What is the minimum size of an object to cast a shadow on the entire Earth?

When I fly up(don't ask me how), I see how my Sun's shadow on Earth is getting bigger but dimmer as the Sun's light is reflected from other objects.

As I flew higher and higher in the direction of the Sun, the Earth got smaller, and eventually, I could cast a shadow on the entire Earth. At this point, no reflection will make my shadow dimmer as I will be covering everything.

However, the closer I get to the Sun, the bigger its angular diameter gets, and I am running into another problem where I am not big enough to cover all its light.

What is the minimum size of an object to cast a shadow on the entire Earth? How far away from Earth should it be?

Assume the Earth is at a distance of 1 AU from the Sun at 0.527 degrees.

Bonus question: Would it be realistic to terraform Venus by temporarily freezing it, by blocking the Sun?

• If the sun is overhead, as you move toward the sun the umbral shadow that you cast on the Earth is always smaller than your cross-section, even without the light of reflecting objects, unless you happen to be larger than the Sun. Commented Jun 15, 2022 at 22:46

The minimum area such an object needs to cover is the cross sectional area of the earth, since the partial cone covering both earth and sun (which contains all the light rays that start anywhere on the sun's surface and reach any point on the earth's surface) has its minimum cross section at the earth itself, and an object casting a shadow on the entire earth must be able to cover some cross section of this cone.

This minimum is not realistic, for sure, since such an object would actually have to be inside the earth to cover said minimal cross section (since that cross section is the sun's cross section). But given that the earth's radius is merely $$r\approx0,00004~\mathrm{AU}$$, while earth's distance from the sun already varies by $$\approx0,03~\mathrm{AU}$$, the error should be smaller than the error due to the assumption that earth's distance to the sun is constant.

• I don’t think this is correct. I think the minimum is the size of the earth
– Dale
Commented Jun 15, 2022 at 22:52
• @Dale You are absolutely right, I was thinking of a situation with a small light source, which is obviously not the case when the sun is involved... I corrected it. Thanks. Commented Jun 15, 2022 at 23:01
• @Dale, I agree. This is what I understood from this answer. Also Earth minimum is unrealistic since it needs to be inside the Earth. Commented Jun 16, 2022 at 15:04

A shawdow can never be smaller than the object itself so cover the entire earth the size has to as big as earth as long as there is no glass or lens in between the light source and the object