the_great_triangle_in_the

In [1]:

import math

# Constants
R_Earth = 6366  # Earth's radius in km (Eratosthenes' estimate)
D_Moon = 60 * R_Earth  # Distance from Earth to Moon (60 times Earth's radius)

# Function to calculate distance to the Sun based on an angle
def calculate_distance_to_sun(angle_deg):
    # Convert angle to radians
    angle_rad = math.radians(angle_deg)
    
    # Compute the distance ratio (Distance from Earth to Sun / Distance from Earth to Moon)
    distance_ratio = 1 / math.cos(angle_rad)
    
    # Compute the distance to the Sun
    D_Sun = distance_ratio * D_Moon
    return D_Sun

# Compute distance to the Sun using the old angle (87 degrees)
D_Sun_old = calculate_distance_to_sun(87)
print(f"Distance to the Sun using the old angle (87°): {D_Sun_old:.2f} km")

# Compute distance to the Sun using the modern angle (89.85 degrees)
D_Sun_modern = calculate_distance_to_sun(89.85)
print(f"Distance to the Sun using the modern angle (89.85°): {D_Sun_modern:.2f} km")

# Modern known value for distance to the Sun (1 AU)
D_Sun_known = 149600000  # in km
print(f"Modern known distance to the Sun (1 AU): {D_Sun_known} km")

# Compare the computed values with the known distance
print(f"Factor by which the computed old distance is off: {D_Sun_known / D_Sun_old:.2f}")
print(f"Factor by which the computed modern distance is off: {D_Sun_known / D_Sun_modern:.2f}")

Distance to the Sun using the old angle (87°): 7298232.94 km
Distance to the Sun using the modern angle (89.85°): 145898139.61 km
Modern known distance to the Sun (1 AU): 149600000 km
Factor by which the computed old distance is off: 20.50
Factor by which the computed modern distance is off: 1.03

Explanation¶

The first step, The Triangle of Aristarchus¶

When the Moon appears half-illuminated — at its first or last quarter — the angle between the Moon and the Sun (as seen from Earth) forms a right triangle in the sky:

The Moon–Sun–Earth triangle is right-angled at the Moon.
The angle $\theta$ between the Sun and the Moon, as seen from Earth, can be measured.
Aristarchus estimated this angle to be: $\theta = 87^\circ$

But really, $\theta = 89.85^\circ$, Aristarchus did not have the tools for precise measurements.

This gives us a triangle with:

$ \angle \text{Moon} = 90^\circ $
$ \angle \text{Earth} = \theta $
$ \angle \text{Sun} = 180^\circ - 90^\circ - \theta $

The second step, Measuring the Distance to the Moon¶

The ancients had two clever methods to estimate how far the Moon was from the Earth, I prefer the Lunar Eclipse method:

Lunar Eclipse Geometry (Aristotle & Hipparchus)¶

During a lunar eclipse, the Moon passes through Earth’s shadow. Observing this, the ancients noticed:

Earth’s shadow at the Moon is roughly 2.5 times wider than the Moon itself.
The angular size of the Moon and the Sun are about the same:

$$ \theta_{\text{Moon}} \approx \theta_{\text{Sun}} \approx 0.5^\circ $$

From this, and some geometry, they reasoned that:

$$ \text{Distance from Earth to Moon} \approx 60 \times R_{\text{Earth}} $$

This is close to the modern value.

The third step, Measuring the Radius of the Earth¶

The earliest known measurement was made by Eratosthenes (c. 240 BC) using shadows in two cities:

At noon on the summer solstice:
- In Syene, the Sun was directly overhead.
- In Alexandria, the Sun cast a shadow forming angle $\theta$.

He measured:

$$ \theta \approx 7.2^\circ = \frac{1}{50} \text{ of a full circle} $$

Knowing the distance between the cities:

$$ d \approx 5000 \text{ stadia} \approx 800 \text{ km} $$

Then:

$$ \frac{d}{C_{\text{Earth}}} = \frac{\theta}{360^\circ} \quad \Rightarrow \quad C_{\text{Earth}} = \frac{360^\circ}{\theta} \cdot d = 50 \cdot d $$

And finally:

$$ R_{\text{Earth}} = \frac{C_{\text{Earth}}}{2\pi} \approx \frac{50 \cdot 800}{2\pi} \approx 6366 \text{ km} $$

This is close to the modern value of 6371 km.

The final step: completing the great triangle¶

From the previous steps, we have:

The Earth-to-Moon distance $ D_{\text{Moon}} \approx 60 \times R_{\text{Earth}} $.
The Moon-to-Sun distance $ D_{\text{Sun}} $ is roughly 19 times the Earth-to-Moon distance, according to Aristarchus' estimation:

$$ \frac{D_{\text{Sun}}}{D_{\text{Moon}}} \approx 19.1 $$

Thus, combining these:

$$ D_{\text{Sun}} \approx 19.1 \times D_{\text{Moon}} = 19.1 \times 60 \times R_{\text{Earth}} $$

Using Eratosthenes' measurement of Earth’s radius, ( R_{\text{Earth}} \approx 6366 \text{ km} ), we get:

$$ D_{\text{Sun}} \approx 19.1 \times 60 \times 6366 \text{ km} $$

This gives:

$$ D_{\text{Sun}} \approx 726,336 \text{ km} $$

This value is off by many orders, the source of error is the incorrect measurement of angle between the moon and sun.

Adjusting for Modern Estimates¶

Using the precise angle measurement for Earth to Sun distance:

$$ D_{\text{Sun}} \approx 149,600,000 \text{ km} $$

The Great Triangle in the Sky¶