Why Do Tithis Appear Earlier in America? A Mathematical, Astronomical and Cultural Explanation of the Hindu Calendar

Background

More than a decade has passed since the first Bhutanese family was resettled. Despite growing concerns that the Bhutanese community is losing its cultural and linguistic bonds, moral values, and religious (and/or spiritual) beliefs, many believe that the community has managed well through the transition. It is not difficult to imagine how the circumstances of exiled Bhutanese people have changed after resettlement. For much of the resettled population, perhaps the greatest success has been the opportunity to leave behind past pains and miseries, troubles and adversities, uncertainties and misfortunes, and start life over again from scratch.

However, they have been carrying their cultural bags along with them as they carried them when they migrated to Bhutan hundreds of years ago. For these people, a celebration at the festival is not merely partying but observing the rituals rooted deeply in their cultural practices. These values and the norms that govern cultural events have always been vital for them.

The past few years have seen increased engagements regarding the celebration dates of major festivals. Some of the Bhutanese astrologers have proposed different dates for Dashain and Tihar (and other tithis) celebrations in America (in fact, many existing online Panchangas had mentioned this earlier). The discussions, agreements, and disagreements regarding this proposal that have flooded on social media conspicuously reveal how concerned (and confused?) the community is towards such issues.

The underlying questions are both scientific and cultural. If the Sun and Moon are the same celestial bodies observed throughout the world, why should a tithi appear on different civil dates in different countries? Does a tithi actually occur earlier in America? More importantly, does a difference in the astronomical date necessarily justify observing a religious festival on a different day?

Answering these questions requires distinguishing between astronomical events, civil-time conventions, and the religious rules governing festival observances. Although these concepts are closely related, they are not interchangeable. The author does not claim sufficient expertise in Hindu astrology to resolve every religious disagreement about festival dates. Instead, this article explains the astronomical principles underlying the Hindu Panchanga, examines the mathematical definition of a tithi, and discusses how time zones, local sunrise and sunset, and calendrical traditions influence the dates assigned to religious festivals.

Calendars and Their Astronomical Foundations

Calendars have played an essential role in human civilization by providing systems for organizing agricultural activities, religious celebrations, administrative affairs, and social events. Different cultures have developed calendrical systems based on recurring astronomical phenomena, particularly the apparent motion of the Sun and the changing phases of the Moon. Calendars are commonly classified as solar, lunar, or lunisolar systems (U.S. Naval Observatory, n.d.). Solar calendars are organized principally around the Earth’s annual revolution around the Sun and the recurring seasonal cycle. Lunar calendars follow successive cycles of lunar phases, while lunisolar calendars combine lunar months with a solar-year framework (Dershowitz & Reingold, 2011). The Gregorian calendar, widely adopted for civil purposes, is primarily solar. In contrast, the Hindu calendar incorporates both solar and lunar measurements and therefore belongs to the broader family of lunisolar calendars.
However, there is no single, universally identical Hindu calendar. Different regions and traditions follow distinct conventions regarding the beginning of months, the calculation of astronomical positions, and the assignment of religious observances (Chakravarty, 2008; Lian, 2001; Time and Date AS, n.d.-a). The Hindu calendar represents more than a simple system of counting days. Its structure incorporates complex astronomical relationships and calendrical rules developed through a long history of mathematical and cultural traditions (Dershowitz & Reingold, 2011).

Hindu Panchanga: A Multidimensional Calendar

The Sanskrit term Panchanga combines pancha, meaning five, and anga, meaning limb or component. It refers to five principal elements used in traditional Hindu timekeeping.
These components are:

1. Tithi: A lunar day determined by the changing angular separation between the Moon and the Sun.
2. Nakshatra: A division of the ecliptic used to describe the position of the Moon relative to the traditional lunar mansions.
3. Vara: The weekday.
4. Yoga: A calendrical quantity based on the combined ecliptic longitudes of the Sun and Moon.
5. Karana: Half of a tithi, corresponding to a 6° increase in the Moon’s longitude relative to the Sun.

