AST 443/PHY 517
Homework 1 Solutions
Use the following ephemeris and stellar data for questions 1-10:
On the night of 2-3 September 2013, at Cerro Tololo, Chile,
• sunset is at 18:36 CST (Chilean Standard Time) and sunrise is at 06:49 CST.
• Astronomical Twilight ends at 19:49 CST (when the Sun is 18o below the horizon), and
begins at 05:23 CST.
• The Local Mean Sideral time at midnight is 22:06:33.
• At midnight the moon is at 9h 1m 26s , +12o 1’ 36”.
• Cerro Tololo is at latitude −30.165 degrees; the Chilean Standard Time zone is 4 hours
west of Greenwich.
The Five
name
Sirius
Canopus
α Centauri
Arcturus
Vega
Brightest Stars in the Sky
RA (2000) DEC
h
m
6 45
8.9s −16o 42’ 58”
6h 23m 57.2s −52o 41’ 44”
14h 39m 36.2s −60o 50’ 7”
14h 15m 39.6s +19o 10’ 57”
18h 36m 56.2s +38o 47’ 1”
1. What is the Julian date at midnight CST on 2-3 September 2013? 2456538.667
What is the CST of local midnight? 0h 42.5m
Local midnight is half way between sunset and sunrise.
2. At midnight CST:
Compute the hour angles of the 5 brightest stars.
See column (1) below. HA = ST-RA; by convention, -12h < HA <12h
Which, if any, are observable (zenith distance <60o )? None
Which, if any, are above the horizon?
Canopus, α Cen, and Vega are above the horizon. See column (2) below.
The horizon is at 90o zenith distance.
The Five Brightest Stars in the Sky
name
HA
ZD Best day min ZD
SD
(1)
(2)
(3)
(4)
(5)
o
Sirius
-8:39 112.6
Jan 2
13 .4 100o .1
Canopus
-8:17 84.0
Dec 28
22o .5 139o .7
α Centauri 7:27 73.6
May 3
30o .7
360o
o
Arcturus
7:51 123.0
Apr 26
49 .3 78o .3
Vega
3:30 84.5
Jul 2
68o .9 62o .2
3. Which of these 5 stars can be observed at some time on this night from Cerro Tololo?
At what times?
Sirius, Canopus, and α Cen are observable at some time this night. Vega is
never observable above the 60o zenith distance limit.
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4. Which of these 5 stars is closest to the moon? What is the angular distance?
Sirius, at about 44.3 degrees
5. The sidereal time at midnight advances by 3m 56s each day. What are the best days to
observe these targets?
See column (3) above.
6. What is the minimum zenith distance for each star?
See column (4) above. The minimum ZD occurs when the star is on the meridian,
and equals the diffence between the declination and the observer’s latitude.
7. Determine the length of the semi-diurnal arc for each star. Which stars, if any, are
circumpolar? At the right time of year (i.e., when it transits at local midnight), which
star is observable for the longest uninterrupted time?
See column (5) above. α Cen is circumpolar. The star visible for the longest
uninterrupted time is also α Cen. To compute the time, set the altitude a = 30o ,
and compute the corresponding hour angle. The time visible is twice this hour angle
(to compute the semi-diurnal arc you set a=0).
8. How long after Canopus transits does Sirius transit? 21m 12s
This is the difference is right ascensions
9. At the start of astronomical twilight, what is the azimuth and elevation of Sirius?
elevation a = 43o ; Azimuth A = 81o (East of North)
10. Show that the angular distance D between two points on the surface of a sphere, with
coordinates (α1 ,δ1 ),(α2 ,δ2 ), is given by the expression
cosD = sinδ1 sinδ2 + cosδ1 cosδ2 cos(α2 − α1 )
Use the law of cosines.
Use the identities cos(90-x)=sin(x) and sin(90-x)=cos(x)
11. If you were to pilot a plane from New York City to Tokyo on a great circle route, what
heading should you take out of JFK airport? What is the great circle distance?
The distance is about 97o .6, or about 10540 km. Your initial heading is
-26o .9, or a bit West of North.
12. Plot the hour angle of the horizon as a function of declination for Stony Brook. Use
these data to determine the length of the day (the time when the Sun is above the horizon) for the equinoxes and solstices. You may neglect refraction in your calculations.
How much longer is the first day of summer than the first day of winter?
The inclination of the ecliptic is 23o 26’ 21”.448
When the Sun is on the horizon, it’s elevation a = 0o . Set the latitude φ =
40o . The declination of the Sun ranges ± the inclination of the ecliptic. On
the Summer solstice the day is about 14.8 hours long; on the winter solstice
the day is about 9.2 hours long. The difference is 5.6 hours.
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13. (PHY 517 only) The Gemini-N telescope, at a latitude of +19.8o on Mauna Kea, is
on an alt-az mount. The azimuth of a source at declination +19.8o changes instantawill exceed
neously from 90o to 270o as it crosses the zenith. The change in azimuth dA
dt
the maximum slew rate for upper culminations (or meridian transits) at high altitude.
Assume that the maximum azimuthal slew rate is 1o /second. Can you observe the
galaxy M64 through upper culmination? How close to the zenith can you get?
Work in alt-az coordinates. Compute dA/dt as a function of H. M64 is
at +21o 40’52”, so the maximum A is 88o .12. Solving numerically, I got
dA/dt ¡ 0o .12/sec, so you can indeed observe M64 through uper culmination. You can get to within about 0.23 degrees of the zenith, but this is not
recommended.
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