Stars/Sun/Locating the Sun

This laboratory is an activity for you to create a method for locating the Sun. While it is part of the astronomy course principles of radiation astronomy, it is also independent.

Some suggested locating entities to consider are electromagnetic radiation, neutrinos, mass, time, Euclidean space, Non-Euclidean space, radar, sonar, line of sight, and spacetime.

More importantly, there are your locating entities. You are being asked to find a way to quantitatively and directly locate the Sun relative to a familiar Earth location.

You may choose to define your locational entities or use those already available.

Usually, research follows someone else's ideas of how to do something. But, in this laboratory you can create these too as long as they solve the problem of locating the Sun.

This is an astronomy Sun locator laboratory, but you may create what a locator is.

Yes, this laboratory is structured.

I will provide an example of a technique to locate the Sun experimentally. The rest is up to you.

Questions, if any, are best placed on the discussion page.

When you register as a user on Wikiversity you can also use subpages to complete the various lessons, laboratories, and problem sets associated with the course principles of radiation astronomy. A subpage can look like your username/your Locating the Sun laboratory report, for example, where the "/" separates the page from the subpage. The subpage is in your user space. Once completed you can ask for evaluation, if you wish or move your subpage to, for example, Stars/Sun/Locating the Sun/your Locating the Sun laboratory report.

Notations
You are free to create your own notation or use that already presented. A method to statistically assess your locator is also needed.

Control groups
For creating a solar locator, what would make an acceptable control group? Think about a control group to compare your locator or your process of location for the Sun to.

Introduction
Generally, the Sun is not transparent to any known wavelength of electromagnetic radiation. Metal atoms in the photosphere prevent most wavelengths of radiation from traveling through the Sun. A laser oriented on the surface of Mercury to transmit a beam to the Earth, even when the Sun occults Mercury relative to Earth, would not work.

Experimentation
As the animation at right suggests, if Earth and Mercury were coplanar in their orbits around the Sun, Mercury would transit across the Sun at least three times in an Earth year.

This can also be directly derived from the known synodic period for Mercury with the Earth of 115.88 d. The Sun could then be approximately located by the intersection of the three rays from Earth through Mercury into the Sun. As long as the three rays are extended far enough past their mutual intersection, this intersection would be the location of the Sun.

Coplanarity
Although the eccentricity of Mercury's orbit may change the coplanar transit events, the actual reason for the rarity of transits is the lack of coplanarity. The figure at right shows the approximate locations of the poles of the various planet orbits.

Mercury's orbital pole is closest to the Sun's north pole, while Earth's is in the grouping above and to the right of Mercury's.

Synodic period
Using the last visible transit of 2006 11 08, and an approximate synodic period of 116 d, the two previous passes of Mercury occurred on 2006 07 15 and 2006 03 21. On these two previous days there is no direct observation of Mercury in front of the Sun by line of sight, but the dates are based on observations of the synodic period.

To verify the synodic period, the last three visible transits should be approximate multiples of the period.

The synodic period varies by about ±4.0 d.

The three rays
The three rays even without visible transits of Mercury across the Sun should have occurred on 2006 03 21, 2006 07 15, and 2006 11 08.

Relative to the fixed background stars, each of these days has a specific right ascension. The first occurs 80 d into the year so (80 d/365 d)*24 h = 5h 15m 37s. The second occurs 196 d into the year or (196 d/365 d)*24 h = 12h 53m 16s. The third occurs 312 d into the year or (312 d/365 d)*24 h = 20h 30m 54s.

The ecliptic for each of these rays should be in Taurus, Virgo, and Capricornus, respectively. The declinations are approximately +23.2°, -5.0°, and -18.5°.

In summary, the three rays whose approximate intersection or closest pass locates the Sun within a small volume are
 * 1) RA 5h 15m 37s Dec +23.2°, with a range of RA 5h 01m 48s Dec +23.2° to RA 5h 29m 25s Dec +23.2°,
 * 2) RA 12h 53m 16s Dec -5.0°, with a range of RA 13h 07m 4s Dec -5.0° to RA 12h 39m 27s Dec -5.0°, and
 * 3) RA 20h 30m 54s Dec -18.5°, with a range of RA 20h 44m 43s Dec -18.5° to RA 20h 17m 6s Dec -18.5°.