These five components form the mathematical and calendrical foundation of the Panchanga. Their astronomical calculations are used in determining religiously significant dates and periods (Dershowitz & Reingold, 2011; Time and Date AS, n.d.-b). Indian calendrical astronomy has evolved over many centuries. Astronomical and mathematical traditions associated with the Vedanga Jyotisha, Surya Siddhanta, and the work of mathematicians such as Aryabhata have contributed to its development (Chakravarty, 2008; Mohan, 2015). The historical origins of these traditions, however, require careful interpretation. Abhyankar (1993) proposed that certain early Vedic calendrical practices could be traced to approximately 7000 BCE through astronomical reconstruction. Although such studies are important contributions to the historical discussion, a proposed astronomical date should not automatically be treated as an independently established historical chronology.

Dershowitz and Reingold (2011) and Ohashi (2008) demonstrated that Indian calendrical systems have undergone substantial mathematical development and have incorporated different methods of calculating solar and lunar positions. Therefore, the Hindu Panchanga is best understood as a historically evolving calendrical tradition that combines astronomical computation with cultural and religious conventions.

Tithi: The Astronomical Definition of a Lunar Day

Among the five components of the Panchanga, the tithi is particularly important because many Hindu festivals are associated with specific lunar dates. A tithi is commonly described as a lunar day. Unlike a conventional civil day, however, its duration is not fixed at 24 hours. To understand a tithi, it is necessary to consider the Moon’s orbital motion relative to the Sun. The Moon completes one revolution around the Earth relative to distant stars in approximately 27.3 days. However, because the Earth simultaneously revolves around the Sun, the Moon requires approximately 29.53 days to return to the same phase relative to the Sun as observed from Earth. This interval is known as the synodic month (Positional Astronomy Centre, n.d.).
The Hindu lunar month is divided into 30 tithis, grouped into two fortnights:

• Shukla Paksha: The waxing fortnight, during which the angular separation between the Sun and Moon progresses from new Moon toward full Moon.
• Krishna Paksha: The waning fortnight, extending from full Moon toward the following new Moon.

Each fortnight consists of 15 tithis (see Table 1). The waxing fortnight concludes with Purnima, while the waning fortnight concludes with Amavasya. Although tithis are closely related to lunar phases, they represent mathematically defined intervals rather than distinct visible appearances of the Moon.

Drawing Parallels Between Tithis and Lunar Phases

Although a tithi is defined mathematically rather than visually, its progression is closely related to the familiar cycle of lunar phases. As the Moon revolves around the Earth, its appearance changes in a regular pattern over approximately 29.53 days, producing the sequence of phases described in modern astronomy (Chaisson & McMillan, 2005). At astronomical new Moon, the geocentric ecliptic longitudes of the Sun and Moon coincide, and the illuminated side of the Moon is directed largely away from the Earth, making the Moon generally invisible. As the Moon continues in its orbit, the illuminated portion visible from Earth gradually increases, producing the waxing crescent, first quarter, waxing gibbous, and eventually the full Moon (Figure 1).

After full Moon, the illuminated portion gradually decreases through the waning gibbous, third quarter, and waning crescent phases until the next new Moon. In Hindu calendrical terminology, the waxing half of the lunar month is called Shukla Paksha, and the waning half is called Krishna Paksha. Thus, the broad sequence of lunar waxing and waning corresponds closely to the two fortnights recognized in the Panchanga. Figure 2 illustrates this relationship by showing the progression of tithis between Purnima (full Moon) and Amavasya (new Moon).

It is important, however, not to equate tithis directly with the eight principal lunar phases used in modern astronomy. The visible lunar phases describe the changing appearance of the Moon as seen from Earth, whereas tithis divide the synodic month into 30 mathematically defined angular intervals. In this sense, tithis are not 30 distinct visible phases of the Moon; rather, they are 30 successive lunar dates whose progression underlies the phase cycle.

Mathematical Calculation of Tithis

A complete circle contains 360°. Because the lunar month is divided into 30 tithis, each tithi corresponds to an increase of 12° in the Moon’s ecliptic longitude relative to the Sun (Bhujle & Vahia, 2006).

The angular difference is represented mathematically as:

Δλ = (λMoon − λSun) mod 360°

where:

• Δλ is the angular difference between the geocentric ecliptic longitudes of the Moon and Sun.
• λMoon represents the geocentric ecliptic longitude of the Moon.
• λSun represents the geocentric ecliptic longitude of the Sun.

