Is it permissible for someone who is unable to determine the Qibla from his home to approximate it using the direction of the sun?

The fundamental principle is that a Muslim should seek to determine the direction of the Qibla. If they know it by seeing a house [known to be oriented towards the Qibla] or by using instruments, they should do so. If they do not know it, they should look to the mihrabs (prayer niches) of mosques. If they do not find any, they should follow a trustworthy person who knows. If they do not find such a person, they should exert their own effort (ijtihad) and pray in the direction that their effort leads them to, in accordance with the statement of Al-Nawawi: "If someone outside the land of Mecca does not know the Qibla, and does not find a mihrab, nor anyone to inform them as previously mentioned, then they are obliged to exert their own effort (ijtihad) in [determining] the Qibla, and they should face the direction that their effort leads them to." Determining the Qibla by the sun is not for everyone due to the varying direction of its rising, except for those who are knowledgeable about constellations, longitudes, latitudes, and the sun's orbits. The most helpful tools available today are reliable modern applications.