True Position
Calculating true positions requires applying the manda and śīghra corrections a total of four times (two each). This page first walks through how each correction is calculated using an example, then applies the full sequence.
In this page, we will be using Lord Caitanya's appearance day as the example to calculate true positions.
Ahargaṇa: 714,403,972,030
| Graha | Mean Position | Mandocca | Śīghroca |
|---|---|---|---|
| Sūrya (Sun) | 321.64431° | 77.27793° | — |
| Candra (Moon) | 139.91534° | 221.26217° | — |
| Budha (Mercury) | 321.64431° | 220.46066° | 81.92843° |
| Śukra (Venus) | 321.64431° | 79.85450° | 80.87690° |
| Maṅgala (Mars) | 263.63927° | 130.03798° | 321.64431° |
| Bṛhaspati (Jupiter) | 259.86122° | 171.34402° | 321.64431° |
| Śani (Saturn) | 224.31668° | 236.62491° | 321.64431° |
| Rāhu | 329.08646° | — | — |
| Ketu | 149.08646° | — | — |
Step 1: The Kendra (Anomaly)
The anomaly is the angular distance between a graha's ucca and its mean position.
The kendra is the angular distance between a graha's ucca and its mean position. There are two: the manda kendra and the śīghra kendra.
śīghra kendra = śīghroca − mean position
(SS 2.29)
Using Budha as an example:
= 220.46066° − 321.64431° = −101.18365° + 360° = 258.81635°
śīghra_kendra = śīghroca − mean
= 81.92843° − 321.64431° = −239.71588° + 360° = 120.28412°
If the raw result is negative, add 360° to bring it into the range [0°, 360°). The sign of the correction then depends on whether this kendra falls below or above 180°.
The kendra determines how the final correction will apply to the mean position to get true position.
kendra > 180° → subtract
(SS 2.45)
Step 2: The Bhuja (Reference Angle)
The reference angle is the kendra reduced to within 0° to 90°, used as input to the sine table.
Take the kendra and find which quadrant it falls into. Find the angle's position within its quadrant, measured from 0° to 90°. The quadrant determines which formula you use to find that angle: (SS 2.29–30)
| Quadrant | Formula |
|---|---|
| 1 | given |
| 2 | 180° − deg |
| 3 | deg − 180° |
| 4 | 360° − deg |
For Budha:
manda_bhuja = 258.81635° − 180° = 78.81635°
Śīghra kendra is in Q2:
sighra_bhuja = 180° − 120.28412° = 59.71588°
Step 2.5: Sine Lookup
The full lookup method is on the Trigonometry page.
manda cosine (11.18365°) = 667.0
śīghra sine (59.71588°) = 2968.984
Step 3: The Paridhi (Circumference)
The paridhi is the circumference of the epicycle — the orbit the graha traces around its mean position. The manda and śīghra corrections each use a separate epicycle, and each graha has a different size. The SS gives two values: one at the even quadrant ends (kendra = 0° or 180°) and one at the odd quadrant ends (kendra = 90° or 270°).
Manda Paridhi:
| Graha | Even | Odd |
|---|---|---|
| Sūrya (Sun) | 14° | 13°40' |
| Candra (Moon) | 32° | 31°40' |
| Maṅgala (Mars) | 75° | 72° |
| Budha (Mercury) | 30° | 28° |
| Bṛhaspati (Jupiter) | 33° | 32° |
| Śukra (Venus) | 12° | 11° |
| Śani (Saturn) | 49° | 48° |
(SS 2.34–35)
Śīghra Paridhi:
| Graha | Even | Odd |
|---|---|---|
| Maṅgala (Mars) | 235° | 232° |
| Budha (Mercury) | 133° | 132° |
| Bṛhaspati (Jupiter) | 70° | 72° |
| Śukra (Venus) | 262° | 260° |
| Śani (Saturn) | 39° | 40° |
(SS 2.36–37)
correction = (sin(bhuja) × diff) / radius
corrected_paridhi = paridhi_even − correction
(SS 2.38)
For Budha:
manda_correction = (3372.655 × 2°) / 3438 = 1.96198°
manda_corrected_paridhi = 30° − 1.96198° = 28.03802°
śīghra_diff = 133° − 132° = 1°
śīghra_correction = (2968.984 × 1°) / 3438 = 0.86355°
śīghra_corrected_paridhi = 133° − 0.86355° = 132.13645°
Step 4: Epicycle Components
Convert the corrected paridhi to an epicycle radius in arc-minutes, then find its sine and cosine components.