The expression mod 360° ensures that the angular separation is expressed within a complete cycle. When the geocentric longitudes of the Moon and Sun are equal, their longitudinal separation is 0°. This corresponds to the astronomical new Moon and marks the end of the Amavasya tithi and the beginning of Shukla Pratipada under the stated mathematical convention. As the Moon moves eastward relative to the Sun, the longitudinal difference increases. When it reaches 12°, the first tithi ends and the second begins. At 24°, the second tithi ends, and the process continues through successive multiples of 12°. When the separation reaches 180°, the astronomical full Moon occurs, corresponding to the boundary between Purnima and Krishna Pratipada.

Bhujle and Vahia (2006) examined the mathematical formulation of tithi calculations in the Surya Siddhanta tradition and proposed methods for improving the derivation of the computational constants used in traditional calculations. The essential astronomical principle remains that a tithi is determined by the relative geocentric positions of the Sun and Moon.

Duration of a Tithi

Although each tithi represents an angular difference of 12°, the time required for the separation to increase by that amount is not constant. The apparent orbital motions of the Sun and Moon vary because of the characteristics of their respective orbital movements. Consequently, individual tithis do not have identical durations. According to the Positional Astronomy Centre (n.d.), the mean duration of a tithi is approximately 23 hours, 37 minutes, and 28 seconds. Its actual duration can vary from approximately 19 hours and 59 minutes to 26 hours and 47 minutes. This variation has significant implications for calendar construction. A tithi may begin at any time during a civil day. It may continue across two consecutive dates, extend across two successive sunrises, or begin and end between successive sunrises.

As Dershowitz and Reingold (2011) explained, the relationship between lunar intervals and sunrise-based civil days is an important aspect of Indian calendrical calculations. Consequently, the tithi associated with a particular civil date cannot always be determined simply by dividing the lunar month into equal 24-hour periods.

Are Tithis Different in Different Parts of the World?

The mathematical definition provides the basis for answering the central question. A tithi calculated from the geocentric longitudes of the Sun and Moon begins and ends at the same astronomical instant throughout the world. The term geocentric means that celestial positions are calculated with reference to the Earth’s centre. The angular separation between these geocentric positions does not depend on whether a person lives in Kathmandu, New York, Oslo, or Adelaide (Table 2). Therefore, under a common computational method, tithi boundaries are global astronomical instants rather than geographical events that move progressively from one country to another (Bhujle & Vahia, 2006; Dershowitz & Reingold, 2011).

What differs between countries is the civil-time representation of these instants. Time zones divide the world into regions using different local clock times. Because geographical locations are assigned different civil-time offsets from Coordinated Universal Time (UTC), the same astronomical event may appear on different local calendar dates.

The New Moon of October 2019: An Illustration

The astronomical new Moon of October 2019 provides a useful example. According to the astronomical calculations published by Espenak (n.d.), the new Moon occurred on October 28, 2019, at 03:38 UTC. This single astronomical event corresponded to different local times in different parts of the world.

Table 2 demonstrates that the astronomical new Moon occurred on October 27 in New York but October 28 in Kathmandu. Nevertheless, these were not separate astronomical events. They were different civil-time representations of precisely the same instant. The distinction is particularly important for understanding the expression that tithis come earlier in America. In astronomical terms, a tithi does not begin earlier in America. Rather, because North American time zones are generally behind Nepal’s time zone, the relevant instant may fall on an earlier American civil date. Thus, the appearance of an earlier tithi is a consequence of civil-time conventions rather than a difference in the underlying astronomical event. Oslo observed Central European Summer Time (UTC+2) during the full Moon of 13 October 2019 but had returned to Central European Time (UTC+1) before the new Moon of 28 October 2019 (Time and Date AS, n.d.-c).

Astronomical Tithis and Religious Festival Dates

The global simultaneity of a tithi does not, however, mean that every religious festival must be observed on the same civil date throughout the world. This distinction is fundamental to understanding the disagreements regarding occurrence of tithis. The astronomical interval of a tithi and the religious date assigned to a festival are two related but different concepts. The first is established through astronomical calculation. The second is determined by applying calendrical and religious conventions to the astronomical information.

