Isaac Newton and the astronomical refraction

Waldemar H. Lehn

doi:10.1364/AO.47.000H95

Applied Optics
Vol. 47,
Issue 34,
pp. H95-H105
(2008)
•https://doi.org/10.1364/AO.47.000H95

Isaac Newton and the astronomical refraction

Waldemar H. Lehn

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Waldemar H. Lehn, "Isaac Newton and the astronomical refraction," Appl. Opt. 47, H95-H105 (2008)

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Abstract

In a short interval toward the end of 1694, Isaac Newton developed two mathematical models for the theory of the astronomical refraction and calculated two refraction tables, but did not publish his theory. Much effort has been expended, starting with Biot in 1836, in the attempt to identify the methods and equations that Newton used. In contrast to previous work, a closed form solution is identified for the refraction integral that reproduces the table for his first model (in which density decays linearly with elevation). The parameters of his second model, which includes the exponential variation of pressure in an isothermal atmosphere, have also been identified by reproducing his results. The implication is clear that in each case Newton had derived exactly the correct equations for the astronomical refraction; furthermore, he was the first to do so.

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Apparent Elevation	Calculated Refraction	Newton’s Table	Difference
$0 °$	$33^{'} 20^{''}$	$33^{'} 20^{''}$	$0^{''}$
$1 °$	$23^{'} 55^{''}$	$23^{'} 12^{''}$	$43^{''}$
$2 °$	$17^{'} 42^{''}$	$17^{'} 29^{''}$	$13^{''}$
$3 °$	$13^{'} 40^{''}$	$13^{'} 40^{''}$	$0^{''}$
$4 °$	$10^{'} 59^{''}$	$11^{'} 04^{''}$	$- 5^{''}$
$5 °$	$9^{'} 06^{''}$	$9^{'} 13^{''}$	$- 7^{''}$

Altitudo apparens	Refractio aestiva	Refractio verna et autumnalis	Refractio hyberna	Calculated Values for Spring–Fall	Difference
° ′	′ $''$	′ $''$	′ $''$	′ $''$	$''$
0.00	31.30	33.20	35.10	31.32	$- 108$
0.30	26.06	27.45	29.24	27.00	$- 45$
1.00	21.50	23.12	24.34	23.12	0
1.30	18.51	20. 2	21.13	20. 3	1
2.00	16.27	17.29	18.31	17.29	0
2.30	14.31	15.23	16.15	15.23	0
3.00	12.52	13.40	14.28	13.40	0
3.30	11.32	12.15	12.58	12.15	0
4.00	10.25	11. 4	11.43	11. 4	0
4.30	9.29	10. 5	10.41	10. 4	$- 1$
5.00	8.40	9.13	9.46	9.13	0
6.00	7.24	7.52	8.20	7.52	0
7.00	6.27	6.51	7.15	6.51	0
8.00	5.42	6. 3	6.24	6. 3	0
9.00	5. 5	5.24	5.43	5.24	0
10.00	4.36	4.53	5.10	4.53	0
11.00	4.11	4.27	4.43	4.27	0
12.00	3.51	4. 5	4.19	4. 5	0
13.00	3.33	3.46	3.39	3.46	0
14.00	3.18	3.30	3.42	3.30	0
15.00	3. 4	3.16	3.28	3.15	$- 1$

Altitudo apparens	Refractio	Calculated Values	Diff.	Biot’s values	Diff.
° ′	′ $''$	′ $''$	$''$	′ $''$	$''$
0. 0	33.45	33.45	0	33.33	$- 12$
0.30	27.35	27.35	0	27.25	$- 10$
1. 0	23. 7	23. 5	$- 2$	22.57	$- 10$
1.30	19.46	19.41	$- 5$	19.36	$- 10$
2. 0	17. 8	17. 4	$- 4$	17. 0	$- 8$
3. 0	13.20	13.20	0	13.18	$- 2$
4. 0	10.48	10.50	2	10.50	2
5. 0	9. 2	9. 5	3	9. 5	3
6. 0	7.45	7.47	2	7.48	3
7. 0	6.47	6.47	0	6.48	1
8. 0	6. 0	6. 1	1	6. 2	2
9. 0	5.22	5.23	1	5.24	2
10. 0	4.52	4.52	0	4.53	1

Apparent Elevation	Calculated Refraction	Newton’s Table	Difference
$0 °$	$33^{'} 20^{''}$	$33^{'} 20^{''}$	$0^{''}$
$1 °$	$23^{'} 55^{''}$	$23^{'} 12^{''}$	$43^{''}$
$2 °$	$17^{'} 42^{''}$	$17^{'} 29^{''}$	$13^{''}$
$3 °$	$13^{'} 40^{''}$	$13^{'} 40^{''}$	$0^{''}$
$4 °$	$10^{'} 59^{''}$	$11^{'} 04^{''}$	$- 5^{''}$
$5 °$	$9^{'} 06^{''}$	$9^{'} 13^{''}$	$- 7^{''}$

Altitudo apparens	Refractio aestiva	Refractio verna et autumnalis	Refractio hyberna	Calculated Values for Spring–Fall	Difference
° ′	′ $''$	′ $''$	′ $''$	′ $''$	$''$
0.00	31.30	33.20	35.10	31.32	$- 108$
0.30	26.06	27.45	29.24	27.00	$- 45$
1.00	21.50	23.12	24.34	23.12	0
1.30	18.51	20. 2	21.13	20. 3	1
2.00	16.27	17.29	18.31	17.29	0
2.30	14.31	15.23	16.15	15.23	0
3.00	12.52	13.40	14.28	13.40	0
3.30	11.32	12.15	12.58	12.15	0
4.00	10.25	11. 4	11.43	11. 4	0
4.30	9.29	10. 5	10.41	10. 4	$- 1$
5.00	8.40	9.13	9.46	9.13	0
6.00	7.24	7.52	8.20	7.52	0
7.00	6.27	6.51	7.15	6.51	0
8.00	5.42	6. 3	6.24	6. 3	0
9.00	5. 5	5.24	5.43	5.24	0
10.00	4.36	4.53	5.10	4.53	0
11.00	4.11	4.27	4.43	4.27	0
12.00	3.51	4. 5	4.19	4. 5	0
13.00	3.33	3.46	3.39	3.46	0
14.00	3.18	3.30	3.42	3.30	0
15.00	3. 4	3.16	3.28	3.15	$- 1$

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