Q.(a) The earth-moon distance is about 60 earth radius. What will be the diameter of the earth (approximately in degrees) as seen from the moon?

Using $\theta \approx D/d$ (small-angle approximation):

1. $\theta_E = \dfrac{2R_E}{60R_E} = \dfrac{1}{30}\ \text{rad} \approx 2^\circ$
2. Same base distance for both, so $\dfrac{D_M}{D_E} = \dfrac{\theta_M}{\theta_E} = \dfrac{0.5^\circ}{2^\circ} = \dfrac{1}{4}$ …

The tool: small-angle angular diameter

where $D$ is the object's actual diameter and $d$ is its distance from the observer — valid whenever $D \ll d$, true for every body in this problem.

(a) Earth's angular diameter seen from the Moon

Earth's diameter is $D_E = 2R_E$; the Earth–Moon distance is $d_{EM} = 60R_E$.

Converting to degrees ($1\ \text{rad} = 180/\pi$ degrees):

(b) Relative size of the Moon

The Moon's angular diameter as seen from Earth, over the same distance $d_{EM}$, is given as $\theta_M = \left(\tfrac{1}{2}\right)^\circ$. Since both $\theta_E$ and $\theta_M$ use the same distance $d_{EM}$, their ratio equals the ratio of the actual diameters:

(c) Ratio of Sun's diameter to Earth's diameter

A total solar eclipse tells us the Sun and Moon have (almost) the same angular diameter as seen from Earth: $\theta_S \approx \theta_M$. Since $\theta = D/d$ for both:

$\frac{D_S}{d_{SE}} \approx \frac{D_M}{d_{EM}} \implies \frac{D_S}{D_M} \approx \frac{d_{SE}}{d_{EM}} = 400$ …

Concept: Angular Size and Small-Angle Approximation

When an object's physical size is much smaller than its distance from the observer, the angular diameter (in radians) is given by:

To convert radians to degrees: $1 \text{ radian} \approx 57.3^\circ$

(a) Method: Small-Angle Formula

Step 1: Identify given data

• Distance from Moon to Earth = $60 \, R_E$ (where $R_E$ = Earth's radius)
• Earth's actual diameter = $2 R_E$

Step 2: Apply the small-angle formula

Step 3: Convert to degrees

Answer: The Earth's diameter as seen from the Moon is approximately $2^\circ$.

(b) Method: Ratio Using Angular Size Formula

Step 1: For Moon as seen from Earth

Step 2: Write the angular size relation

where $D_M$, $D_E$ are diameters, $d_{EM}$ is Earth-Moon distance.

Step 3: Take the ratio

Answer: The Moon's diameter is $\frac{1}{4}$ of Earth's diameter.

(c) Method: Combined Angular Size and Distance Ratios

Step 1: Given

• Sun's distance $d_S = 400 \times d_{EM}$
• From (a), Earth's angular size from Moon = $2^\circ$
• From (b), Moon's angular size from Earth = $0.5^\circ$ …

Here are the common mistakes students make on this classic angular size and astronomical scaling problem, and how to avoid each.

Mistake 1: Confusing Diameter with Radius in the Angular Size Formula

The Mistake:

For part (a), students often plug the Earth’s radius ($R_E$) into the formula for angular diameter instead of the Earth’s diameter ($D_E = 2R_E$). They might write:

This gives half the correct answer.

Why it happens:

The formula $\theta \approx \frac{\text{arc length}}{\text{radius}}$ is usually taught for small angles, where the "arc length" is the linear size of the object. Students forget that the linear size of a sphere seen from a distance is its diameter, not its radius.

How to Avoid:

Always define the variables clearly before plugging in.

• Linear size of the object = Diameter ($D$).
• Distance to the object = $d$.
• For small angles (in radians): $\theta \approx \frac{D}{d}$.

For part (a):

• $D_{\text{Earth}} = 2 R_E$
• $d = 60 R_E$
• $\theta \approx \frac{2 R_E}{60 R_E} = \frac{1}{30} \text{ rad}$

Then convert to degrees:

Key takeaway: The angular diameter of a sphere is determined by its full width, not its half-width.

Mistake 2: Forgetting to Convert Radians to Degrees

The Mistake:

Students stop at $\theta = \frac{1}{30}$ rad and write the answer as $\frac{1}{30}^\circ$ or just leave it in radians without converting.

Why it happens:

The problem explicitly asks for the answer "in degrees," but students often treat the radian measure as if it were already in degrees.

How to Avoid:

Always check the unit requested in the question. Use the conversion factor:

So $\frac{1}{30}$ rad $\approx 1.9^\circ$.

Pro tip: Memorize that $1$ rad $\approx 57^\circ$. For quick checks, $\frac{1}{60}$ rad $\approx 1^\circ$.

Mistake 3: Mixing Up Which Object is the "Observer" and Which is the "Observed"

The Mistake:

In part (b), students might calculate the Moon's diameter as seen from Earth, but the question asks for the relative size of the Moon compared to Earth. They might invert the ratio or use the wrong angular diameter.

Why it happens:

The problem has three parts with shifting perspectives:

• (a) Earth seen from Moon.
• (b) Moon seen from Earth.
• (c) Sun seen from Earth.

Students lose track of which angular diameter belongs to which object.

How to Avoid:

Draw a simple diagram or write a clear statement for each part.

For part (b):

• Given: Angular diameter of Moon as seen from Earth = $\theta_M = 0.5^\circ$.
• From part (a): Angular diameter of Earth as seen from Moon = $\theta_E \approx 2^\circ$.
• Distance between Earth and Moon is the same in both directions ($d$).

Use the angular size formula for both:

Divide the two equations:

Key takeaway: When distances are equal, the ratio of diameters equals the ratio of angular diameters.

Mistake 4: Using the Wrong Distance in Part (c)

The Mistake:

For part (c), students use the Earth-Moon distance ($d_{EM}$) as the distance to the Sun, instead of the given "400 times the earth-moon distance."

Why it happens:

The problem states: "the sun is found to be at a distance of about 400 times the earth-moon distance." Students might misread this as "the Sun is 400 times farther than the Moon" but then forget to multiply the distance correctly in the angular size formula.

How to Avoid:

Write the given data explicitly:

• $d_{\text{Sun}} = 400 \times d_{\text{Earth-Moon}}$
• Angular diameter of Sun as seen from Earth = $\theta_{\text{Sun}} \approx 0.5^\circ$ (same as Moon, a famous coincidence).

Now, for the Sun:

For the Moon (from part b):

$\theta_{\text{Moon}} \approx \frac{D_{\text{Moon}}}{d_{\text{EM}}}$ …