r = 1682.28 / (2π) = 267.74'
sine = 267.74 × (3372.655 / 3438) = 262.7'
cosine = 267.74 × (667.0 / 3438) = 51.9'
Step 5: The Karṇa (True Distance)
The hypotenuse of the correction triangle, which represents the true distance from Earth to the graha.
= √(3386.1² + 262.7²)
= √11,534,104
= 3396.2
Step 6: Ucca Correction
The sine of the correction angle, then its arc via inverse table lookup.
inverse lookup: between entry 1 (225) and entry 2 (449)
arc = 225 + ((265.9 − 225) × 225) / (449 − 225)
= 225 + 41.1
= 266.1 kālā → 266.1 / 60 = 4.43°
(SS 2.33)
Step 7: Apply Sign
The manda kendra determines whether to add or subtract the correction from the mean position.
corrected mean = 321.64431° − 4.43° = 317.21°
(SS 2.45)
Four Corrections
There are four corrections applied to the mean position to arrive at the true position. This is because there are two types of corrections: manda and śīghra. For all planets except the Sun and Moon, both corrections are interdependent. Both corrections are calculated from the mean position. However, applying one changes the basis for calculating the other.
The way this is resolved is refining the position piece by piece instead of all at once.
There are two half corrections: half śīghra and half manda. After that, the manda correction is recalculated and applied in full, followed by the full śīghra correction.
Steps 1 & 2:
= 81.92843° − 321.64431° = −239.71588° + 360° = 120.28412°
Q2 → add
śīghra_bhuja = 180° − 120.28412°
= 59.71588°
Step 2.5:
śīghra cosine (30.28412°) = 1733.8
Steps 3 & 4:
= 133° − 132° = 1°
correction = (śīghra sine × diff) / trijyā
= (2968.984 × 1°) / 3438 = 0.86355°
corrected_paridhi = paridhi_even − correction
= 133° − 0.86355° = 132.13645°
corrected_paridhi = 132.13645° × 60
= 7928.19'
r = 7928.19' / 2π
= 1261.95'
sine = r × (śīghra sine / trijyā)
= 1261.95 × (2968.984 / 3438) = 1089.7'
cosine = r × (śīghra cosine / trijyā)
= 1261.95 × (1733.8 / 3438) = 636.4'
Step 5:
= √((3438 − 636.4)² + 1089.7²)
= √(2801.6² + 1089.7²)
= √9,036,450
= 3006.1'
Steps 6 & 7:
Since this is the first of four corrections, only half is applied:
= (3438 × 1089.7) / 3006.1 = 1246.3'
→ inverse lookup: 21.27°
kendra < 180° → add
half_correction = full_correction / 2
= 21.27° / 2 = 10.64°
half_sighra_mean = mean + half_correction
= 321.64431° + 10.64° = 332.28°
Steps 1 & 2:
= 220.46066° − 332.28° = −111.82° + 360° = 248.18°
Q3 → subtract
manda_bhuja = 248.18° − 180°
= 68.18°
Step 2.5:
manda cosine (21.82°) = 1277
Steps 3 & 4:
= 30° − 28° = 2°
correction = (manda sine × diff) / trijyā
= (3191 × 2°) / 3438 = 1.856°
