Deimos Calculator

Understanding the Deimos Orbital Period Calculator

This calculator helps determine the orbital period of a celestial body, specifically using Deimos (one of Mars' moons) as a conceptual example, around a central body like a planet. The orbital period is the time it takes for an object to complete one full orbit around another object.

The Physics Behind the Calculation

The calculation is based on Kepler's Third Law of Planetary Motion, specifically its generalized form derived from Newton's Law of Universal Gravitation. The formula used to calculate the orbital period (T) is:

$T = 2\pi \sqrt{\frac{a^3}{GM}}$

Where:

• T is the orbital period in seconds.
• π (pi) is a mathematical constant, approximately 3.14159.
• a is the semi-major axis of the orbit, which for simplicity is often approximated by the orbital radius when orbits are nearly circular. This is the value entered as 'Orbital Radius' in kilometers.
• G is the gravitational constant, approximately $6.67430 \times 10^{-11} \text{ N} (\text{m/kg})^2$.
• M is the mass of the central body (e.g., a planet) in kilograms. This is the value entered as 'Central Body Mass'.

How the Calculator Works

The calculator takes two primary inputs:

• Orbital Radius (a): The average distance from the center of the central body to the center of the orbiting body. It is entered in kilometers (km) and then converted to meters (m) for the calculation.
• Central Body Mass (M): The mass of the object being orbited (e.g., Mars). It is entered in kilograms (kg).

The calculator then performs the following steps:

1. Converts the orbital radius from kilometers to meters ($a_{meters} = a_{km} \times 1000$).
2. Calculates the cube of the orbital radius in meters ($a^3$).
3. Calculates the product of the gravitational constant (G) and the central body's mass ($GM$).
4. Divides the cubed orbital radius by the $GM$ product ($\frac{a^3}{GM}$).
5. Takes the square root of the result.
6. Multiplies the square root by $2\pi$. This gives the orbital period in seconds.
7. Converts the orbital period from seconds to more understandable units like hours or days.

Example Calculation

Let's calculate the orbital period of Deimos around Mars.

• Orbital Radius of Deimos (approximate): 23460 km
• Mass of Mars (M): Approximately $5.972 \times 10^{24}$ kg

1. Convert radius to meters: $23460 \text{ km} \times 1000 \text{ m/km} = 23,460,000 \text{ m}$ 2. Calculate $a^3$: $(2.346 \times 10^7 \text{ m})^3 \approx 1.292 \times 10^{22} \text{ m}^3$ 3. Calculate $GM$: $(6.67430 \times 10^{-11} \text{ N} (\text{m/kg})^2) \times (5.972 \times 10^{24} \text{ kg}) \approx 3.986 \times 10^{14} \text{ m}^3/\text{s}^2$ 4. Calculate $\frac{a^3}{GM}$: $(1.292 \times 10^{22} \text{ m}^3) / (3.986 \times 10^{14} \text{ m}^3/\text{s}^2) \approx 32409 \text{ s}^2$ 5. Calculate $\sqrt{\frac{a^3}{GM}}$: $\sqrt{32409 \text{ s}^2} \approx 180.03 \text{ s}$ 6. Calculate T: $2\pi \times 180.03 \text{ s} \approx 1131.1 \text{ s}$ 7. Convert to hours: $1131.1 \text{ s} / (3600 \text{ s/hr}) \approx 0.314 \text{ hours}$. This is approximately 18.8 hours.

Therefore, the calculated orbital period for Deimos is approximately 1131 seconds, or about 0.314 Earth days (roughly 7.5 hours). *Note: Actual orbital period of Deimos is closer to 30.3 hours, which suggests its orbit is not perfectly circular or there are other factors. This calculator uses a simplified model.*

Use Cases

This calculator is useful for:

• Students and educators learning about orbital mechanics.
• Amateur astronomers estimating orbital periods.
• Anyone curious about the dynamics of celestial bodies.

It provides a simplified way to understand the relationship between distance, mass, and orbital time.