Consider the planet COROT-11b. It was discovered using the transit method, and astronomers have followed up with radial velocity measurements, so both its size (radius 1.43 Jupiter radii, or \(R_{\mathrm{J}}\) ) and its mass \(\left(2.33 \mathrm{M}_{\mathrm{J}}\right)\) are known. Using this information, you can find the density, which 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?

Step by step solution

01

Calculate Mass in Kilograms

Convert the mass of the planet from Jupiter masses \(\text{M}_{\text{J}}\) to kilograms. Given that 1 Jupiter mass \(\text{M}_{\text{J}}\) is approximately \2.00 \times 10^{27}\ kg, calculate the mass of COROT-11b. \[2.33 \text{M}_{\text{J}} \times 2.00 \times 10^{27} \text{ kg/M}_{\text{J}} = 4.66 \times 10^{27} \text{ kg}\]

02

Convert Radius to Meters

Convert the radius of the planet from Jupiter radii \(\text{R}_{\text{J}}\) to meters. Given that 1 Jupiter radius \(\text{R}_{\text{J}}\) is approximately \7.15 \times 10^{7}\ meters, calculate the radius of COROT-11b. \[1.43 \text{R}_{\text{J}} \times 7.15 \times 10^{7} \text{ m/R}_{\text{J}} = 1.02145 \times 10^{8} \text{ m}\]

03

Find the Planet's Volume

Calculate the volume of the planet using the formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \). Substitute the radius you found in Step 2. \[V = \frac{4}{3} \pi \(1.022 \times 10^8 \text{ m} \)^3 \approx 4.45 \times 10^{24} \text{ m}^3 \]

04

Calculate the Density

Density is mass divided by volume. Use the mass from Step 1 and the volume from Step 3 in the formula \( \rho = \frac{m}{V} \). \[ \rho = \frac{4.66 \times 10^{27} \text{ kg}}{4.45 \times 10^{24} \text{ m}^3} \approx 1048 \text{ kg/m}^3 \]

05

Compare Density with Water

Compare the calculated density with the density of water (1000 kg/m³). Since the planet's density (1048 kg/m³) is slightly higher than that of water, COROT-11b is likely to be gaseous because it is less dense than rocky planets.

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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 one of the most effective ways to discover exoplanets. It involves observing the slight dimming of a star that occurs when a planet passes in front of it. This temporary reduction in brightness, or 'transit,' provides crucial data about the planet's size and orbit.
If done multiple times, we can confirm the planet's existence and estimate its radius based on how much light it blocks.
Detecting small dimming requires highly sensitive instruments, but it allows astronomers to measure very small planets, even Earth-sized ones.

radial velocity

Radial velocity measurements help determine a planet's mass by observing the 'wobble' of its star. As a planet orbits, its gravitational pull causes the star to move in a small orbit or wobble.
These movements cause changes in the star's light spectrum due to the Doppler effect. When the star moves towards us, the light shifts to blue; when it moves away, it shifts to red.
By measuring these shifts, astronomers can calculate the planet's mass. This method gives us vital information about the planet's gravitational influence on its star.

planetary mass conversion

To understand a planet's properties, we often convert its mass into more familiar units. For instance, the mass of COROT-11b is given as 2.33 Jupiter masses.Given that 1 Jupiter mass (\text{M}_{\text{J}}) is approximately 2.00 \times 10^{27}\text{ kg}, we multiply:\[2.33 \text{M}_{\text{J}} \times 2.00 \times 10^{27} \text{ kg/M}_{\text{J}} = 4.66 \times 10^{27}\text{ kg}\]This conversion helps in making realistic comparisons with other known masses, like the mass of Earth or other celestial objects.

planetary radius conversion

Converting a planet’s radius from Jupiter radii to meters is crucial for further calculations. For example, the radius of COROT-11b is given as 1.43 Jupiter radii:1 \text{R}_{\text{J}}\text{ is approximately } 7.15 \times 10^{7}\text{ meters}Calculate as follows:\[1.43 \text{R}_{\text{J}}\times 7.15 \times 10^{7} \text{ m/R}_{\text{J}} = 1.02145 \times 10^{8} \text{ m}\]This radius in meters can now be used to find other physical properties like the planet's volume.

volume of a sphere

To find the volume of a spherical planet, we use the formula:\[V = \frac{4}{3} \times \text{π} \times r^3\]Substitute the radius from the previous calculation (1.02145 \times 10^{8} meters):\[V = \frac{4}{3} \times \text{π} \times (1.02145 \times 10^8 \text{ m})^3 \ \text{= } 4.45 \times 10^{24}\text{ m}^3\]This gives us the planet's volume, a value that can be directly used to calculate its density.

density comparison

Density is crucial to understanding what a planet might be made of. It is calculated as mass divided by volume:\[ρ = \frac{m}{V}\]For COROT-11b:\[ρ = \frac{4.66 \times 10^{27} \text{ kg}}{4.45 \times 10^{24} \text{ m}^3} ≈ 1048 \text{ kg/m}^3\]Comparing this to the density of water (1000 kg/m³), we see that COROT-11b has a density slightly higher than water.
This suggests it may be composed mainly of gas, as rocky planets tend to have much higher densities.

gaseous vs rocky planets

Determining whether a planet is gaseous or rocky involves comparing its density to known materials:

• **Gaseous planets** (like Jupiter) have low densities because they are composed mainly of hydrogen and helium.
  They are much less dense than water.
• **Rocky planets** (like Earth) have higher densities, often around 3000-5500 kg/m³.
  This is due to their solid iron and silicon-based components.

Given COROT-11b's density (1048 kg/m³), it is less dense than rocky planets but more dense than expected for a completely gaseous planet.
Therefore, it is likely to have a combination of gas and some heavier elements.