corrected_paridhi = paridhi_even − correction
= 30° − 1.856° = 28.144°
corrected_paridhi = 28.144° × 60
= 1688.6'
r = 1688.6' / 2π
= 268.8'
sine = r × (manda sine / trijyā)
= 268.8 × (3191 / 3438) = 249.5'
cosine = r × (manda cosine / trijyā)
= 268.8 × (1277 / 3438) = 99.8'
Step 5:
= √((3438 − 99.8)² + 249.5²)
= √(3338.2² + 249.5²)
= √11,205,829
= 3347.5'
Steps 6 & 7:
Since this is the second of four corrections, only half is applied:
= (3438 × 249.5) / 3347.5 = 256.2'
→ inverse lookup: 4.27°
kendra ≥ 180° → subtract
half_correction = full_correction / 2
= 4.27° / 2 = 2.14°
half_manda_mean = half_sighra_mean − half_correction
= 332.28° − 2.14° = 330.14°
Steps 1 & 2:
= 220.46066° − 330.14° = −109.68° + 360° = 250.32°
Q3 → subtract
manda_bhuja = 250.32° − 180°
= 70.32°
Step 2.5:
manda cosine (19.68°) = 1157
Steps 3 & 4:
= 30° − 28° = 2°
correction = (manda sine × diff) / trijyā
= (3236 × 2°) / 3438 = 1.882°
corrected_paridhi = paridhi_even − correction
= 30° − 1.882° = 28.118°
corrected_paridhi = 28.118° × 60
= 1687.1'
r = 1687.1' / 2π
= 268.5'
sine = r × (manda sine / trijyā)
= 268.5 × (3236 / 3438) = 252.7'
cosine = r × (manda cosine / trijyā)
= 268.5 × (1157 / 3438) = 90.4'
Step 5:
= √((3438 − 90.4)² + 252.7²)
= √(3347.6² + 252.7²)
= √11,270,283
= 3357.1'
Steps 6 & 7:
The full correction is applied to the original mean, not the preliminary position:
= (3438 × 252.7) / 3357.1 = 258.8'
→ inverse lookup: 4.32°
kendra ≥ 180° → subtract
full_manda_mean = mean − correction
= 321.64431° − 4.32° = 317.32°
Steps 1 & 2:
= 81.92843° − 317.32° = −235.39° + 360° = 124.61°
Q2 → add
śīghra_bhuja = 180° − 124.61°
= 55.39°
Step 2.5:
śīghra cosine (34.61°) = 1952
Steps 3 & 4:
= 133° − 132° = 1°
correction = (śīghra sine × diff) / trijyā
= (2829 × 1°) / 3438 = 0.823°
corrected_paridhi = paridhi_even − correction
= 133° − 0.823° = 132.177°
corrected_paridhi = 132.177° × 60
= 7930.6'
r = 7930.6' / 2π
= 1262.3'
sine = r × (śīghra sine / trijyā)
= 1262.3 × (2829 / 3438) = 1038.9'
cosine = r × (śīghra cosine / trijyā)
= 1262.3 × (1952 / 3438) = 716.7'
Step 5:
= √((3438 − 716.7)² + 1038.9²)
= √(2721.3² + 1038.9²)
= √8,484,787
= 2912.9'
Steps 6 & 7:
= (3438 × 1038.9) / 2912.9 = 1225.9'
→ inverse lookup: 20.91°
kendra < 180° → add
true_position = full_manda_mean + correction
= 317.32° + 20.91° = 338.23°
Śrī Caitanya's Appearance True Positions
Corrections
| Graha | Manda Position |
Śīghra Position |
½ Śīghra Corr. |
½ Manda Corr. |
Full Manda Corr. |
Full Śīghra Corr. |
|---|---|---|---|---|---|---|
| Sūrya (Sun) | 77.27793° | — | — | — | 1.9972° | — |
