Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Topic: Seeing the sun rise twice
Replies: 2   Last Post: Jun 17, 1996 6:55 PM

 Messages: [ Previous | Next ]
 Michael Slavutin Posts: 7 Registered: 12/12/04
Re: Seeing the sun rise twice
Posted: Jun 17, 1996 11:10 AM

In article <31B3F61D.7E2E@wi.leidenuniv.nl>
"N.R.Bruin" <nbruin@wi.leidenuniv.nl> writes:

>In the book "Staring at the Sun", Julian Barnes describes a WW II
>nightfighter pilot returning from a mission over France:
>
>While he is still above enemy territory, flying west to Brittain, to his
>great dismay he sees the sun rising behind him. The dark colour of his
>plane, being the perfect camouflage at night, makes him all the more
>visible in daylight. He goes in a steep dive.
>
>When he is finally above the Channel, he levels off and returns to
>normal flight. After a while, he looks back and sees the sun rising
>behind him ... again.
>
>I wondered whether this phenomenon can occur on the lattitude of, say,
>London using WW II technology. Does anyone have the necessary data to do
>these calculations ?

The answer is definitely yes. Yes, it is possible.

First, the sun rising in the east. That means that Earth is rotating from
west to east. The pilot was flying from France to England or to the WEST.
That means he was flying AGAINST the Earth rotation. Now the condition to
see the two dawns is just to move faster than the Earth rotates. Let's
check it :
take the latitude phi = 50 degrees North Latitude
the radius of the Earth r = 6 400 km ( approximately )
so the required velocity is v = 2*pi/(24*3600) * r*cos(phi) = 300 m/s or
1000 km/h. I'm not an expert but as far as I know the planes could reach
this velocity.

Second, the higher the plane, the farer the horizon. The formula is
l = sqrt(2*h*r+h^2) where h is an altitude and r is again the Earth
radius. For a man 1.80 m high it gives l=4.8 km. For a plane at h = 1000 m
l = 113 km and for a plane at h = 5000 m l = 253 km. So the pilot flying
at 5000 m will see the sunrise earlier than the one flying at 1000 m.
For how earlier? For the latitude of 50 degrees NL we have
t= (253-113)/6400 * 24/(2*pi) * 1/cos(50) = 0.13 h or almost 8 min.

Now consider the following scenario. The pilot flying at approximately
5000 m with the speed, say, 600 km/h. Suddenly he sees the sunrise behind
him. The pilot takes a steep dive ( your own words ! ) to, say, 1000 m and
also accelerates. We've seen that at this altitude he has the whole 8
minutes till the sunrise. And the fact that he accelerated to something
like 700 km/h gives him even more time ( much more, in fact :
8 min * 1000 km/h / (1000-700) km/h = 26 minutes ). Above the
channel he rises again and again expands his horizon and sees the sunrise
again.

So, as you see, this phenomena was possible then.
Michael.

Date Subject Author
6/17/96 Michael Slavutin
6/17/96 David Chan