The mass of the Sun can be calculated using the orbital properties of a planet (like Earth) and Newton's Law of Universal Gravitation.
Here's the breakdown:
The Fundamental Principle:
The gravitational force between the Sun and a planet provides the centripetal force that keeps the planet in its orbit. By equating these two forces and knowing the planet's orbital radius and period, we can solve for the Sun's mass.
The Formula:
The key equation is derived from equating gravitational force and centripetal force:
G M_sun m_planet / r^2 = m_planet * v^2 / r
Where:
- G is the gravitational constant (approximately 6.674 × 10^-11 N⋅m²/kg²)
- M_sun is the mass of the Sun (what we want to find)
- m_planet is the mass of the planet (e.g., Earth)
- r is the average distance between the planet and the Sun (orbital radius)
- v is the orbital speed of the planet
We can simplify this since v = 2πr/T, where T is the orbital period. Substituting that in and simplifying even more, we get:
M_sun = (4π^2 r^3) / (G T^2)
Notice that the planet's mass cancels out. This is important because it means we don't need to know the planet's mass to calculate the Sun's mass!
Steps for Calculation:
-
Determine the Earth's Orbital Radius (r): This is the average distance between the Earth and the Sun, approximately 1.496 x 10^11 meters (1 astronomical unit or 1 AU).
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Determine the Earth's Orbital Period (T): This is the time it takes for Earth to complete one orbit around the Sun, approximately 365.25 days, which is about 3.156 x 10^7 seconds.
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Apply the Formula: Plug the values of G, r, and T into the formula:
M_sun = (4 π^2 (1.496 x 10^11 m)^3) / ((6.674 x 10^-11 N⋅m²/kg²) * (3.156 x 10^7 s)^2)
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Calculate: Solve for M_sun. The result should be approximately 1.989 × 10^30 kg.
In Summary:
By using the easily observable orbital characteristics of Earth (its distance from the Sun and orbital period), along with the gravitational constant, we can accurately determine the mass of the Sun.
