Universesandbox erika (talk | contribs) (→Properties: Added link to Surface Map) |
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− | At every [[ |
+ | At every [[N-Body_Simulation#Time_Step|time step]] during the simulation, Universe Sandbox calculates the change in the surface temperature across the surface of every [[Planetary Body]] object in the simulation. These calculations consider the effects of incoming heat from sources like nearby [[Star|stars]], the effects of cooling, and the spread of heat across the surface of the object. |
==Related Properties & Settings== |
==Related Properties & Settings== |
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===Properties=== |
===Properties=== |
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− | * The [[Effective Temperature]] and [[Average Albedo]] properties are listed in the Radiation section of the object's [[Surface Tab|Surface]] tab. |
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* The temperature of each point on the object's surface can be seen by hovering over that spot on the Temperature [[Surface Map|surface map]]. |
* The temperature of each point on the object's surface can be seen by hovering over that spot on the Temperature [[Surface Map|surface map]]. |
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* The [[Average Surface Temperature]] property of an object represents the average temperature over the entire surface of the object. |
* The [[Average Surface Temperature]] property of an object represents the average temperature over the entire surface of the object. |
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* The [[Minimum Temperature]] and [[Maximum Temperature]] represent the lowest and highest temperatures on the Temperature map, respectively. |
* The [[Minimum Temperature]] and [[Maximum Temperature]] represent the lowest and highest temperatures on the Temperature map, respectively. |
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+ | * Several properties related to the movement of energy into and out of the object, and the effect of this energy flow on temperature, are listed in the Energy Flow section of the object's properties panel. |
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===Settings=== |
===Settings=== |
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==Models== |
==Models== |
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− | === |
+ | ===Energy Flow=== |
− | + | The temperature of an object depends on the flow of energy in and out of the object. The rate at which energy is absorbed and radiated by the object, also known as the power, will affect the rate at which the temperature changes. |
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+ | |||
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− | + | For each point on the [[Surface Map|surface map]] of the object, Universe Sandbox performs the following calculations in each time step. First, Universe Sandbox calculates the rate of energy absorbed from various sources in the simulation (listed below). The total [[Energy Absorption Rate]], or absorption power, is the sum of the the rates from each source: |
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+ | :<math>P_{abs} = P_{stars} + P_{atmo} + P_{tidal} + P_{nova} + P_{clouds}. </math> |
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The power absorbed by an object can come from a number of sources, including: |
The power absorbed by an object can come from a number of sources, including: |
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* [[Stellar Heating]] from [[Star|stars]] in the simulation |
* [[Stellar Heating]] from [[Star|stars]] in the simulation |
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− | * [[ |
+ | * [[Atmosphere Heating]] from the object's atmosphere radiating heat back towards the surface |
− | * [[Radiogenic Heating]] due to radioactive decay within the object |
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* [[Tidal Heating]] created by internal tidal forces from [[gravity]] |
* [[Tidal Heating]] created by internal tidal forces from [[gravity]] |
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− | * [[ |
+ | * [[Supernova Heating]] from [[Supernova|supernovae]] in the simulation |
+ | * [[Supernova Drag Heating]] due to friction from passing [[Supernova Cloud]] objects |
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+ | |||
+ | In each time step, the each point on the object's surface will also radiate energy out into space. The [[Energy Radiation Rate]], or radiation power, is calculated from the temperature of the point using the [https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law Stefan-Boltzmann law]: |
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+ | where σ is the [https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_constant Stefan-Boltzmann constant] and ''R'' is the radius of the object. |
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===Surface Temperature=== |
===Surface Temperature=== |
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⚫ | |||
− | For each point on the surface map of the object, Universe Sandbox performs the following calculations in each time step. First, Universe Sandbox calculates the incoming energy from heat sources (listed above). Some of the energy from stars will be reflected back into space and some will be absorbed. The fraction of incoming stellar energy that is absorbed is calculated using the object's [[Average Albedo]]. |
