astronomy assignment

Duration of day:

Sunrise equation

The sunrise equation as follows can be used to derive the time of sunrise and sunset for any solar declination and latitude in terms of local solar time when sunrise and sunset actually occur:

where:

is the hour angle at either sunrise(when negative value is taken) or sunset(when positive value is

taken);

is the latitude of the observer on the Earth;

is the sun declination.

Theory:

The Earth rotates at an angular velocity of 15/hour, therefore the expression gives the interval of time before and after local solar noon that sunrise or sunset will occur. The sign convention

is typically that the observer latitude is 0 at the Equator , positive for the Northern Hemisphere and

negative for the Southern Hemisphere , and the solar declination is 0 at the vernal and autumnal equinoxes when the sun is exactly above the Equator, positive during the Northern Hemisphere summer and negative during the Northern Hemisphere winter. The expression above is always applicable for latitudes between the Arctic circle and Antarctic circle. North of the Arctic Circle or south of the Antarctic Circle, there is at least one day of the year with no sunrise or sunset. . Formally, there

is a sunrise or sunset when during the Northern Hemisphere

summer, and when during the Northern Hemisphere winter. Out of these latitudes, it is either 24-hour daytime or 24-hour night time.

Generalized Equation Also note that the above equation neglects the influence of atmospheric refraction (which lifts the solar disc by approximately 0.6 when it is on the horizon) and the non-zero angle subtended by the solar disc (about 0.5). The times of the rising and the setting of the upper solar limb as given in astronomical almanacs correct for this by using the more general equation.

with the altitude(a) of the center of the solar disc set to about 0.83 (or 50 arc minutes).

Calculation: The generalized equation relies on a number of other variables which need to be calculated before it can itself be calculated. These equations have the solar-earth constants substituted with angular constants expressed in degrees.

Calculate current Julian Cycle

is the Julian date;

is the longitude west (west is positive, east is negative) of the observer on the Earth;

is the Julian cycle since Jan 1st, 2000.

Approximate solar Noon

is an approximation of solar noon at .

Solar mean Anomaly

M is the solar mean anomoly.

C is the equation of center.

Ecliptic longitude

Jtransit is the hour angle for solar transit (or solar noon).

Declination of the Sun

is the declination of the sun.

Hour Angle

This is the equation from above with corrections for astronomical refraction and solar disc diameter.

o is the hour angle;

is the north latitude of the observer (north is positive, south is negative) on the Earth.

This is the main equation from above with the solar disc correction.

Calculate sunrise and sunset

Jset is the actual Julian Date of sunset.

Jrise is the actual Julian Date of sunrise.

MATLAB CODE:

clear all;

clc

close all;

L=70;%for tromso,norway

J=linspace(1,365,365);

for i=1:365

P(i)=asin(.39795*cos(.2163108 + 2*atan(.9671396*tan(.00860*(i-186))))); % approx. estimation of

declination of sun at a given day

D1(i) = 24 - (24/pi)*acos((sin(0.8333*pi/180) +

sin(L*pi/180)*sin(P(i)))/cos(L*pi/180)*cos(P(i))); % duration of day

end

L=-23;%for sao paulo,brazil

for i=1:365



D2(i) = 24 - (24/pi)*acos((sin(0.8333*pi/180) +


end

L=13; % for bengaluru

for i=1:365



D3(i) = 24 - (24/pi)*acos((sin(0.8333*pi/180) +


end

figure(1)

plot (J,D1);

figure(2)

plot (J,D2);

figure(3)

plot (J,D3);

RESULTS:

Day time for Tromso, Norway:

Day time for sao paulo,brazil:

Day time for bengaluru:

Analysis:

Norway being away from the equator the day time varies considerably from 2 hours minimum in winter

to 24 hours maximum in summer(i.e., sun does not set in summer).but in bengaluru and sao paulo

which are relatively closer to equator average day time does not vary considerably and it is nearly 12

hours throughout the year in bengaluru.

References:

http://en.wikipedia.org/wiki/Sunrise_equation

http://www.esrl.noaa.gov/gmd/grad/solcalc/solareqns.PDF

astronomy assignment

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