For the recorded transit days, the corresponding RAs are
 * 1) 1999 11 15 319 d/365 d)*24 h = 20h 58m 31s
 * 2) 2003 05 07 127 d/365 d)*24 h = 8h 21m 2s
 * 3) 2006 11 08 312 d/365 d)*24 h = 20h 30m 54s
 * 4) 2016 05 09 130 d/365 d)*24 h = 8h 31m 29s

The ecliptic for each of these rays should be in Capricornus, Aries, Capricornus, and Aries, respectively. The declinations are approximately -18.5°, +18°, -18.5°, and +18°. A supplemental ray can be
 * 1) RA 12h 53m 16s Dec -5.0°, with a range of RA 13h 07m 4s Dec -5.0° to RA 12h 39m 27s Dec -5.0°,

Intersection volume
The measurement of the solar parallax would use
 * $$d = \frac{1AU}{sin{p}}.$$

The actual distance between each measurement is the chord between the two measurement points rather than 1 AU. This displacement (d1, d2, or d3) is given by
 * $$d1 = 1AU \times sin({\pi \times \frac{\alpha}{360}}),$$

where $$\alpha$$ is the angle between the first and second passes.

Using the Universal coordinate converter to obtain the angle between passes yields 114.38°, 111.27°, and 134.34°. d1 = 0.8405 AU, d2 = 0.8255 AU, and d3 = 0.9217 AU.

Results
The ranges for the angles then yield the approximate intersection volume for the Sun's location.

The distance to the Sun varies for the error (standard deviation) in d1 from 0.9829 to 1.019 AU.

The distance to the Sun varies for the error in d2 from 0.9818 to 1.020 AU.

The distance to the Sun varies for the error in d3 from 0.9889 to 1.012 AU.

These suggest that the position of the Sun along each ray varies from about 0.9845 to 1.017 AU, or at the extreme limits for the standard deviation in the synodic period from 0.9818 to 1.020.

Discussion
For an AU = 1.496 x 108 km, the extremes from these back-of-the-envelope calculations suggest that because of the variations in the synodic period of Mercury the intersection or location volume of the Sun may vary from 0.9818 (1.469 x 108 km) to 1.020 (1.526 x 108 km. The location of the Sun may vary from about 2.7 million to 3.0 million kilometers during a given year, or about 2 %, from a central position.

The durations of passes or transits may be upwards of 8 h. This suggests that the variations in the synodic period relate directly to the variations of the orbit of Mercury. While the Sun may move throughout the year, including changes in its diameter, there is no direct evidence that the Sun is outside Mercury's orbit (about 50 million kilometers in diameter) at any time.

A greater number of transits would likely reduce the error on the Sun's location using line of sight.

The mean diameter of the Sun's photosphere is 1,392,684 km. The large variations suggested here would amount to more than four solar radii in either direction from a possible central position. It may be the case that the coronal cloud around the Sun affects Mercury's orbit regarding its synodic period.

Conclusions
A series of elementary back-of-the envelope calculations has verified using recent transits of Mercury between the Sun and the Earth that the location of the Sun is well within the orbit of Mercury and probably within 2 % of a central position. At the extreme the position of the Sun may be well away from the barycenter for the Solar System.

Report
Title:

The location of the Sun using Mercury Transits

by --Marshallsumter (discuss • contribs) 23:44, 22 February 2014 (UTC)

Abstract

A series of simple back-of-the-envelope calculations starting from three recent transits of the Sun by Mercury demonstrate that the Sun is highly likely to be well within the orbit of Mercury and within a few solar diameters of the Solar System barycenter.

Introduction

The Sun is probably the only astronomical object in the Solar System whose position cannot be verified by direct radiation astronomy. This presents certain theoretical limitations on the forces controlling the Solar System. Use of transits such as Mercury between the Sun and the Earth may help to impose practical limits on the Sun's real location.

Experiment

Starting from historical records of Mercury transits and the known synodic period, a number of experiments regarding coplanarity and actual transit dates are used to calculate rays with RA and Dec that should limit the Sun's locational volume.

Results

The Sun's position along each ray tested may vary from about 0.98 to 1.12 AU.

Discussion

Limitations on the actual solar position may further refine these calculations by considering variations in Mercury's orbit and the Sun's coronal cloud.

Conclusion

The Sun's position is demonstrated with several Mercury transits to be around 2 % of the barycenter of the Solar System.

Evaluation
To assess your locator, including your justification, analysis and discussion, I will provide such an assessment of my example for comparison and consideration.

Evaluation

Variations in the Earth's orbit and Mercury's orbit during 2006 which are likely available in archives or databases could have possibly reduced the uncertainty in the Sun's location.

Hypotheses

 * 1) The Sun is not at the barycenter of the Solar System.
 * 2) The May 9, 2016 transit of Mercury can be combined with the above to improve the location of the Sun.