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+ | Heat flows from areas of high temperature to areas of low temperature, so the heat in a single point on the surface map of an object may spread or [https://en.wikipedia.org/wiki/Diffusion diffuse] into nearby points, and the heat in those points may spread to the first point. Universe Sandbox calculates this diffusion by using the [https://en.wikipedia.org/wiki/FTCS_scheme FTCS method] to solve the [https://en.wikipedia.org/wiki/Heat_equation heat equation]. The speed of diffusion, which in reality depends on the material that makes up the surface, is determined by the simulation's [[Sim Settings Menu#Heat Diffusion Rate|Heat Diffusion Rate]]. The temperatures of each point on the surface map are then corrected based on the results of this diffusion calculation. |
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+ | |||
+ | The [[Average Surface Temperature]] is calculated by averaging the surface temperatures of each point on the surface, weighted by the area of each point. |
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+ | |||
+ | ===Equilibrium and Effective Temperatures=== |
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+ | |||
+ | As described above, the change of temperature in each time step depends on the flow of energy into and out of the object. When the absorbed power exactly equals the radiated power, the object is in [https://en.wikipedia.org/wiki/Thermal_equilibrium thermal equilibrium], and its temperature will not change. Since the radiated power depends on the temperature of the object, there is a certain temperature where this equilibrium occurs, <math>T_{eq}</math>. At the [[Equilibrium Temperature]], the radiated power equals the absorbed power: |
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+ | :<math>P_{rad} = P_{abs}</math> |
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+ | :<math>\sigma T_{eq}^4 4 \pi R^2 = P_{stars} + P_{atmo} + P_{tidal} + P_{nova} + P_{clouds}</math> |
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+ | |||
+ | The [[Effective Temperature]] is the equilibrium temperature that the object would have if it were a perfect blackbody. In the case of the Universe Sandbox simulation, this means the equilibrium temperature the object would have without an atmosphere: |
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+ | :<math>\sigma T_{eff}^4 4 \pi R^2 = P_{stars} + P_{nova} + P_{tidal} + P_{clouds}</math> |
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+ | |||
+ | If the object does not have an atmosphere, the Equilibrium Temperature will equal the Effective Temperature. |
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+ | |||
+ | ==Limitations== |
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+ | |||
+ | ===Heating/Cooling=== |
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+ | |||
+ | The rate at which an object radiates energy into space depends strongly on its temperature (in fact, it depends on temperature to the fourth power). But the rate at which the temperature changes depends on the energy radiation rate. This makes it challenging to accurately predict what the temperature of an object will be far in the future. Instead, Universe Sandbox approximates the temperature change as a linear change in each time step. This is similar to how Universe Sandbox approximates the motion of objects as being linear in each time step in the [[N-Body Simulation]]. As with the N-body simulation, this approximation works best for small time steps and slow rates of change. |
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+ | At large time steps and/or large heating and cooling rates, the temperature calculations described above will become less accurate. Most notably, the errors in the temperature calculations for each point may affect the Average Surface Temperature, leading to errors in the Heating Rate. |
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− | Some of this absorbed energy will be re-radiated back into space. If the object has no atmosphere, the output power, or the rate at which energy is radiated away, is calculated from the temperature of the point using the [[https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law|Stefan-Boltzmann law]]. If the object does have an atmosphere, the output power is calculated from the temperature of the point and the [[Greenhouse Effect]] of the object <ref name="Kleeman">Kleeman, Richard (2017). [https://math.nyu.edu/faculty/kleeman/zero_dim_ebm.html Zero-Dimensional Energy Balance Model]</ref>. |
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+ | ===Heat Diffusion=== |
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⚫ | |||
⚫ | |||
− | + | The simulation of heat spreading from hotter regions of the surface to cooler regions also depends on the time step. At higher time steps, the [https://en.wikipedia.org/wiki/FTCS_scheme FTCS method] that Universe Sandbox uses to solve the [https://en.wikipedia.org/wiki/Heat_equation heat equation] becomes unstable and extremely inaccurate. At these higher time steps, Universe Sandbox will artificially lower the speed of heat diffusion to avoid instability in the solution. |
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[[Category:Heating]] |
[[Category:Heating]] |
Revision as of 19:23, 30 October 2020
At every time step during the simulation, Universe Sandbox calculates the change in the surface temperature across the surface of every Planetary Body object in the simulation. These calculations consider the effects of incoming heat from sources like nearby stars, the effects of cooling, and the spread of heat across the surface of the object.
Related Properties & Settings
Properties
- The temperature of each point on the object's surface can be seen by hovering over that spot on the Temperature surface map.
