by observers on Earth.

**Mass**: The mass of a planet directly affects its gravitational pull on its star. A more massive planet will exert a stronger force, resulting in a more noticeable gravitational wobble. This increased wobble makes it easier to detect the presence of the planet. For instance, a planet five times the mass of Jupiter will cause a significant wobble compared to a planet the size of Earth.

**Distance**: The distance between a planet and its star also plays a significant role. A planet that is further away will have less gravitational influence, causing a smaller wobble. However, if the system's center of mass remains within or near the star, the wobble might be harder to detect. This is particularly true for small planets located at greater distances from their subject stars.

In conclusion, larger planets closer to their stars are more easily detectable due to the observable impact on their star's movements, emphasizing the importance of planetary mass and distance in exoplanetary studies.