| Candra (Moon) | 221.26217° | — | — | — | 5.03437° | — |
| Budha (Mercury) | 220.46066° | 81.92843° | 10.63553° | -2.13628° | -4.3165° | 20.90918° |
| Śukra (Venus) | 79.8545° | 80.8769° | 22.14128° | 0.873° | 1.74756° | 43.90098° |
| Maṅgala (Mars) | 130.03798° | 321.64431° | 11.10296° | -4.00971° | -8.67103° | 25.28282° |
| Bṛhaspati (Jupiter) | 171.34402° | 321.64431° | 4.55835° | -2.54744° | -5.0816° | 9.6567° |
| Śani (Saturn) | 236.62491° | 321.64431° | 3.18795° | 0.71105° | 1.3131° | 6.3743° |
True Positions
| Graha | Mean Position |
½ Śīghra Corrected |
½ Manda Corrected |
Full Manda Corrected |
True Position |
|---|---|---|---|---|---|
| Sūrya (Sun) | 321.64431° | — | — | 323.6415° | 323.6415° |
| Candra (Moon) | 139.91534° | — | — | 144.94971° | 144.94971° |
| Budha (Mercury) | 321.64431° | 332.27984° | 330.14356° | 317.32781° | 338.23698° |
| Śukra (Venus) | 321.64431° | 343.78559° | 344.65859° | 323.39187° | 7.29285° |
| Maṅgala (Mars) | 263.63927° | 274.74223° | 270.73252° | 254.96824° | 280.25106° |
| Bṛhaspati (Jupiter) | 259.86122° | 264.41957° | 261.87213° | 254.77962° | 264.43632° |
| Śani (Saturn) | 224.31668° | 227.50463° | 228.21567° | 225.62977° | 232.00407° |
| Rāhu | 329.08646° | — | — | — | 329.08646° |
| Ketu | 149.08646° | — | — | — | 149.08646° |
Zodiac Positions
| Graha | True Position | Rāśi | Nakṣatra |
|---|---|---|---|
| Sūrya (Sun) | 323.6415° | Kumbha 23°38′ | Pūrva Bhādrapadā 3°38′ |
| Candra (Moon) | 144.94971° | Siṃha 24°57′ | Pūrva Phalgunī 11°37′ |
| Budha (Mercury) | 338.23698° | Mīna 8°14′ | Uttara Bhādrapadā 4°54′ |
| Śukra (Venus) | 7.29285° | Meṣa 7°18′ | Aśvinī 7°18′ |
| Maṅgala (Mars) | 280.25106° | Makara 10°15′ | Śravaṇa 0°15′ |
| Bṛhaspati (Jupiter) | 264.43632° | Dhanus 24°26′ | Pūrva Āṣāḍhā 11°06′ |
| Śani (Saturn) | 232.00407° | Vṛścika 22°00′ | Jyeṣṭhā 5°20′ |
| Rāhu | 329.08646° | Kumbha 29°05′ | Pūrva Bhādrapadā 9°05′ |
| Ketu | 149.08646° | Siṃha 29°05′ | Uttara Phalgunī 2°25′ |
Ravi (the Sun) was in Kumbha-rāśi (Aquarius) and Pūrvabhādrapāda;
Candra (the Moon) was in Pūrvaphalgunī;
Budha (Mercury) was in Mīna-rāśi (Pisces) and Uttarabhādrapāda;
Śukra (Venus) was in Meṣa-rāśi (Aries) and the nakṣatra of Aśvinī;
Maṅgala (Mars) was in Makara-rāśi (Capricorn) and Śravaṇā;
Bṛhaspati (Jupiter) was in Dhanu-rāśi (Sagittarius) and Pūrvāṣāḍhā;
Śani (Saturn) was in Vṛścika-rāśi (Scorpio) and Jyeṣthā;
Rāhu was in Pūrvabhādrapāda;
Ketu was in Siṁha-rāśi (Leo) and Uttaraphalgunī.
The lagna was Siṁha.
— CC Ādi 13.89, purport (Bhaktivinoda Ṭhākura's horoscope), Śrīla A.C. Bhaktivedānta Swāmī Prabhupāda