Physics 1135 Astronomy Group Project Answers

Part 1: Planetary distances

1. Distance to the moon:

tan A = tan (.952) = 0.0166
d = 6378/ 0.0166 = 383,823
rounding to three significant digits: d = 384,000 = 384 thousand kilometers
From the formula for the small angle approximation: d=180*6378/(0.952*3.14159) = 383,858
rounding to three significant digits: d = 384,000 = 384 thousand kilometers
Click here for diagram.

2. Distance to Venus:

d=206,265*6,378/31.8 = 41,369,754
round to 2 significant digits: d= 41,000,000 = 41 million kilometers

3. Distance to sun

distance from Venus to sun = sin 46.3 = 0.723 astronomical units
distance from Venus to Earth during transit = 1 - 0.723 = 0.277
number of kilometers per astronomical unit = 41,369,754 / 0.277 = 149,331,326
More precise measurements would give the value 149.6 million kilometers,
which rounded to two signifiant digits is 150,000,000 = 150 million kilometers
Click here for diagram.

4. Distance to planets.

Here are the calculations for Mars:
d = 206,265 * 0.0172 / 6,781.7 = 0.523 astronomical units
d = 0.523 * 150 million = 78.5 million kilometers
S = 78,500,000 * 17.84 / 206,265 = 6,789.5 kilometers
rounded to three significant digits: S = 6,790 kilometers
Here are the rest of the results, rounded to three significant digits:

Planet	Parallax angle (A) (seconds)	Distance from Earth (astronomical units)	Distance from Earth (km)	Angular size (seconds)	Size (km)
Mars	6,781.7	0.523	78.5 million	17.84	6,790
Jupiter	844.1	4.20	631 million	46.79	143,000
Saturn	415.3	8.54	1,280 million	19.41	121,000
Uranus	194.8	18.2	2,730 million	3.86	51,100
Neptune	121.8	29.1	4,370 million	2.34	49,500

Part 2: Gravity

Here are sample calculations for Mercury. The sequence of buttons on your calculator is probably something like this:

acceleration of gravity:
6.67 EXP -11 TIMES 3.30 EXP 23 = DIVIDE 2.44 EXP 6 X SQUARED =
result: 3.7
Note: the key for exponential (scientific) notation might be labelled EE or EXP.
fall distance:
3.7 TIMES 5 X SQUARED = DIVIDE 2 = result: 46.3
escape velocity:
2 TIMES 6.67 EXP -11 TIMES 3.3 EXP 23 = DIVIDE 2.44 EXP 6 SQRT
result: 4,247.5 (round to three significant digits: 4,250
On some calculators, the key sequence would be like this:
SQRT ( 2 TIMES 6.67 EXP -11 TIMES 3.3 EXP 23 DIVIDE 2.44 EXP 6 ) =

Results: (acceleration of gravity rounded to one decimal place (giving 2 or 3 significant digits; fall distance rounded to whole numbers; escape velocities have been rounded to 3 significant digits)

Planet	acceleration of gravity	fall distance	escape velocity
Mercury	3.7	46	4,250
Venus	8.9	111	10,400
Earth	9.8	122	11,200
Moon	1.6	20	2,380
Mars	3.7	46	5,020
Jupiter	24.8	310	59,500
Saturn	10.4	130	35,500
Uranus	8.9	111	21,300
Neptune	11.1	139	23,400
Pluto	0.7	8	1,220
Mystery object	4 billion billion	(meaningless)	300 million meters per second = the speed of light

Part 3: Star distances and magnitudes

For a star, the distance in parsecs is just one divided by the parallax angle. The distance in light years is 3.26 times the distance in parsecs. Here is a sample of the keystrokes to find the distance and absolute magnitude of Arcturus:
d = 1/.091 = 10.99 (round to 11.0 parsecs)
-0.06 MINUS 5 TIMES 1.1 LOG =
result: -0.27
or the calculator sequence might be like this:
-0.06 MINUS 5 TIMES LOG( 1.1 ) =
Note: you can calculate d/10 = 11/10=1.1 in your head, since dividing by 10 is easy.
Results:

                                   Apparent   Distance     Distance   Absolute
Star                   Parallax   Magnitude  (parsecs) (light years)   Magnitude 
Sirius (Canis_Major)      0.370       -1.46        2.7        8.8         1.4
Canopus (Carina)          0.033       -0.72       30.3       98.8        -3.1
Alpha_Centauri (c)        0.770       -0.01        1.3        4.2         4.4
Arcturus (Bootes)         0.091       -0.06       11.0       35.8        -0.3
Vega (Lyra)               0.130        0.04        7.7       25.1         0.6 
Capella (Auriga)          0.071        0.05       14.1       45.9        -0.7
Procyon (Canis_Minor)     0.290        0.37        3.4       11.2         2.7
Achernar (Eridanus)       0.050        0.51       20.0       65.2        -1.0
Hadar (Centaurus)         0.011        0.63       90.9      296.4        -4.2
Altair (Aquila)           0.200        0.77        5.0       16.3         2.3 
Aldebaran (Taurus)        0.063        0.86       15.9       51.7        -0.1
Spica (Virgo)             0.013        0.91       76.9      250.8        -3.5
Pollux (Gemini)           0.083        1.16       12.0       39.3         0.8
Fomalhaut (P_A)           0.140        1.19        7.1       23.3         1.9
Proxima_Centauri (Cent)   0.772       11.05        1.3        4.2        15.5 
Barnard's Star            0.545        9.54        1.8        6.0        13.2
Wolf 359                  0.421       13.53        2.4        7.7        16.7
Lalande 21185             0.397         7.5        2.5        8.2        10.5
Epsilon Eridani           0.303        3.73        3.3       10.8         6.1
61 Cygni                  0.294        5.22        3.4       11.1         7.6
Epsilon Indi              0.291        4.68        3.4       11.2         7.0

Here is a sample calculation for Deneb: 1.26 PLUS 6.9 = DIVIDE 5 = 1.632
1.632 10 TO THE POWER = 42.85
you can multiply by 10 in your head. The result is 428.5; round
to three significant digits: d=429 parsecs
Note: You can enter 1.26 + 6.9 instead of 1.26 - (-6.9) because subtracting a negative number is the same as adding a positive number.
Or your calculator sequence might look like this:
10 TIMES 10 TO THE POWER ( ( 1.26 + 6.9 ) DIVIDE 5 ) =
Results:

                   Apparent       Absolute
                   Visual           Visual   Distance   Distance
                   Magnitude     Magnitude  (parsecs) (light years) 
Rigel (Orion)              0.14        -6.8        244        797
Betelgeuse (Orion)         0.41        -5.5        152        496
Deneb (Cygnus)             1.26        -6.9        429       1397
Mimosa (Crux)              1.28        -4.6        150        489
Acrux (Crux)               1.39        -4.0        120        390
Antares (Scorpius)         0.92        -4.5        121        396