The umbral shadows of the Galilean moons

[Nik271]

While waiting for a clear night this recent thread on shadow transits 

 

started me thinking, how large the shadows of the Galilean moons on Jupiter really are?

Here I'm interested in the umbral shadow, that is the area on Jupiter where the moon creates a total eclipse of the Sun. This should correspond to the dark dots we see in our telescopes. I'm assuming that the penumbral annulus  (the partial eclipse area)  scatters a lot of light in Jupiter's atmosphere and is not easily visible. Also I'm assuming the shadow cone hits Jupiter surface at close to perpendicular, ( in general the shadow is  going to be an ellipse with a minor axis as given below).

After  a short calculation with similar triangles here are the results:

Io: 2970km

Europa: 1920km

Ganymede: 3520km

Callisto: 1380km

 

The first three tally with my own observations: Largest shadow is by Ganymede, then Io, then Europa. The shadow of Callisto is a surprse: despite being the second largest moon it actualy casts the smallest umbral shadow. The reason is becuase Callisto orbits very far out, almost twice as far as Ganymede and its umbral shadow is decreased a lot. Incidentally I have not yet observed a shadow transit of Callisto to confirm this, it seems to pass either above or below Jupiter.

Another interesting point is that 1 arcsecond angle to Jupiter from Earth at opposition (4AU distance ) spans about  2900km, i.e. the shadow of Io. We can see even Europa's shadow with small scopes , which shows that the classical resolution limits apply mostly to starlike sources and 'contrast'  features can be detected which are much smaller.

I'm attaching my calculations if anyone is interested. I didn't compute the larger penumbral shadow diameters, basic geometry shows that it should be larger by as much above the diameter of the moon G as the umbral diamter is smaler than it (i.e. G+(G-U)=2G-U).

 

 

 

 

 

 

Edited September 30, 2023 by Nik271
typos corrected

[Rodd]

More amazing than your Galilean calculations is the image of Jupiter. It’s huge and you used an 81 mm scope!  I could never get the planet 1/5 as big with a C11 (or as impressive)

[dweller25]

1 hour ago, Rodd said:

More amazing than your Galilean calculations is the image of Jupiter. It’s huge and you used an 81 mm scope!  I could never get the planet 1/5 as big with a C11 (or as impressive)

I think the image is a SkySafari simulation

Edited October 4, 2023 by dweller25