This report collects the solutions to the proposed exercises for the Activida Guiada I of the subject Astronomia classica e instrumentacion astronomica. All exercises have been solved using the Stellarium Astronomy Software with the following configuration:

• Stellarium version: 0.20.4
• Azimuth from South: true

About Stellarium

Stellarium is a popular open-source planetarium software that allows users to explore and visualize the night sky in real-time. It provides a realistic and immersive experience by rendering a 3D representation of the sky, complete with stars, constellations, planets, nebulae, and other celestial objects.

The software is a powerful and user-friendly software that brings the wonders of the night sky to your computer or mobile device. Whether you are an amateur astronomer, a student, or simply fascinated by the cosmos, Stellarium offers a rich and immersive way to explore and appreciate the beauty of our universe.

Solutions to proposed exercises

This section contains the solutions to the proposed exercises.

Exercise 1 - Solution

Place yourself in the city where you live. Look towards the East and advance the time. Estimate the angle formed by the trajectories of the celestial bodies with the horizon of the location. Were you expecting this value?

The following data was used in Stellarium during the simulation:

• Location: Earth, Valencia, 23 meters
• Date and time: 2023-05-20 02:48:00 UTC+02:00

The geometry of the problem is depicted in the figure below:

The following points are declared:

• Point $E$ denotes the East location in the horizontal coordinates system
• Point $Z$ denotes the Zenit location in the horizontal coordinates system
• Point $Q$ denotes the a point in the Equator plane in the horizontal coordinates system
• Point $P$ denotes the North Pole location in the horizontal coordinates system

The following angles are declared:

• Angle $\varphi$ is the altitude of point P over the horizon. This is the latitude of the observer, as it is the same angle between the zenith and the equator.

• Angle ${\varphi}’$ is the complementary angle of $\varphi$, meaning that ${\varphi}’ = 90^{\circ} - \varphi$. This angle is the colatitude.

Expected value

Considering that the latitude of Valencia is $\varphi = 39^\circ 28’ 11.10’’$, thus the colatitude can be computed as:

$$ {\varphi}_{\text{expected}}’=90^\circ - \varphi = 50.53^\circ $$

Observed value

To avoid the use of the equatorial grid, a star raising exactly from the East needs to be found. Such star is the $\eta$ Aqr, which raises above the horizon at 02:48:00 UTC+02:00 time.

The angle measured from the $\eta$ Aqr to the East point is measured:

The angle is denoted by $d$ and has a value of $d = 22^\circ 29’ 19.17’’$. The altitude (height) of the star is $h = 17^\circ 07’ 56.7’’$. By using the law of sines of spherical trigonometry, it is possible to compute:

$$ {\varphi}’_{\text{observed}} = \arcsin{\left(\frac{\sin{(h)}}{\sin{(d)}} \right)} = 54.40^\circ $$

Previous value is close to the theoretical one. A simpler observational method to compute the complementary angle of the altitude of the Northen Star.

Exercise 2 - Solution

From your own city, what is the altitude above the horizon of the North Star or the Southern Celestial Pole, if you live in the southern hemisphere? Please provide the names of three circumpolar stars.

The following data was used in Stellarium during the simulation:

• Location: Earth, Valencia, 23 meters
• Date and time: 2023-05-20 02:00:00 UTC+02:00

The value of the height for the Northen Star provided by Stellarium is $h_{expected}=38^\circ 51’ 51.0’’$. The observed value is $h_{observed}=30^\circ 43’ 3.39’’$.

All stars within an angular distance of $\varphi$ measured from the pole are circumpolar stars, meaning that they never set and can be always observed.

From the simulation, it Merak, Altais, and Errai are found to be circumpolar stars.

Exercise 3 - Solution

From your own city and in the current time period, please indicate the sunrise and sunset times for the dates of the equinoxes and solstices.

The following data was used in Stellarium during the simulation:

• Location: Earth, Valencia, 23 meters

Dates for the equinoxes and solstices are:

From the observations, it can be seen that:

• During the winter, the total daytime is the smallest one. The Sun raises right to the East and sets left to the West.

• On both equinox, the Sun raises exactly from East and sets to the West.

• During the summer, the total daytime is the greatest one. The Sun raises left to the East and sets right to the West.