- The Average Surface Temperature property of an object represents the average temperature over the entire surface of the object.
- The Minimum Temperature and Maximum Temperature represent the lowest and highest temperatures on the Temperature map, respectively.
- Several properties related to the movement of energy into and out of the object, and the effect of this energy flow on temperature, are listed in the Energy Flow section of the object's properties panel.
Settings
- Tidal Heating can be turned off or increased in the Sim Settings Menu.
- Collision Heating can be turned off in the Sim Settings Menu.
- The Heat Diffusion Rate can be set in the Sim Settings Menu, or using the Heating button in the Sim menu in the bottom bar.
Models
Energy Flow
The temperature of an object depends on the flow of energy in and out of the object. The rate at which energy is absorbed and radiated by the object, also known as the power, will affect the rate at which the temperature changes.
For each point on the surface map of the object, Universe Sandbox performs the following calculations in each time step. First, Universe Sandbox calculates the rate of energy absorbed from various sources in the simulation (listed below). The total Energy Absorption Rate, or absorption power, is the sum of the the rates from each source:
The power absorbed by an object can come from a number of sources, including:
- Stellar Heating from stars in the simulation
- Atmosphere Heating from the object's atmosphere radiating heat back towards the surface
- Tidal Heating created by internal tidal forces from gravity
- Supernova Heating from supernovae in the simulation
- Supernova Drag Heating due to friction from passing Supernova Cloud objects
In each time step, the each point on the object's surface will also radiate energy out into space. The Energy Radiation Rate, or radiation power, is calculated from the temperature of the point using the Stefan-Boltzmann law:
where σ is the Stefan-Boltzmann constant and R is the radius of the object.
Surface Temperature
The change in the temperature of each point on the object's surface, ΔT, is calculated using the Energy Absorption Rate, Pabs, the Energy Radiation Rate, Prad, the time step, Δt, and the object's Surface Heat Capacity, C, using
Heat flows from areas of high temperature to areas of low temperature, so the heat in a single point on the surface map of an object may spread or diffuse into nearby points, and the heat in those points may spread to the first point. Universe Sandbox calculates this diffusion by using the FTCS method to solve the heat equation. The speed of diffusion, which in reality depends on the material that makes up the surface, is determined by the simulation's Heat Diffusion Rate. The temperatures of each point on the surface map are then corrected based on the results of this diffusion calculation.
The Average Surface Temperature is calculated by averaging the surface temperatures of each point on the surface, weighted by the area of each point.
Equilibrium and Effective Temperatures
As described above, the change of temperature in each time step depends on the flow of energy into and out of the object. When the absorbed power exactly equals the radiated power, the object is in thermal equilibrium, and its temperature will not change. Since the radiated power depends on the temperature of the object, there is a certain temperature where this equilibrium occurs, . At the Equilibrium Temperature, the radiated power equals the absorbed power:
The Effective Temperature is the equilibrium temperature that the object would have if it were a perfect blackbody. In the case of the Universe Sandbox simulation, this means the equilibrium temperature the object would have without an atmosphere:
If the object does not have an atmosphere, the Equilibrium Temperature will equal the Effective Temperature.
Limitations
Heating/Cooling
The rate at which an object radiates energy into space depends strongly on its temperature (in fact, it depends on temperature to the fourth power). But the rate at which the temperature changes depends on the energy radiation rate. This makes it challenging to accurately predict what the temperature of an object will be far in the future. Instead, Universe Sandbox approximates the temperature change as a linear change in each time step. This is similar to how Universe Sandbox approximates the motion of objects as being linear in each time step in the N-Body Simulation. As with the N-body simulation, this approximation works best for small time steps and slow rates of change.
At large time steps and/or large heating and cooling rates, the temperature calculations described above will become less accurate. Most notably, the errors in the temperature calculations for each point may affect the Average Surface Temperature, leading to errors in the Heating Rate.
Heat Diffusion
The simulation of heat spreading from hotter regions of the surface to cooler regions also depends on the time step. At higher time steps, the FTCS method that Universe Sandbox uses to solve the heat equation becomes unstable and extremely inaccurate. At these higher time steps, Universe Sandbox will artificially lower the speed of heat diffusion to avoid instability in the solution.