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Definitions

JD (Julian Date)

Julian Date (JD) is the continuous count of days since the beginning of the Julian Period used primarily by astronomers. The Julian Date for 12:00 UT (Universal Time) on January 1, 4713 BCE is defined as 0.0. JD is used to simplify the calculation of the number of days between two dates.

dYear (Decimal Year)

A decimal year is a way of expressing a date as a year with a fractional part. The integer part represents the year, and the fractional part represents the portion of the year that has passed. For example, January 1, 2024, would be approximately 2024.0, and July 1, 2024, would be approximately 2024.5.

GMST (Greenwich Mean Sidereal Time)

Greenwich Mean Sidereal Time (GMST) is the hour angle of the vernal equinox at the Prime Meridian (0° longitude). GMST is used in astronomy to relate the rotation of the Earth to the fixed stars, providing a measure of the Earth's rotation angle with respect to the stars rather than the Sun.

Conversions

Julian Date (JD) to Decimal Year (dYear)

To convert JD to a decimal year, we can use the following formula: $\text{dYear} = \text{year} + \frac{\text{JD} - \text{JD}{\text{start}}}{\text{JD}{\text{end}} - \text{JD}_{\text{start}}}$

where:

• $\text{JD}_{\text{start}}$ is the Julian Date at the start of the year.
• $\text{JD}_{\text{end}}$ is the Julian Date at the end of the year.
• $\text{year}$ is the integer part of the year.

Decimal Year (dYear) to Julian Date (JD)

To convert a decimal year to JD, we can use the following formula:

$\text{JD} = \text{JD}{\text{start}} + (\text{dYear} - \text{year}) \times (\text{JD}{\text{end}} - \text{JD}_{\text{start}})$

Julian Date (JD) to GMST (Greenwich Mean Sidereal Time)

To convert JD to GMST, we use the following formula:

$\text{GMST} = 18.697374558 + 24.06570982441908 \times (\text{JD} - 2451545.0)$

The result is in hours. To convert GMST to an angle (degrees), we multiply by 15 since there are 15 degrees per hour: $\text{GMST}_{\text{angle}} = \text{GMST} \times 15$

These equations provide a mathematical framework for converting between Julian Date, Decimal Year, and Greenwich Mean Sidereal Time.

Understanding the Constants in the JD to GMST Conversion Formula

The formula in question is:

$\text{GMST} = 18.697374558 + 24.06570982441908 \times (\text{JD} - 2451545.0)$

1. Epoch J2000.0 (2451545.0)

• 2451545.0: This number represents the Julian Date for the epoch J2000.0, which corresponds to January 1, 2000, 12:00 T (Terrestrial Time). This epoch is a standard reference point used in astronomy.

2. GMST at J2000.0 (18.697374558 hours)

• 18.697374558 hours: This is the GMST in hours at the epoch J2000.0. It is derived from precise astronomical observations and theoretical models of Earth's rotation. At the J2000.0 epoch, the GMST is 18.697374558 hours. This value serves as a baseline for calculating GMST at any other Julian Date.

3. Rate of Change of GMST (24.06570982441908 hours per day)

• 24.06570982441908 hours per day: This number represents the rate at which GMST increases per Julian day. It is derived from the Earth's rotation rate. The exact factor of 24.06570982441908 accounts for the fact that a sidereal day is slightly shorter than a solar day.

Derivation of the Constants

GMST at J2000.0 (18.69737455818.69737455818.697374558 hours)

The GMST at J2000.0 is based on precise astronomical observations and calculations. The value 18.697374558 hours means that at the epoch J2000.0, the vernal equinox had an hour angle of 18.697374558 hours at the Prime Meridian.

Rate of Change of GMST (24.0657098244190824.0657098244190824.06570982441908 hours per day)

To understand this value, consider the following:

1. Length of a Sidereal Day:
   • A sidereal day is the time it takes for the Earth to complete one full rotation relative to the fixed stars, which is approximately 23.9344696 hours.
   • The Earth's rotation period relative to the stars (sidereal day) is shorter than the rotation period relative to the Sun (solar day).
2. Conversion to Hours per Julian Day:

   • The number of sidereal days in a solar day is given by:

     $\text{Number of Sidereal Days} = \frac{24 \text{ hours (solar day)}}{23.9344696 \text{ hours (sidereal day)}} \approx 1.00273790935$

   • Therefore, the rate of change of GMST in hours per Julian day is:

     $24 \text{ hours/day} \times 1.00273790935 \approx 24.06570982441908 \text{ hours/day}$

Step-by-Step Construction of the Formula

1. Starting Point (J2000.0 Epoch):

   • Use the known GMST at J2000.0:

     $\text{GMST at J2000.0} = 18.697374558 \text{ hours}$

2. Rate of Change:

   • Incorporate the rate of change of GMST per Julian day:

     $24.06570982441908 \text{ hours/day}$

3. Time Difference from J2000.0:

   • Calculate the time difference from the J2000.0 epoch: JD−2451545.0.

     $JD−2451545.0\text{JD} - 2451545.0$

4. Formula Construction:

   • Combine these elements into the final formula to obtain GMST for any Julian Date:$\text{GMST} = 18.697374558 + 24.06570982441908 \times (\text{JD} - 2451545.0)$

Summary

• 2451545.0: Julian Date for epoch J2000.0.
• 18.697374558: GMST in hours at epoch J2000.0, derived from observations.
• 24.06570982441908: Rate of increase of GMST per Julian day, derived from the ratio of the length of a solar day to a sidereal day.

These constants are derived from careful astronomical observations and calculations, providing a precise way to convert Julian Date to Greenwich Mean Sidereal Time