# Combining imagination and mathematics improves understanding

Loading new images of a full Moon rising over the Ostrobothnia countryside fueled my imagination. What would I see in the sky if I could make the planet Trappist-1d change places with the Earth? In the compact planet system around the ultra-cool brown dwarf star Trappist-1, the neighbour planets are almost within reach. Ok. Not exactly in a length of an arm, but close to just a 50’th of the distance between our inner planets away.

From Trappist-1d, which fits inside the “habitable zone” the next planet further from the star would be planet “e” just 1 050 000 km away. That’s not even fully three times the distance between the Earth and the Moon. Taking into the count that Trappist-1e has a radius 92% that of the Earth it’s time to wonder whether the Moon or Trappist-1e would look larger in the evening sky. Remember, we are not standing on the unknown surface of planet “e”. In our imagination, we’re standing on the spinning Earth in planet “e”‘s place around Trappist-1.

The planet Trappist-1e would appear with a radius 23% longer than the radius of our Moon seen from Earth. A 23% longer radius means a 51% larger surface and a 86% greater volume.

The Moon has a radius of 1740 km, and the mean distance between the Moon and the Earth is 384000 km. Trappist-1e has a radius of about 5860 km and lies 1050000 km from Trappist-1d.

Solving the equation for x, we find that, assuming Trappist-1e is appearing next to the Moon on the Earth-sky, Trappist-1e would have an apparent radius of 2140 km. It would look to us as it had a radius 23% longer than the radius of the Moon.

Below is another way to concretize the thought. Please keep in mind that the scales are not the same on the x- and y- axis.

If one would turn around and look at the planets closer to the star, how would they appear?

Jan teaches mathematics and interdisciplinary science to pupils 13-16 years of age at Sursik School, Pedersöre, Finland. Space-related science often gives some sort of answer to the question “Why?”, a question quite common in math class. It also triggers curiosity, one key component in progress.