Spring equinox - 2023-03-20 - Sunrise 07:03 - Sunset 19:13

Summer solstice - 2023-06-21 - Sunrise 06:35 - Sunset 21:34

Autumnal equinox - 2023-03-20 - Sunrise 07:50 - Sunset 20:00

Winter solstice - 2023-12-22 - Sunrise 08:17 - Sunset 17:41

Exercise 4 - Solution

Find the Moon and position yourself at a time when the Moon is visible (above the horizon). Advance in time in daily increments. Does the Moon’s position change significantly from one day to the next or only slightly? How many days are needed for it to return (approximately) to the same area of the sky? Does this position change significantly compared to the initial position?

The following data was used in Stellarium during the simulation:

• Location: Earth, Valencia, 23 meters
• Date and time: 2023-05-01 00:00:00 UTC+02:00

The position of the Moon in the night sky was simulated for a month. It can be seen that every day the Moon delays to reach previous position till reaching a point at which it is no longer visible at night.

The estimated visible time of the moon during the month is around 14 days. Note this value is half of the orbital period of the Moon around the Earth, which takes between 27 and 28 days.

Tracking the Moon for two months, it can be seen that its altitude oscillates:

Doing the same for a whole year shows an interesting pattern forming a region with two upper and lower boundaries. The Moon can always be found within this region:

Exercise 5 - Solution

Position yourself in Valencia and change the date to October 3, 2005, at 9 in the morning. Locate and fixate the Sun using the search window. Zoom in and let time advance. Do you observe anything unusual? Would the same phenomenon be observed in your city?

The following data was used in Stellarium during the simulation:

• Location I: Earth, Valencia, 23 meters
• Location II: Cape Town, South Africa, 25 meters
• Date and time: 2005-10-03 11:02:00 UTC+02:00

The simulation indicates that a solar eclipse took place during October 3, 2025. This phenomenon could be totally seen from the city of Valencia and partially from other locations in the globe.

During a solar eclipse, the Moon passes between the Earth and the Sun, casting a shadow on a portion of the Earth’s surface. In the case of a partial solar eclipse, the Moon partially blocks the Sun, resulting in a portion of the Sun’s disk remaining visible.

Total solar eclipse – Valencia, Spain on 2005-10-03 11:02:00 UTC+02:00

The phenomenon is only visible depending on the location of the observer. For example, in Cape Town (South Africa) the Moon is not seen to intercept the Sun trajectory:

Partial solar eclipse – Cape Town, South Africa on 2005-10-03 11:02:00 UTC+02:00

Exercise 6 - Solution

Draw the Analemma from your city (you can use the Astronomical Calculations window, calculating the Sun’s ephemeris from the current day to the same day one year ago).

The following data was used in Stellarium during the simulation:

• Location: Earth, Valencia, 23 meters
• Date and time: 2023-01-01 09:00:00 UTC+01:00 – 2024-01-01 09:00:00 UTC+01:00

By using the Ephemeris calculator it is possible to obtain the trajectory of the Sun in the sky over a year. The sampling frequency is 15 solar days:

The shape described by the trajectory is known as analemma.

Exercise 7 - Solution

Conduct a systematic study of the Sun’s position throughout the year. Note the dates during which the Sun is passing through a specific constellation. To do this, obtain the boundaries of the constellations. When analyzing, indicate if you have detected anything unexpected.

The following data was used in Stellarium during the simulation:

• Location: Earth, Valencia, 23 meters
• Date and time: 2023-01-01 09:00:00 UTC+01:00 – 2024-01-01 09:00:00 UTC+01:00

By plotting the position of the Sun over the course of a year, it is possible to see that the Sun moves along the ecliptic. This line crosses different constellations. For this reasons, these constellations are known as Constellations of the Western Zodiac. The 12 constellations of the Zodiac are: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pisces.

Exercise 8 - Solution

Position yourself at the North Pole. The starting date will be January 1, 2000. Study the positions of Jupiter and Saturn as the days progress. Do their movements resemble that of Mars? How do they differ from the movement of Mars? In what ways?

The following data was used in Stellarium during the simulation:

• Location: North Pole – Latitude 90º 00’ 00" – Longitude 0º 00’ 00"
• Date and time: 2000-01-01 09:00:00 UTC+01:00 – 2001-01-01 09:00:00 UTC+01:00

Positions of Jupiter (brown) and Mars (red) as seen from the North Pole

Positions of Saturn (brown) and Mars (red) as seen from the North Pole

Both Jupiter and Saturn are always visible During the year while Mars sets below the horizon.

