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### BARKER'S TABLES

Available on this Web site are a number of computer-generated Barker's tables (all at intervals of 5" in the true anomaly f):
• Watson's convention (C = 75).
This is an extended version of the table published by Watson [1], which sets the right-hand side of Barker's equation H=100 for f=90°. Entries are dimensionless. To use this table, divide the time from perihelion passage (t-T0, in any system of units) by the perigee distance to the 3/2 power (q3/2). Multiply by K = 75√(GM/2) in consistent units, and find the result in the table. The corresponding angle gives f. The constant K for heliocentric orbits for different sets of units is:
• t in days, q in AU: K = 0.9122791
• t in days, q in miles: K = 8.176030E11
• t in days, q in km: K = 1.669229E12
• t in seconds, q in meters: K = 6.109450E11
• t in seconds, q in cm: K = 6.109450E14
• Oppolzer's convention (C = √2/k).
This is an extended version of the table published by Oppolzer [2], which uses astronomical units and assumes the Sun as the central body (k is the Gaussian gravitational constant = 0.01720209895). To use this table, divide the time from perihelion passage (t-T0, in days) by the perigee distance (in AU) to the 3/2 power (q3/2). Find the result in the table. The corresponding angle gives f.
• Unity convention (C = 1).
This is a new tabulation that can be used for any central body and any system of units. Entries are dimensionless. To use this table, divide the time from perihelion passage (t-T0, in any system of units) by the perigee distance to the 3/2 power (q3/2). Multiply by K = √(GM/2) in consistent units, and find the result in the table. The corresponding angle gives f. The constant K for heliocentric orbits for different sets of units is:
• t in days, q in AU: K = 0.01216372
• t in days, q in miles: K = 1.090137E10
• t in days, q in km: K = 2.225638E10
• t in seconds, q in meters: K = 8.145933E09
• t in seconds, q in cm: K = 8.145933E12
• SI convention (C = √(2/GM), for Sun, SI units).
This is a new tabulation, intended for use with calculations in SI units and the Sun as the central body. Entries have units of s/m3/2. To use this table, divide the time from perihelion passage (t-T0, in seconds) by the perigee distance (in meters) to the 3/2 power (q3/2). Find the result in the table. The corresponding angle gives f.
• CGS convention (C = √(2/GM), for Sun, CGS units).
This is a new tabulation, intended for use with calculations in CGS units and the Sun as the central body. Entries have units of s/cm3/2. To use this table, divide the time from perihelion passage (t-T0, in seconds) by the perigee distance (in cm) to the 3/2 power (q3/2). Find the result in the table. The corresponding angle gives f.

#### Example

For example, Comet Barnard (1889 III) has a perihelion distance of 1.102 AU. Find its position 3 years after perihelion passage, assuming a parabolic orbit.

Solution. Here t-T0 is 1095.75 days. Dividing this by (1.102 AU)3/2 gives 947.1943. Looking this up in Oppolzer's version of Barker's tables gives roughly f = 142° 33' 55".

#### References

[1] J.C. Watson. Theoretical Astronomy, Table VI. Lippincott, 1868.
[2] T.R. Oppolzer. Lehrbuch zur Bahnbestimmung der Kometen und Planeten I, Tafel IV. Leipzig, Berlin, 1882.

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