1. What is Black Hole Calculator?

Definition: This calculator computes the event horizon radius of a black hole before and after a falling object merges with it, the final black hole mass, the event horizon growth percentage, and the energy released due to mass conversion during the merger.

Purpose: It is used in astrophysics to understand the effects of mass accretion on a black hole, including changes in its event horizon and the energy released as radiation during the process.

2. How Does the Calculator Work?

The calculator uses the following equations:

To calculate the final black hole mass after the merger (assuming 5.5% of the total mass is converted to energy):

The event horizon radius after the merger is calculated using the same equation with the final mass:

The event horizon growth percentage is:

The energy released due to the 5.5% mass conversion is calculated using:

Where:

• \( r_s \): Event horizon radius before the merger (m)
• \( r_{s,final} \): Event horizon radius after the merger (m)
• \( M_{bh} \): Initial black hole mass (kg)
• \( M_{obj} \): Mass of the falling object (kg)
• \( M_{final} \): Final black hole mass after the merger (kg)
• \( G \): Gravitational constant (\( 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \))
• \( c \): Speed of light (\( 299,792,458 \, \text{m/s} \))
• \( E_{released} \): Energy released due to mass conversion (J)

Unit Conversions:

• Mass (\( M_{bh} \), \( M_{obj} \), \( M_{final} \)):
  • 1 kg = 1 kg
  • 1 Solar Mass = \( 1.989 \times 10^{30} \, \text{kg} \)
  • 1 Earth Mass = \( 5.972 \times 10^{24} \, \text{kg} \)
• Event Horizon Radius (\( r_s \), \( r_{s,final} \)):
  • 1 meter = 1 m
  • 1 kilometer = 1000 m
  • 1 mile = 1609.344 m
• Energy Released (\( E_{released} \)):
  • 1 J = 1 J
  • 1 × 10¹² J = 10¹² J
  • 1 × 10¹⁵ J = 10¹⁵ J
  • 1 × 10²⁴ J = 10²⁴ J
  • 1 × 10⁴⁴ J = 1 foe (bethe)

Steps:

• Enter the initial black hole mass (\( M_{bh} \)) and the mass of the falling object (\( M_{obj} \)) with their respective units (kg, solar masses, or Earth masses).
• Convert both masses to kilograms.
• Calculate the event horizon radius before the merger (\( r_s \)).
• Calculate the final black hole mass (\( M_{final} \)) after the merger.
• Calculate the event horizon radius after the merger (\( r_{s,final} \)).
• Calculate the event horizon growth percentage.
• Calculate the energy released due to the 5.5% mass conversion.
• Convert the results to the selected units for display (meters, kilometers, or miles for radii; kg, solar masses, or Earth masses for final mass; various powers of 10 for energy).
• Display the results with 4 decimal places.

3. Importance of Black Hole Calculations

These calculations are crucial for:

• Understanding Black Hole Growth: They show how a black hole's event horizon changes as it accretes mass, which is key to studying black hole evolution.
• Energy Release in Astrophysics: The energy released as radiation during mass accretion can power phenomena like quasars and active galactic nuclei.
• Space Exploration: Understanding the gravitational effects and energy release near black holes is important for planning missions in extreme environments.

4. Using the Calculator

Example: Calculate the effects of a 1 solar mass object falling into a black hole with an initial mass of 10 solar masses.

1. Enter the black hole mass as 10 solar masses and the falling object mass as 1 solar mass.
2. The calculator computes:
   • Initial black hole mass: \( M_{bh} = 10 \times 1.989 \times 10^{30} = 1.989 \times 10^{31} \, \text{kg} \)
   • Falling object mass: \( M_{obj} = 1 \times 1.989 \times 10^{30} = 1.989 \times 10^{30} \, \text{kg} \)
   • Event horizon radius before: \( r_s = \frac{2GM_{bh}}{c^2} = \frac{2 \times (6.67430 \times 10^{-11}) \times (1.989 \times 10^{31})}{(299,792,458)^2} \approx 29532.5010 \, \text{m} \approx 29.5325 \, \text{km} \)
   • Total mass before conversion: \( M_{bh} + M_{obj} = 1.989 \times 10^{31} + 1.989 \times 10^{30} = 2.188 \times 10^{31} \, \text{kg} \)
   • Final black hole mass: \( M_{final} = (M_{bh} + M_{obj}) \times 0.945 = 2.188 \times 10^{31} \times 0.945 \approx 2.068 \times 10^{31} \, \text{kg} \)
   • Event horizon radius after: \( r_{s,final} = \frac{2GM_{final}}{c^2} = \frac{2 \times (6.67430 \times 10^{-11}) \times (2.068 \times 10^{31})}{(299,792,458)^2} \approx 30748.8888 \, \text{m} \approx 30.7489 \, \text{km} \)
   • Event horizon growth: \( \left| \frac{M_{final}}{M_{bh}} - 1 \right| \times 100 = \left| \frac{2.068 \times 10^{31}}{1.989 \times 10^{31}} - 1 \right| \times 100 \approx 3.9839\% \)
   • Energy released: \( E_{released} = ((M_{bh} + M_{obj}) \times 0.055) \times c^2 = (2.188 \times 10^{31} \times 0.055) \times (299,792,458)^2 \approx 1.082 \times 10^{46} \, \text{J} \)
   • In \( 10^{44} \, \text{J} \): \( 1.082 \times 10^{46} / 10^{44} \approx 108.2235 \, \text{foe} \)

5. Frequently Asked Questions (FAQ)

Q: What is the event horizon?
A: The event horizon is the boundary around a black hole beyond which nothing—not even light—can escape due to the immense gravitational pull. Its radius is the Schwarzschild radius.

Q: Why is 5.5% of the total mass converted to energy?
A: During the merger, a portion of the total mass (black hole + falling object) is converted to energy (e.g., as gravitational waves or radiation). The 5.5% value is an approximation based on typical efficiencies in such processes.

Q: What does the event horizon growth percentage indicate?
A: It shows the relative change in the event horizon radius due to the added mass, reflecting how the black hole's size changes after the merger.

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