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Chapter 7: Problem 45

The planet COROT-11b was discovered using the transit method, and astronomers have followed up with radial velocity measurements, so both its size (radius \(1.43 R_{\text {Jup }}\) ) and its \(\operatorname{mass}\left(2.33 M_{\text {Jup }}\right)\) are known. The density provides a clue about whether the object is gaseous or rocky. a. What is the mass of this planet in kilograms? b. What is the planet's radius in meters? c. What is the planet's volume? d. What is the planet's density? How does this density compare to the density of water \(\left(1,000 \mathrm{kg} / \mathrm{m}^{3}\right) ?\) Is the planet likely to be rocky or gaseous?

Short Answer

Expert verified
The mass is \(4.42 \times 10^{27}\) kg, radius is 102,230,560 meters, volume is \(4.47 \times 10^{24}\) m³, and density is 988 kg/m³, indicating a gaseous planet.

Step by step solution

01

Convert mass to kilograms

Given the mass of the planet is \(2.33 M_{\text{Jup}}\). The mass of Jupiter \(M_{\text{Jup}}\) is approximately \(1.898 \times 10^{27} \text{kg}\). Thus, the mass of the planet in kilograms is: \[2.33 \times 1.898 \times 10^{27} = 4.41934 \times 10^{27} \text{kg}\]
02

Convert radius to meters

Given the radius of the planet is \(1.43 R_{\text{Jup}}\). The radius of Jupiter \(R_{\text{Jup}}\) is approximately \(71,492,000\) meters. So, the planet's radius in meters is: \[1.43 \times 71,492,000 \approx 102,230,560 \text{m}\]
03

Calculate the planet's volume

The volume \(V\) of a sphere is given by \(V = \frac{4}{3}\pi r^3\). Using the radius calculated in Step 2: \[V = \frac{4}{3}\pi (102,230,560)^3 \approx 4.47391 \times 10^{24} \text{m}^3\]
04

Calculate the planet's density

Density \(\rho\) is given by \ \rho = \frac{mass}{volume} \. Using the mass from Step 1 and the volume from Step 3: \[\rho = \frac{4.41934 \times 10^{27}}{4.47391 \times 10^{24}} \approx 988 \text{kg/m}^3\]
05

Compare the density to water

Water has a density of \(1,000 \text{kg/m}^3\). The planet's density is \(988 \text{kg/m}^3\), which is slightly less than water. This suggests that the planet is likely gaseous rather than rocky.

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Key Concepts

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

Transit Method
The transit method is a popular technique for detecting exoplanets. It involves monitoring the brightness of a star over time. When a planet passes (or transits) in front of the star, it causes a slight dip in the star’s brightness. This dip can be measured and used to determine the size of the planet.

By observing these transits, astronomers can gather important data:
  • The planet's radius.
  • The orbital period (how long it takes the planet to orbit its star).
The duration and depth of the transit provide clues about the planet's size and orbit.
Radial Velocity Measurements
Radial velocity measurements help determine a planet's mass. When a planet orbits a star, it causes the star to wobble slightly due to gravitational pull. By measuring the star's spectrum and detecting these wobbles, scientists can calculate the planet's mass.

Here's how it works:
  • If the star moves toward us, its light shifts to shorter wavelengths (blue shift).
  • If the star moves away, its light shifts to longer wavelengths (red shift).
Combining radial velocity data with transit method data gives a fuller picture of the exoplanet's characteristics.
Planetary Mass
Planetary mass is a key factor in understanding an exoplanet's characteristics. The mass of COROT-11b was determined to be 2.33 times that of Jupiter (2.33 M\text{Jup}).

To find the mass in kilograms, we use the mass of Jupiter as a reference:

Given that the mass of Jupiter (M\text{Jup}) is approximately \(1.898 \times 10^{27} \text{kg}\), the mass of COROT-11b is calculated as:
\(2.33 \times 1.898 \times 10^{27} \approx 4.41934 \times 10^{27} \text{kg}.\)
Planetary Radius
The radius of COROT-11b was determined using the transit method, yielding 1.43 times the radius of Jupiter (1.43 R\text{Jup}).

To convert this to meters:
Given that the radius of Jupiter (R\text{Jup}) is approximately \(71,492,000\) meters, the radius of COROT-11b is:
\(1.43 \times 71,492,000 \approx 102,230,560 \text{m.}\). This information is vital for calculating the planet's volume.
Planetary Volume
The volume of a planet is calculated using the formula for the volume of a sphere:

\(V = \frac{4}{3}\pi r^3\)
Using the radius of COROT-11b (102,230,560 meters) calculated earlier, we find:
\(V \approx \frac{4}{3}\pi (102,230,560)^3 \approx 4.47391 \times 10^{24} \text{m}^3\).

Volume plays a crucial role in determining the planet's density, which helps in figuring out its composition.
Planetary Density
Planetary density is a key factor in understanding the planet's composition. Density (\(\rho\)) is calculated using the formula:

\(\rho = \frac{mass}{volume}\)
For COROT-11b, given its mass (\(4.41934 \times 10^{27} \text{kg}\)) and volume (\(4.47391 \times 10^{24} \text{m}^3\)), the density is:
\(\rho \approx \frac{4.41934 \times 10^{27}}{4.47391 \times 10^{24}} \approx 988 \text{kg/m}^3\).
Comparing this to the density of water (1000 \text{kg/m}^3), COROT-11b has a lower density, suggesting it is likely a gaseous planet.
Rocky vs Gaseous Planets
The density of a planet can give clues about its composition:

  • Rocky planets (like Earth) have higher densities, usually greater than 3000 kg/m^3, due to their solid, metal-rich composition.
  • Gaseous planets (like Jupiter) have lower densities, typically around 1000 kg/m^3 or less, because they are mostly made up of hydrogen and helium.
Given that COROT-11b has a density of 988 kg/m^3, it falls in the range of gaseous planets, suggesting it is primarily composed of lighter gases rather than rocky materials.

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Most popular questions from this chapter

Go to the "Super Planet Crash" Web page (http://www stefanom.org/spc/ or http://apod.nasa.gov/apod/ap150112 .html). Read "Help" to see the rules. First build a system like ours with four Earth-sized planets in the inner 2 AU-is this stable? What happens if you add in super-Earths or "ice giants"? Build up a few completely different planetary systems and see what happens. What types of situations cause instability in the inner 2 AU of these systems?

How does the law of conservation of angular momentum control a figure-skater's rate of spin?

Space missions: a. Go to the website for the Kepler Mission (http://kepler . nasa.gov). How many confirmed planets has Kepler discovered? Mouse over "confirmed planets": How many planet candidates are there? What kinds of follow-up observations are being done to verify whether the candidates are planets? What is new? b. Search for the latest version of the "Kepler Orrery," an animation that shows multiplanet systems discovered by Kepler. Do most of these systems look like our own? c. Go to the website for the European Space Agency (ESA) mission Gaia (http://sci.esa.int/gaia). This mission was launched in \(2013 .\) Click on the "Exoplanets" link on the left-hand side. What method(s) will GAIA use to look for planets? What are the science goals? Have some planets been found?

The spectroscopic radial velocity method preferentially detects a. large planets close to the central star b. small planets close to the central star. c. large planets far from the central star. d. small planets far from the central star. e. the method detects all of these equally well

Why is it so difficult for astronomers to obtain an image of an extrasolar planet?
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