A simple temperature model for the solar nebula disk is
Tdisk(a) = 20 � SQRT 100 AU/ a K, (1) where a should be expressed in AU in this equation.
(a) Find the temperature in the solar nebula disk at the orbit of Mer- cury, Earth and Jupiter.
(b) Where is the water ice line located in this disk?
(c) Mercury is mostly made of metal, the Earth mostly made of rock and Jupiter mostly made of gas. Are these three planetary bulk compositions consistent with the condensation sequence, if one follows the above temperature model for the Solar nebula?
To find the temperature in the solar nebula disk at the orbits of Mercury, Earth, and Jupiter, we need to substitute the respective values of "a" into the temperature model equation (1).
(a) The orbits of Mercury, Earth, and Jupiter are approximately 0.39 AU, 1 AU, and 5.2 AU from the Sun, respectively.
For Mercury (a = 0.39 AU):
Tdisk(0.39) = 20 * sqrt(100 / 0.39) K
To calculate this, you need to find the square root of (100 / 0.39) and multiply it by 20.
For Earth (a = 1 AU):
Tdisk(1) = 20 * sqrt(100 / 1) K
To calculate this, find the square root of (100 / 1) and multiply it by 20.
For Jupiter (a = 5.2 AU):
Tdisk(5.2) = 20 * sqrt(100 / 5.2) K
To calculate this, find the square root of (100 / 5.2) and multiply it by 20.
(b) The water ice line represents the distance from the Sun where the temperature drops low enough for water vapor to condense into ice. This occurs when the temperature in the disk reaches a certain threshold. To find the water ice line location, we need to know the temperature at which water vapor freezes into ice.
Typically, it is accepted that water vapor condenses into ice when the temperature drops below 150 Kelvin (K). So, we can set up the equation Tdisk(a) = 150 K and solve for "a" to find the location of the water ice line.
150 = 20 * sqrt(100 / a)
To solve this, divide both sides of the equation by 20 and then square both sides. Finally, divide both sides by 10 and square them. The resulting value of "a" will represent the location of the water ice line.
(c) To determine if the three planetary bulk compositions are consistent with the condensation sequence, we need to understand the condensation temperatures of different materials.
Metals tend to condense at higher temperatures, followed by rocks, and then gases. So, if the temperature at a particular location in the solar nebula is higher than the condensation temperature of a specific material, that material would not condense at that location.
Considering the temperature model for the solar nebula disk, if the temperatures at the orbits of Mercury, Earth, and Jupiter are below the condensation temperatures of metals, rocks, and gases respectively, then the three planetary compositions are consistent with the condensation sequence.