Derive the look angles from 39N 77W to the GPS satellite given on the slide below week 801 and time of week 500,000 seconds. Note the distinction between the time of almanac applicability, time of week, and the time for which the az/el to the satellite is desired at GPS time. Orbital parameters for a typical GPS satellite: Week 801 almanac for PRN-01 ID: 01 Note that the right ascension Health: 000 varies with time. Eccentricity: 0.3765106201E-002 Time of Applicability (s): 503808.0000 Orbital Inclination (rad): 0.9617064849 Rate of Right Ascension (r/s): -0.7817468486E-008 SQRTIA (m^(1/2)): 5153.614258 Note that sqrtla); rather Right Ascension at Week (rad): 0.7017688714E+000 Than a, is provided! Argument of Perigee (rad): 0.434909394 Mean Anomaly (rad): 0.4480223834E+000 Afo (s): -0.1049041748E-004 These terms refer to drift Af1 (s/ls): 0.0000000000E+000 of the atomic clocks onboard week: 801 each GPS satellite.
The Correct Answer and Explanation is:
To derive the look angles (elevation and azimuth) from a location (39N, 77W) to a GPS satellite, we need to follow a series of steps based on the orbital parameters and the time information provided in the question.
Understanding the Problem
- Location: 39N, 77W (latitude and longitude).
- Time of Week: 500,000 seconds (this is the time at which you want to calculate the look angles).
- Almanac Parameters: These provide the orbital parameters of the GPS satellite at a reference time (Week 801), but the parameters change with time. We need to compute the satellite’s position at the given time of week (500,000 seconds).
Key Orbital Parameters
- Eccentricity (e): 0.03765106201 (affects the satellite’s orbital shape).
- Inclination (i): 0.9617064849 radians (the angle between the orbital plane and the equator).
- Right Ascension of Ascending Node (Ω): Varies with time, and a rate of change is provided (-0.7817468486E-008 radians per second).
- Argument of Perigee (ω): 0.434909394 radians (defines the orientation of the ellipse).
- Mean Anomaly (M): 0.4480223834 radians (this gives the satellite’s position in its orbit).
- SQRTIA (m^(1/2)): 5153.614258 (affects the orbital semi-major axis).
Deriving the Satellite Position
- Correct the Mean Anomaly:
The mean anomaly evolves over time as: M(t)=M0+n(t−t0)M(t) = M_0 + n(t – t_0)M(t)=M0+n(t−t0) where M0M_0M0 is the initial mean anomaly, nnn is the mean motion (calculated from the semi-major axis), and t0t_0t0 is the reference time. - Solve Kepler’s Equation:
Use Kepler’s equation to solve for the eccentric anomaly (E): M=E−esin(E)M = E – e \sin(E)M=E−esin(E) This equation requires an iterative method to solve for E (eccentric anomaly). - Compute the True Anomaly:
Once you have the eccentric anomaly, you can compute the true anomaly ν\nuν, which gives the position of the satellite along its orbit. - Calculate the Satellite’s Position in Orbital Plane:
Using the true anomaly, you can compute the satellite’s position in its orbital plane (x, y, z coordinates). - Account for Orbital Inclination and Right Ascension of Ascending Node:
Apply rotations to convert the satellite’s position in the orbital plane to Earth-centered, Earth-fixed coordinates. - Calculate the Elevation and Azimuth:
Using the satellite’s position in Earth-centered coordinates and the observer’s position (39N, 77W), calculate the line-of-sight vector and the elevation/azimuth angles. The elevation angle is the angle between the satellite and the observer’s local horizon, while the azimuth angle is the angle measured clockwise from north.
Final Notes
To calculate these angles precisely, you would use detailed GPS orbital mechanics and numerical methods. In practice, this calculation is typically performed using specialized software or GPS receivers that apply these steps with more accuracy and efficiency. However, these are the basic steps and principles needed for understanding how to derive the look angles based on the given orbital parameters and time of week.
