B Use Surface-Volume to approximate gravity for planets and protons

[Owe Kristiansen]

TL;DR Summary

Approximate surface gravity using the gravitational constant surface area and mass only

g ≈ (4π G M) / A

Title: Gravity Proportional to Mass over Surface Area — A Dimensional Approximation

I derived an approximation for surface gravity using the gravitational constant and surface area:
g ≈ (4π G M) / A

What Each Symbol Means:

• g: Surface gravity — the acceleration due to gravity at the surface (in m/s²)
• G: Gravitational constant ≈ 6.674 × 10⁻¹¹ m³/kg·s²
• M: Mass of the object (in kilograms)
• A: Surface area of the object (in square meters)
• 4π: A geometric factor that comes from the surface area of a sphere


It matches known values well for spheres.

Main Idea

Starting from the units of the gravitational constant:
[G] = m³ / (kg·s²)

I looked for a combination of measurable quantities:

• g: surface gravity (m/s²)
• A: surface area (m²)
• M: mass (kg)

The combination (g × A) / M gives:
(m/s² × m²) / kg = m³ / (kg·s²)

To match the geometry of a sphere, I introduced the factor 4π, as my calculations was 4pi off, leading to:
G ≈ (g × A) / (4π M)
→ g ≈ (4π G M) / A

This can also be written using:

• α = A / V: specific surface area (1/m)
• v = V / M: specific volume (m³/kg)


Then:
G ≈ (g × α × v) / (4π)

Validation Table (Using Surface Area)

Body | Mass (kg) | Surface Area (m²) | g (calc) | g (known)
---------|----------------|--------------------|----------|----------
Earth | 5.972e24 | 5.10e14 | 9.80 | 9.81
Moon | 7.35e22 | 3.79e13 | 1.62 | 1.62
Sun | 1.989e30 | 6.09e18 | 274 | 274
Jupiter | 1.898e27 | 6.15e16 | 25.9 | 24.8
Proton | 1.6726e-27 | 8.88e-30 | 1.58e-7 | —

---

Reverse Approximation of G

Using:
G ≈ (g × A) / (4π M)

Body | g (m/s²) | Surface Area (m²) | Mass (kg) | G (approx)
---------|----------|--------------------|----------------|-------------
Earth | 9.81 | 5.10e14 | 5.972e24 | 6.67e-11
Moon | 1.62 | 3.79e13 | 7.35e22 | 6.67e-11
Sun | 274 | 6.09e18 | 1.989e30 | 6.67e-11
Jupiter | 24.8 | 6.15e16 | 1.898e27 | 6.68e-11
Proton | 1.58e-7 | 8.88e-30 | 1.6726e-27 | 6.66e-11

---

Feedback Welcome

• Is this formulation already known or used?
• Could the specific volume form be useful for non-spherical bodies?
• Is the reverse approximation of G meaningful or just numerically coincidental?

Note: This article was created with the help of Microsoft Copilot. Some values may contain approximations or errors. Feedback and corrections are welcome.

 

[Drakkith]

Your formula: g ≈ (4π G M) / A
Surface gravity formula: g = GM/r²
Surface area of sphere formula: A=4πr²

Isolating r² of SA formula: r² = A/4π
Therefore: g = GM/A/4π
Simplifying by multiplying by reciprocal of denominator: g = 4πGM/A

The formulas are identical.

Edit: Note that real bodies are not spherical and are not point masses, so this formula will always be an approximation for the real world.

 

[Owe Kristiansen]

Thanks so much — I really appreciated your comment. I was aware that surface area and radius are connected through A = 4πr², but I hadn’t explicitly walked through how that substitution ties the formula directly back to Newton’s law. The way you laid it out made me feel like I hadn’t missed the mark — just approached it from a different angle. It was reassuring and really well explained. Thanks again for taking the time.

 

[leoherry]

surface area and radius are connected through A = 4πr²
it took me about 2 weeks to figure this out

thank you, and good effort

 

[Dale]

TL;DR Summary: Approximate surface gravity using the gravitational constant surface area and mass only

g ≈ (4π G M) / A

This article was created with the help of Microsoft Copilot

You were lucky this time. Modern LLM’s are not trained to do physics and they hallucinate a lot. They are not considered acceptable sources here at this time.

 

[Owe Kristiansen]

You were lucky this time. Modern LLM’s are not trained to do physics and they hallucinate a lot. They are not considered acceptable sources here at this time.

Yes, I was, I did not use it for much though.

 

[Owe Kristiansen]

surface area and radius are connected through A = 4πr²
it took me about 2 weeks to figure this out

thank you, and good effort

Thanks, ratios and units is what make me understand physics more than before.