Exercise 9 - Solution

Take a screenshot of Stellarium from the Celestial North Pole at the Kitt Peak Observatory on the date 1000 AD. Locate the North Star (Polaris) and explain its position. How many degrees is it from the Celestial North Pole?

The following data was used in Stellarium during the simulation:

• Location: Earth, Kitt Peak National Observatory, 2096 meters
• Date and time I: 1000-01-01 00:00:00 UTC-07:06 (LMTS)
• Date and time I: 2023-01-01 00:00:00 UTC-07:06 (LMTS)

Polaris in 1000-01-01

The latitude of Kitt Peak is $\varphi = 31^\circ 57’ 30’’$. At the observation time, the altitude of polaris is $h = 29^\circ 49’ 27.6’’$.

Applying the cosine rule under the framework of spherical trigonometry, the angular distance between the Northen Star and the North Pole is:

$$ d = \arccos{\left(\cos{(\Delta a)} \cdot \cos{(\Delta h)} \right)} = 6^\circ 16’ 46’' $$

Polaris in 2023-01-01

Today, polaris is less than one degree close to the North Pole.

Previous movements of the North Pole are explained through the Earth axial precession.

Exercise 10 - Solution

Choose an observatory and indicate its geographical location. Obtain, for each of the following astronomical objects, a range of dates in which they are visible at night. Find, for each object and a specific date, the altitude at the moment of its upper culmination.

The following data was used in Stellarium during the simulation:

• Location: Earth, Gran Telescopio de Canarias, 2267 meters
• coordinates: 28°45'37.7"N, 17°52'58.5"W
• Date and time: 2023-01-01 – 2024-01-01
• Objects: M31, M51, Barnard star, Júpiter, and the Pleiades

The lack of parametric analysis in Stellarium together with the lack of an existing application programming interface (API) makes very difficult to obtain high quality data for this exercise in a reasonable amount of human time.

M31 (Andromeda) Visible date range: M31 is visible for a large part of the year from the Northern Hemisphere. Generally, it is visible from August to March in the night sky. Culmination height is $h_{max} \simeq 77^\circ$.

M51 (Whirlpool Galaxy) Visible date range: M51 is also visible for a large part of the year from the Northern Hemisphere. Generally, it is visible from March to October in the night sky. Culmination height is $h_{max} \simeq 71^\circ$.

Barnard Star Visible date range: Barnard Star, a nearby red dwarf, is visible throughout the year from the Northern Hemisphere. However, due to its low magnitude and proximity to the celestial equator, it is best observed during the boreal winter. Culmination height is $h_{max} \simeq 66^\circ$.

Jupiter Visible date range: Jupiter is a bright planet and visible to the naked eye for a large part of the year. From the Northern Hemisphere, it is visible for most of the night from March to September. Culmination height is $h_{max} \simeq 76^\circ$, around mid August 2023.

The Pleiades Visible date range: The Pleiades, an open star cluster, are visible from the Northern Hemisphere for a large part of the year. Generally, they are visible from September to April. Culmination height is $h_{max} \simeq 85^\circ$, around mid August 2023.

Conclusion

Stellarium is an incredibly useful tool for simulating the night sky and providing a realistic and interactive stargazing experience. With its accurate rendering of celestial objects, constellations, and planetary motion, Stellarium allows users to explore and learn about the universe from the comfort of their own computer or mobile device. Its user-friendly interface and extensive catalog of astronomical data make it an invaluable resource for astronomers, educators, and astronomy enthusiasts.

However, one limitation of Stellarium is its lack of an Application Programming Interface (API) to automate computations and retrieve specific data programmatically. While Stellarium provides a wealth of visual information and real-time sky simulations, users cannot easily integrate it into their own software or systems to automate complex calculations or gather data for scientific research purposes. This limitation restricts the ability to develop custom applications and workflows that require seamless interaction with Stellarium’s data and functionalities.

Despite this drawback, Stellarium remains a highly valuable tool for visualizing and exploring the night sky. Its stunning graphics, accurate simulations, and educational features make it an essential resource for anyone interested in astronomy and celestial observation.