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 answer these questions, we need to substitute the relevant values into the temperature model equation (Tdisk(a) = 20 × √(100 AU/a K)). Here are the steps for each question:
(a) Finding the temperature at the orbits of Mercury, Earth, and Jupiter:
- First, we need to determine the values of "a" for each planet's orbit.
- The distance between the Sun and Mercury's orbit is approximately 0.39 AU.
- The distance between the Sun and Earth's orbit is approximately 1 AU.
- The distance between the Sun and Jupiter's orbit is approximately 5.2 AU.
- Now, substitute these values into the temperature model equation and calculate the temperatures.
- For Mercury: Tdisk(0.39) = 20 × √(100/0.39) K
- For Earth: Tdisk(1) = 20 × √(100/1) K
- For Jupiter: Tdisk(5.2) = 20 × √(100/5.2) K
(b) Finding the location of the water ice line:
- The water ice line represents the distance from the Sun where the temperature in the solar nebula disk falls below the freezing point of water, allowing water to condense into solid ice.
- To find this location, we need to set the temperature (Tdisk) equal to the freezing point of water (0 degrees Celsius or 273.15 Kelvin) and solve for "a" in the temperature model equation: 273.15 = 20 × √(100/a).
- Square both sides and solve for "a": a = (20^2 × 100) / 273.15^2 AU.
(c) Determining if the planetary bulk compositions are consistent with the condensation sequence:
- The condensation sequence determines the order in which different elements and compounds condense from a gaseous state to solid state at various temperatures. This sequence is influenced by the temperature distribution in the solar nebula disk.
- To assess the consistency of planetary compositions, we need to compare the temperatures at which different materials condense with the temperatures at the respective planet's orbital distances.
- Consult a condensation sequence table or chart to find the condensation temperatures for the relevant materials (metal, rock, and gas) and compare them with the temperatures at each planet's orbit obtained from the temperature model equation.
- If the condensation temperatures for the observed materials are consistent with or lower than the corresponding temperatures at the planet's orbit, then the bulk compositions are consistent with the condensation sequence.
Remember to substitute the values and perform the calculations to get the specific answers to each question.