On January 20th, 2019, a total lunar eclipse was possible to be observed in Santiago, Chile. Using a smartphone attached to a telescope, photographs of the phenomenon were taken. With Earth's shadow on those images, and using textbook geometry, a simple open-source software and analytical procedures, we were allowed to calculate the ratio between the radii of the Moon and the Earth. The results are in very good agreement with the correct value for such ratio. This shows the strength of the smartphone technology to get powerful astronomical results in a very simple way and in a very short amount of time.

Non-Euclidean geometry can be taught to students using astronomical images. By using photographs of the Moon taken with a smartphone through a simple telescope, we were able to introduce these concepts to high-school students and lower-level college students. We teach students how to calculate lengths of mountain ranges or areas of craters on the Moon's surface and introduce ideas of geodesics and spherical triangles. Students can see that accurate measurements cannot be obtained using flat geometry. Instead, by using three-dimensional curved geometry, estimates of lengths and areas can be computed with less than 4% error.

With only five photographs of the Sun at different dates we show that the mass of Sun can be calculated by using a telescope, a camera, and the Kepler's third law. With these photographs we are able to calculate the distance between Sun and Earth at different dates in a period of time of about three months. These distances allow us to obtain the correct elliptical orbit of Earth, proving the Kepler's first law. The analysis of the data extracted from photographs is performed by using an analytical optimization approach that allow us to find the parameters of the elliptical orbit. Also, it is shown that the five data points fit an ellipse using an geometrical scheme. The obtained parameters are in very good agreement with the ones for Earth's orbit, allowing us to foresee the future positions of Earth along its trajectory. The parameters for the orbit are used to calculate the Sun's mass by applying the Kepler's third law and Newton's law for gravitation. This method gives a result wich is in excellent agreement with the correct value for the Sun's mass. Thus, in a span of time of about three months, any student is capable to calculate the mass of the sun with only five photographs, a telescope and a camera.

We revisit the known experiment of using the solar projection along a tube in order to calculate the Sun’s radius. However, we propose several features that greatly improve the precision in the measurements. We discuss how the shape of the performation hole affect the results, and what is the best way to construct it. We also discuss how the examination of the measurements can be performed through photographs taken with smartphones, and their analysis using different free softwares. We also propose how a general theory can be formulated to determine the radius of the Sun with only measurements made with this kind of experiment, without any external parameter to the apparatus. Its usefulness and applicability is also considered. With all these additions to this known experience, this work is focused in taking advantage of the strength of smartphones and software technologies to get powerful and precise astronomical results as an educational tool.