The Significance of Local Sunrise

In many Hindu calendrical traditions, the tithi prevailing at local sunrise is used to identify the lunar date associated with that civil day. This convention is commonly known as Udaya Tithi. Because sunrise occurs at different astronomical instants in different geographical locations, the relationship between a tithi and local sunrise varies across the world. For example, a particular tithi may be present at sunrise in Kathmandu but may have ended before sunrise in New York. Consequently, the two locations may assign different tithi designations to their respective civil dates even though the astronomical boundaries of the tithi remain globally identical.

Dershowitz and Reingold (2011) described the importance of sunrise calculations and the conventions used to determine dates and holidays in Indian lunisolar calendars. The relationship between tithis and sunrise also explains why some tithis may be omitted or repeated in a sunrise-based calendar. If a tithi begins and ends between consecutive sunrises, it may not appear as the sunrise tithi of any day. Conversely, a tithi extending across two consecutive sunrises may be assigned to two successive days (Time and Date AS, n.d.-b). However, the sunrise convention does not determine the dates of all Hindu festivals.

Prescribed Ritual Periods

Different religious observances may depend on the tithi prevailing during specific parts of the day or night. For example, some observances emphasize sunrise, whereas others are associated with midday, sunset, evening twilight, or particular nighttime periods. Laxmi Puja during Diwali provides an important example. According to the calendrical rules described by Drik Panchang (n.d.-b), Diwali Laxmi Puja is traditionally associated with Amavasya during Pradosha Kala, a period beginning after sunset. The recommended observance time may also incorporate additional considerations, such as a suitable Sthira Lagna. Therefore, identifying the appropriate date for Laxmi Puja requires more than determining the exact instant of astronomical new Moon.

It requires examining when the Amavasya tithi overlaps the prescribed ritual period at the relevant geographical location. A difference in the civil date of an astronomical event does not automatically mean that a festival must be observed on different dates. Conversely, different festival dates may result when the relevant tithi overlaps the prescribed local ritual period on different civil days. This distinction is central to the correct interpretation of geographically calculated Panchangas.

Re-examining Laxmi Puja in October 2019

The October 2019 example illustrates the difference between the astronomical occurrence of a lunar phase and the religious date selected for a festival. As shown earlier, the exact astronomical new Moon occurred at 11:38 p.m. on October 27 in New York and 9:23 a.m. on October 28 in Kathmandu (Espenak, n.d.). A simple comparison of the local dates might suggest that Laxmi Puja should have occurred on October 27 in the United States and October 28 in Nepal. However, the actual festival observances demonstrate why such reasoning is incomplete. The Nepalese holiday calendar identified October 27, 2019, as Laxmi Puja (Office Holidays, n.d.; Time and Date AS, n.d.-a).

Similarly, the Hindu festival calendar published by Drik Panchang (n.d.-a) identified October 27, 2019, as the date for Diwali and Laxmi Puja in Seattle, Washington. Thus, despite the difference in the local civil dates of the exact new Moon, the festival was assigned the same date in Nepal and this American location. This example establishes that the local civil date of the astronomical new Moon is not, by itself, sufficient to determine the festival date. The observance depends on the tithi prevailing during the religiously prescribed period, interpreted through the relevant calendrical tradition. The distinction is also mathematically significant. The exact astronomical new Moon marks the boundary at which Amavasya ends and the subsequent tithi begins under the standard geocentric angular definition. It should therefore not be confused with the beginning of Amavasya.

Although different astronomical algorithms and calendrical systems may produce modest differences in calculated transition times, these differences should be examined separately from the religious rules used to assign festival dates (Bhujle & Vahia, 2006; Dershowitz & Reingold, 2011).

Why Do Different Panchangas Sometimes Disagree?

Another important question concerns differences between published almanacs. Gautam (2019) compared information from sources including Timeanddate.com, Mooncalc.org, Drik Panchang, and Marutinandan Patro, an almanac produced for Bhutanese communities in the United States. Although the sources broadly reflected the same astronomical framework, differences were reported in their calculated or displayed timings. Such discrepancies may arise for several reasons. First, calendars may use different time standards. A particularly important example involves the distinction between Eastern Standard Time (EST), which is UTC−5, and Eastern Daylight Time (EDT), which is UTC−4. Failing to account for daylight-saving time can produce an apparent one-hour difference. Second, different computational methods may generate somewhat different values for the positions of the Sun and Moon. Traditional algorithms and modern astronomical ephemerides may employ different mathematical formulations or approximations.

