🌌 Black Hole Temperature Calculator
Calculate Hawking temperature and understand quantum effects at black hole event horizons
Stellar: 3-50 M☉ | Intermediate: 10²-10⁵ M☉ | Supermassive: 10⁶-10¹⁰ M☉
How to Use This Calculator
Enter Black Hole Mass
Input the mass of the black hole in solar masses (M☉). For stellar black holes, typical values range from 3-50 solar masses. For supermassive black holes at galaxy centers, values can be millions to billions of solar masses.
Calculate Temperature
Click "Calculate" to determine the Hawking temperature. This temperature represents the black body temperature of the Hawking radiation emitted by the black hole.
Interpret Results
Understand that larger black holes have lower temperatures. Stellar mass black holes are extremely cold (less than 10⁻⁸ K), making Hawking radiation negligible. Only microscopic black holes would have significant Hawking radiation.
Formula
T = ħc³ / (8πGMk)
Where:
- T = Hawking temperature (K)
- ħ = Reduced Planck constant = h/(2π) = 1.054571817 × 10⁻³⁴ J·s
- c = Speed of light = 299,792,458 m/s
- G = Gravitational constant = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻²
- M = Black hole mass (kg)
- k = Boltzmann constant = 1.380649 × 10⁻²³ J/K
Example Calculation: 10 Solar Mass Black Hole
Given:
- M = 10 M☉ = 10 × 1.989 × 10³⁰ kg = 1.989 × 10³¹ kg
- ħ = 1.054571817 × 10⁻³⁴ J·s
- c = 2.998 × 10⁸ m/s
- G = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻²
- k = 1.380649 × 10⁻²³ J/K
Calculation:
T = ħc³ / (8πGMk)
T = (1.055×10⁻³⁴ × (2.998×10⁸)³) / (8π × 6.674×10⁻¹¹ × 1.989×10³¹ × 1.381×10⁻²³)
T = (2.838×10⁻¹⁷) / (1.455×10¹¹)
T ≈ 6.17 × 10⁻²⁸ K
This is an extremely low temperature - much colder than the cosmic microwave background (2.7 K), meaning the black hole would actually gain mass from absorbing CMB radiation rather than losing it through Hawking radiation.
Key Insights:
- Temperature is inversely proportional to mass: T ∝ 1/M
- Larger black holes are colder and emit less Hawking radiation
- Smaller black holes are hotter and evaporate faster
- For a black hole to have T = 1 K, mass ≈ 2.8 × 10²³ kg (about 0.000014 solar masses)
- Hawking radiation becomes significant only for microscopic black holes
About the Black Hole Temperature Calculator
The Black Hole Temperature Calculator determines the Hawking temperature of a black hole, which is the temperature at which it emits Hawking radiation. This quantum mechanical effect, predicted by Stephen Hawking in 1974, shows that black holes are not completely black but emit thermal radiation due to quantum fluctuations near the event horizon.
When to Use This Calculator
- Quantum Physics Education: Understand Hawking radiation and quantum effects in black holes
- Astrophysics Research: Calculate theoretical black hole temperatures
- Cosmology Studies: Understand black hole evaporation and lifetime
- Theoretical Physics: Explore connections between gravity and quantum mechanics
- Black Hole Thermodynamics: Study black hole entropy and information paradox
Why Use Our Calculator?
- ✅ Accurate Formula: Uses Hawking's exact temperature formula
- ✅ Educational Tool: Learn about quantum effects in black holes
- ✅ Comprehensive Results: Shows temperature and Schwarzschild radius
- ✅ Real Physics: Based on established quantum field theory
- ✅ Free to Use: No registration required
- ✅ Mobile Friendly: Works on all devices
Understanding Hawking Radiation
Hawking radiation is a quantum mechanical phenomenon where black holes emit thermal radiation:
- Quantum Fluctuations: Virtual particle pairs form near the event horizon
- One Particle Escapes: One particle falls into the black hole, the other escapes
- Energy Conservation: The escaping particle carries energy away, reducing black hole mass
- Thermal Spectrum: Radiation follows black body spectrum at Hawking temperature
- Evaporation: Over time, black holes lose mass and eventually evaporate (if isolated)
Key Properties
- Inverse Mass Relationship: Temperature ∝ 1/Mass - larger black holes are colder
- Negligible for Stellar Mass: A 10 solar mass black hole has T ≈ 6×10⁻²⁸ K
- CMB Absorption: Stellar black holes absorb more CMB radiation than they emit
- Microscopic Black Holes: Only tiny black holes (mass < 10¹² kg) would have significant Hawking radiation
- Lifetime: Black hole lifetime ∝ M³ - larger black holes live much longer
Real-World Applications
- Black Hole Information Paradox: Hawking radiation raises questions about information conservation
- Primordial Black Holes: Theoretical microscopic black holes from early universe
- Black Hole Thermodynamics: Connects black holes to thermodynamics and entropy
- Quantum Gravity: Tests theories that unify general relativity and quantum mechanics
- Cosmology: Understanding black hole evolution over cosmic time
Tips for Using This Calculator
- Remember that larger black holes have lower temperatures - this is counterintuitive
- Hawking radiation is negligible for stellar and supermassive black holes
- The cosmic microwave background (2.7 K) is much hotter than Hawking radiation from stellar black holes
- Only hypothetical microscopic black holes would have detectable Hawking radiation
- Black hole evaporation time scales are extremely long for astrophysical black holes
Frequently Asked Questions
What is Hawking radiation?
Hawking radiation is thermal radiation predicted to be emitted by black holes due to quantum effects near the event horizon. It was discovered by Stephen Hawking in 1974 and shows that black holes are not completely black but slowly lose mass over time.
Why do larger black holes have lower temperatures?
Hawking temperature is inversely proportional to mass: T ∝ 1/M. This means as mass increases, temperature decreases. A 10 solar mass black hole has a temperature of about 6×10⁻²⁸ K, while a 10¹⁰ solar mass supermassive black hole has a temperature of about 6×10⁻³⁸ K - extremely cold!
Can we observe Hawking radiation from real black holes?
Not from astrophysical black holes. The Hawking temperature for stellar mass black holes is much lower than the cosmic microwave background (2.7 K), so they actually absorb more radiation than they emit. Only hypothetical microscopic black holes (mass < 10¹² kg) would have detectable Hawking radiation.
How long does it take for a black hole to evaporate?
The lifetime of a black hole is proportional to M³. For a 10 solar mass black hole, the evaporation time is about 10⁶⁷ years - far longer than the age of the universe (13.8 billion years). Only microscopic black holes would evaporate in observable time scales.
What happens to a black hole as it evaporates?
As a black hole emits Hawking radiation, it loses mass. As mass decreases, temperature increases (T ∝ 1/M), so the black hole radiates faster. This creates a runaway process where smaller black holes evaporate faster, eventually disappearing in a burst of radiation.
Is Hawking radiation related to the black hole information paradox?
Yes. Hawking originally showed that Hawking radiation appears to be random and thermal, which would violate quantum mechanics (information would be lost). This "black hole information paradox" has been a major topic in theoretical physics, with recent work suggesting information may be preserved through various mechanisms.