Unfortunately, in iOS 8.2 and earlier airplane mode disables the GPS receiver in the device so it is impossible to determine your location. Without location data the app cannot perform the calculations it needs to predict the sun and moon path.
That being said, if you have iOS 8.3 or later your GPS receiver should still work in airplane mode and so should the app. Lumos does not require cellular service or WiFi to function.
In a future version, we also plan to allow users to enter GPS coordinates or select a location on a map ahead of time and save that location for future use even where a GPS signal is not available.