Bhujle and Vahia (2006) examined the derivation of traditional tithi calculations, while Dershowitz and Reingold (2011) distinguished among several approaches to Indian astronomical and calendrical computation. Third, differences may arise from geographical conventions. The calculated times of sunrise and sunset depend on location, and these local events influence the civil dates assigned to religious observances. Fourth, regional traditions may apply different rules for selecting the relevant tithi when a festival is associated with a particular part of the day. Accordingly, differences between calendars should not automatically be attributed to the geographical movement of astronomical events. A rigorous comparison should establish whether the disagreement concerns the calculated astronomical instant, its conversion into local clock time, the sunrise-based lunar date, or the application of a religious observance rule.

Likewise, claims about the average numerical difference between traditional and modern tithi calculations should specify the algorithms, dates, locations, and statistical procedures used in the comparison. Without such information, a general claim about the precision or superiority of one calendar over another cannot be evaluated reliably.

Implications for Bhutanese Communities Living Abroad

The astronomical and calendrical distinctions discussed above have practical significance for Bhutanese communities living outside South Asia. Religious observances have historically been organized within specific geographical and cultural settings. When communities relocate across multiple time zones, some of the assumptions embedded in familiar calendar practices become more visible. For example, a festival date printed in a Panchanga prepared for Kathmandu may not always produce the same result when the relevant religious rule is applied to sunrise, sunset, or ritual periods in an American city. At the same time, a recommendation to observe a festival one day earlier in the United States should not be justified solely by pointing to an earlier American civil date for the corresponding lunar phase.

An appropriate determination requires the identification of the tithi’s astronomical interval, accurate conversion into the local time zone, calculation of relevant local solar events, and application of the religious rules governing the particular festival. This approach is consistent with the distinction between astronomical computation and calendrical convention emphasized by Dershowitz and Reingold (2011). The challenge is not necessarily a conflict between scientific knowledge and religious tradition. Astronomy provides a mathematical description of celestial phenomena, whereas religious conventions establish the significance and appropriate timing of particular observances. Both dimensions contribute to the construction and practical use of the Panchanga.

Community organizations and religious authorities can help reduce confusion by explaining the geographical basis of their calculations and identifying the rules used to determine festival dates.
Such transparency would also support cultural education among younger generations who are increasingly familiar with scientific explanations but may have limited knowledge of the mathematical and religious foundations of inherited traditions. Furthermore, religious observance and community celebration need not always be treated as identical activities. Families and community groups may organize gatherings around work schedules, school commitments, and the availability of shared facilities while separately recognizing the prescribed ritual timing. This distinction can help communities accommodate contemporary circumstances without disregarding the traditions they seek to preserve.

Conclusion

Tithis are mathematically defined by the changing geocentric ecliptic longitude of the Moon relative to the Sun, with each tithi corresponding to a 12° increase in their angular separation. Consequently, tithi boundaries occur at the same astronomical instant worldwide, although their corresponding civil dates may differ because of time zones. Thus, tithis do not actually occur earlier in America; they may only appear on earlier local calendar dates.

However, the astronomical occurrence of a tithi does not necessarily determine the date of a religious observance. Hindu festival dates may also depend on local sunrise, sunset, prescribed ritual periods, and regional calendrical conventions. The October 2019 Laxmi Puja example illustrates this distinction: although the astronomical new Moon occurred on different civil dates in New York and Kathmandu, the festival was assigned the same date in Nepal and the American calendar examined.

For Bhutanese communities living abroad, understanding the distinction between astronomical calculations and religious calendrical conventions can help resolve disagreements about festival dates while supporting the informed preservation and transmission of cultural traditions.

*Gautam holds a PhD in mathematics didactics and teaches physics and mathematics in an upper secondary school in Norway. He can be reached at rameshgautam@live.com.

Originally published on 6 October 2019. Revised and updated on 10 October 2026.