Equivalent potential temperature calculator
Compare moist air parcels on one scale: air at 300 K and 1000 hPa holding 10 g/kg of vapour has an equivalent potential temperature of 331.127577766795 K. Humidity and lifting condensation level come with it. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- Creativity
- 300
- Pressure
- 1,000
- Mixing ratio
- 0.01
- Temperature c
- 26.85 K
- Mixing ratio gkg
- 10
- Vapor pressure
- 15.8227848101
- Dry pressure
- 984.1772151899
Equivalent potential temperature, often written theta-e or θe, is the temperature a moist air parcel would reach if all of its water vapor condensed out through a moist-adiabatic process and the parcel were then brought dry-adiabatically to a standard reference pressure. Forecasters and students use θe to compare parcels from different heights, to sketch convective available potential energy ideas, and to diagnose warm moist tongues that feed thunderstorms. This equivalent potential temperature calculator implements the widely cited Bolton 1980 analytic approximation: it accepts absolute temperature in kelvin, station or parcel pressure in hectopascals, and water-vapor mixing ratio in kilograms per kilogram of dry air, then returns θe in kelvin together with vapor pressure, relative humidity, lifting-condensation-level temperature, and the intermediate dry potential temperature of the lifted parcel.
What equivalent potential temperature measures and why Bolton’s formula is used
Potential temperature alone removes the effect of dry-adiabatic pressure change so that two dry parcels can be compared on an equal footing. Moist air carries latent heat that is released when vapor condenses, so a second conserved quantity is needed when condensation matters. Equivalent potential temperature folds that latent-heat contribution into a single Kelvin-scale number: larger θe means a warmer and/or moister parcel in the sense relevant to deep convection.
Bolton’s 1980 Monthly Weather Review paper gave a compact algebraic path that avoids iterative moist-adiabatic integration for routine analysis. The steps used here follow that tradition. First the vapor pressure is recovered from the mixing ratio with e equals p times r over epsilon plus r, where epsilon is about 0.622.
Saturation vapor pressure at the given absolute temperature uses Bolton’s exponential formula in Celsius. Relative humidity is e over es. The lifting condensation level temperature TL is then obtained from Bolton’s closed form involving the logarithm of relative humidity.
A dry potential temperature of the lifted parcel at the dry-air pressure p minus e is raised to a kappa-scaled exponent that weakly depends on mixing ratio in grams per kilogram. Finally θe multiplies that intermediate potential temperature by an exponential of a linear function of one over TL times the mixing ratio. When the mixing ratio is exactly zero the moist terms vanish and θe collapses to ordinary potential temperature T times one thousand over p to the power kappa.
Where equivalent potential temperature helps forecasters
Synoptic and mesoscale labs need a deterministic θe oracle so students can check hand calculations of parcel buoyancy proxies without spreadsheet rounding disagreements. Research notebooks that re-process archived soundings often want Bolton’s approximation specifically for comparison with historical papers rather than a full iterative CAPE integrator. Keep temperature in kelvin, pressure in hPa, mixing ratio non-negative in kg/kg, pin the rounded kelvin outputs in tests, and you get reproducible Bolton equivalent potential temperatures for courses, docs, and automation without standing up a meteorological computation stack of your own.
What you can do with it
Convective analysis and parcel comparison
Compute Bolton θe from T, p, and mixing ratio to compare moist parcels and highlight high-θe air feeding thunderstorms.
Sounding homework and lab oracles
Give students a deterministic equivalent potential temperature check against textbook Bolton steps without spreadsheet drift.
Archive reanalysis and dashboards
Batch-evaluate station or model levels for θe columns in research notebooks and operational moisture-return displays.
FAQ
How do I calculate equivalent potential temperature with this tool?
Send absolute temperature in kelvin, pressure in hPa, and water-vapor mixing ratio in kg/kg.
What units should I use for temperature, pressure, and mixing ratio?
Temperature must be absolute kelvin (add 273.15 to Celsius). Pressure is hectopascals (millibars).
Why are non-positive temperature or pressure rejected?
Absolute temperature and pressure must be positive for thermodynamic potential-temperature formulae.
What happens when the mixing ratio is zero?
The moist exponential terms drop out and θe equals ordinary dry potential temperature T·(1000/p)^0.2854, still returned in kelvin.
What does a higher equivalent potential temperature mean?
A warmer or moister parcel in the sense that matters for deep convection. Forecasters watch tongues of high θe air that feed thunderstorms and use it to compare parcels from different heights on one scale.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/earth/equivalent-potential-temperature \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"temperature":300,"pressure":1000,"mixing_ratio":0.01}'const res = await fetch("https://api.kit.forhosting.com/earth/equivalent-potential-temperature", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"temperature": 300,
"pressure": 1000,
"mixing_ratio": 0.01
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/earth/equivalent-potential-temperature",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"temperature": 300,
"pressure": 1000,
"mixing_ratio": 0.01
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/earth/equivalent-potential-temperature", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"temperature":300,"pressure":1000,"mixing_ratio":0.01}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"temperature":300,"pressure":1000,"mixing_ratio":0.01}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/earth/equivalent-potential-temperature", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"temperature": 300,
"pressure": 1000,
"mixing_ratio": 0.01
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "earth.equivalent_potential_temperature",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Non-positive absolute temperature or pressure is rejected with invalid_input so automation never silently emits a nonsense Kelvin value.
What equivalent potential temperature measures and why Bolton’s formula is used
This endpoint returns those intermediates so teaching notebooks can show the pipeline rather than only a black-box number.
Input contract, units, response shape, and error behaviour
Absolute temperature must be strictly positive: zero or negative kelvin is rejected because the thermodynamic formulae are undefined for non-positive absolute temperature. Pressure must likewise be strictly positive. Mixing ratio must be non-negative; a negative moisture content is not physical and is rejected. Example: three hundred kelvin at one thousand hectopascals with 0.01 kg/kg moisture yields a θe several tens of kelvin above the dry potential temperature, as expected for a humid tropical boundary-layer parcel. POST a JSON object with three required finite numbers. temperature is absolute air temperature T in kelvin; convert from Celsius with T_K equals T_C plus 273.15 before calling if your source data is in degrees. pressure is total pressure p in hectopascals (millibars)—convert pascals by dividing by one hundred. mixing_ratio is water-vapor mixing ratio r in kilograms of vapor per kilogram of dry air; ten grams per kilogram is 0.01 kg/kg. Each field may arrive as a JSON number or a numeric string; missing, blank, or non-finite values fail with invalid_input naming the field. If the vapor pressure implied by the mixing ratio meets or exceeds total pressure, the request fails rather than inventing a negative dry-air pressure. On success the body echoes rounded temperature, temperature_c, pressure, and mixing_ratio; reports vapor_pressure and dry_pressure in hPa; relative_humidity as a fraction; lcl_temperature and theta_dl in kelvin; and equivalent_potential_temperature in kelvin under unit K. formula and equation restate the Bolton identity for display, and steps walks through substitution for teaching UIs. Failed validation is not billed; successful analytic calls use the published $0.002 base price with no model tokens and no outbound network.
Where an equivalent potential temperature API earns its keep
Emergency-management briefings and broadcast meteorology tools sometimes plot surface θe to highlight moisture return ahead of severe weather; a tiny parse API can fill that column without shipping a full sounding library to every edge worker.
Technical questions
What does the equivalent potential temperature calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.